Circle illustration showing a radius, a diameter, the center and the circumference

Tycho crater, one of many examples of circles that arise in nature.NASA photo

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point called the center. The common distance of the points of a circle from its center is called its radius.

Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure (also known as the perimeter) or to the whole figure including its interior. However, in strict technical usage, “circle” refers to the perimeter while the interior of the circle is called a disk. The circumference of a circle is the perimeter of the circle (especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Further terminology

Chord, secant, tangent, and diameter.

Arc, sector, and segment

The diameter of a circle is the length of a line segment whose endpoints lie on the circle and which passes through the centre of the circle. This is the largest distance between any two points on the circle. The diameter of a circle is twice its radius.

As well as referring to lengths, the terms “radius” and “diameter” can also refer to actual line segments (respectively, a line segment from the centre of a circle to its perimeter, and a line segment between two points on the perimeter passing through the centre). In this sense, the midpoint of a diameter is the centre and so it is composed of two radii.

A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle’s centre, is the longest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point. A secant is an extended chord: a straight line cutting the circle at two points.

An arc of a circle is any connected part of the circle’s circumference. A sector is a region bounded by two radii and an arc lying between the radii, and a segment is a region bounded by a chord and an arc lying between the chord’s endpoints.

History

The compass in this 13th centurymanuscript is a symbol of God’s act ofCreation. Notice also the circular shape of the halo

The circle has been known since before the beginning of recorded history. It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically “divine” or “perfect” that could be found in circles.^{[citation needed]}

Some highlights in the history of the circle are:

• 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 (3.16049…) as an approximate value of π.^[1]
• 300 BC – Book 3 of Euclid’s Elements deals with the properties of circles.
• 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.^[2]

Analytic results

Length of circumference

The ratio of a circle’s circumference to its diameter is π (pi), a constant that takes the same value (approximately 3.141592654) for all circles. Thus the length of the circumference (c) is related to the radius (r) by

or equivalently to the diameter (d) by

Area enclosed

Area of the circle = π × area of the shaded square

The area enclosed by a circle is π multiplied by the radius squared:

Equivalently, denoting diameter by d,

that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely theisoperimetric inequality.

Equations

Cartesian coordinates

Circle of radius r = 1, center (a, b) = (1.2, -0.5)

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centered at the origin (0, 0), then the equation simplifies to

The equation can be written in parametric form using the trigonometric functions sine and cosine as

where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (x, y) makes with the x-axis. Alternatively, a rational parametrization of the circle is:

In homogeneous coordinates each conic section with equation of a circle is of the form

It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. These points are called the circular points at infinity.

Polar coordinates

In polar coordinates the equation of a circle is:

where a is the radius of the circle, r₀ is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle. For a circle centred at the origin, i.e. r₀ = 0, this reduces to simply r = a. When r₀ = a, or when the origin lies on the circle, the equation becomes

.

In the general case, the equation can be solved for r, giving

,

the solution with a minus sign in front of the square root giving the same curve.

Complex plane

In the complex plane, a circle with a center at c and radius (r) has the equation . In parametric form this can be written z = re^it + c.

The slightly generalized equation  for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with , since . Not all generalised circles are actually circles: a generalized circle is either a (true) circle or aline.

Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x₁, y₁) and the circle has center (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x₁, y₁), so it has the form (x₁−a)x+(y₁−b)y = c. Evaluating at (x₁, y₁) determines the value of c and the result is that the equation of the tangent is

(x₁ − a)x + (y₁ − b)y = (x₁ − a)x₁ + (y₁ − b)y₁

or

(x₁ − a)(x − a) + (y₁ − b)(y − b) = r².

If y₁≠b then slope of this line is

.

This can also be found using implicit differentiation.

When the center of the circle is at the origin then the equation of the tangent line becomes

x₁x + y₁y = r²,

and its slope is

.

Properties

• The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
• The circle is a highly symmetric shape: every line through the center forms a line of reflection symmetry and it has rotational symmetry around the center for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
• All circles are similar.
  • A circle’s circumference and radius are proportional.
  • The area enclosed and the square of its radius are proportional.
    • The constants of proportionality are 2π and π, respectively.
• The circle which is centered at the origin with radius 1 is called the unit circle.
  • Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
• Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord

• Chords are equidistant from the center of a circle if and only if they are equal in length.
• The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
  • A perpendicular line from the center of a circle bisects the chord.
  • The line segment (circular segment) through the center bisecting a chord is perpendicular to the chord.
• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
• If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
• If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
  • For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
• An inscribed angle subtended by a diameter is a right angle.
• The diameter is the longest chord of the circle.

Sagitta

• The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle.
• Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines:

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y/2)². Solving for r, we find the required result.

Tangent

• The line drawn perpendicular to a radius through the end point of the radius is a tangent to the circle.
• A line drawn perpendicular to a tangent through the point of contact with a circle passes through the center of the circle.
• Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

Theorems

Secant-secant theorem

• The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA.
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, thenDC² = DG×DE. (Tangent-secant theorem.)
• If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG = DF×DE. (Corollary of the tangent-secant theorem.)
• The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
• If the angle subtended by the chord at the center is 90 degrees then l = √2 × r, where l is the length of the chord and r is the radius of the circle.
• If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Apollonius circle

Apollonius’ definition of a circle: d₁/d₂constant

Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B. (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points^[3].

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

Stated another way, P is a point on the Apollonius circle if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

Generalized circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

(1)

is not a circle, but rather a line.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying (1) is called a generalized circle. It may either be a true circle or a line. In this sense a lineis a generalized circle of infinite radius.

Bangladesh is a low-lying, riverine country located in South Asia with a largely marshy jungle coastline of 710 kilometers (440 mi) on the northern littoral of the Bay of Bengal. Formed by a delta plain at the confluence of the Ganges (Padma), Brahmaputra (Jamuna), and Meghna Rivers and their tributaries, Bangladesh’s alluvial soil is highly fertile, but vulnerable to flood and drought. Hills rise above the plain only in the Chittagong Hill Tracts in the far southeast and the Sylhet division in the northeast. Straddling the Tropic of Cancer, Bangladesh has a tropical monsoon climate characterized by heavy seasonal rainfall, high temperatures, and high humidity. Natural disasters, such as floods, tornadoes, and tidal bores affect the country yearly. Bangladesh also is affected by major cyclones — on average 16 times a decade. One cyclone struck the southeastern coast in May 1991, killing 136,000 people. The cyclone Sidr struck the southwestern coast on November 15, 2007 affecting not only the coastal districts of the administrative division Khulna but also about half of the tropical forest Sundarbans.

Coordinates: 24°00′N 90°00′E / 24°N 90°E / 24; 90

Physiography

Satellite photographs (from Terra-MODIS) and computer-generated models help visualize Bangladesh’s place in the world. Located in South Asia, it is virtually surrounded by India and the Bay of Bengal to the south. But in many ways, the country’s fate is dominated by the world’s highest mountain range looming to the north-the Himalayas.^[1]

NASA satellite Image of Bangladesh’s physical features (click to enlarge and view national borders)

The physiography of Bangladesh is varied and has an area characterized by two distinctive features: a broad deltaic plain subject to frequent flooding, and a small hilly region crossed by swiftly flowing rivers.The country has an area of 144,000 square kilometers and extends 820 kilometers north to south and 600 kilometers east to west. Bangladesh is bordered on the west, north, and east by a 2,400-kilometer land frontier with India and, in the southeast, by a short land and water frontier (193 km) with Burma (Myanmar). On the south is a highly irregular deltaic coastline of about 580 kilometers, fissured by many rivers and streams flowing into the Bay of Bengal. The territorial waters of Bangladesh extend 12 nautical miles (22 km), and the exclusive economic zone of the country is 200 nautical miles (370 km).

Roughly 80 % of the landmass is made up of fertile alluvial lowland called the Bangladesh Plain. The plain is part of the larger Plain of Bengal, which is sometimes called the Lower Gangetic Plain. Although altitudes up to 105 meters above sea level occur in the northern part of the plain, most elevations are less than 10 meters above sea level; elevations decrease in the coastal south, where the terrain is generally at sea level. With such low elevations and numerous rivers, water—and concomitant flooding—is a predominant physical feature. About 10,000 square kilometers of the total area of Bangladesh is covered with water, and larger areas are routinely flooded during the monsoon season.

The only exceptions to Bangladesh’s low elevations are the Chittagong Hills in the southeast, the Low Hills of Sylhet in the northeast, and highlands in the north and northwest. The Chittagong Hills constitute the only significant hill system in the country and, in effect, are the western fringe of the north-south mountain ranges of Burma and eastern India. The Chittagong Hills rise steeply to narrow ridge lines, generally no wider than 36 meters, with altitudes from 600 to 900 meters above sea level. At 1,052 meters altitude, the highest elevation in Bangladesh is found at Mowdok, in the southeastern part of the hills. Fertile valleys lie between the hill lines, which generally run north-south. West of the Chittagong Hills is a broad plain, cut by rivers draining into the Bay of Bengal, that rises to a final chain of low coastal hills, mostly below 200 meters, that attain a maximum elevation of 350 meters. West of these hills is a narrow, wet coastal plain located between the cities of Chittagong in the north and Cox’s Bazar in the south.

About 67% of Bangladesh’s nonurban land is arable. Permanent crops cover only 2%, meadows and pastures cover 4%, and forests and woodland cover about 16%. The country produces large quantities of quality timber, bamboo, and sugarcane. Bamboo grows in almost all areas, but high-quality timber grows mostly in the highland valleys. Rubber planting in the hilly regions of the country was undertaken in the 1980s, and rubber extraction had started by the end of the decade. A variety of wild animals are found in the forest areas, such as in the Sundarbans on the southwest coast, which is the home of the world-famous Royal Bengal Tiger. The alluvial soils in the Bangladesh Plain are generally fertile and are enriched with heavy silt deposits carried downstream during the rainy season.

Human geography

Urbanization is proceeding rapidly, and it is estimated that only 30% of the population entering the labor force in the future will be absorbed into agriculture, although many will likely find other kinds of work in rural areas. The areas around Dhaka and Comilla are the most densely settled. The Sundarbans, an area of coastal tropical jungle in the southwest and last wild home of the Bengal Tiger, and the Chittagong Hill Tracts on the southeastern border with Burma and India, are the least densely populated.

Climate

Bangladesh has a tropical monsoon climate characterized by wide seasonal variations in rainfall, high temperatures, and high humidity. Regional climatic differences in this flat country are minor. Three seasons are generally recognized: a hot, muggy summer from March to June; a hot, humid and rainy monsoon season from June to November; and a warm-hot, dry winter from December to February. In general, maximum summer temperatures range between 38 and 41 °C. April is the hottest month in most parts of the country. January is the coolest (but still hot) month, when the average temperature for most of the country is 16-20 °C during the day and around 10 °C at night. .

Winds are mostly from the north and northwest in the winter, blowing gently at one to three kilometers per hour in northern and central areas and three to six kilometers per hour near the coast. From March to May, violent thunderstorms, called northwesters by local English speakers, produce winds of up to 60 kilometers per hour. During the intense storms of the early summer and late monsoon season, southerly winds of more than 160 kilometers per hour cause waves to crest as high as 6 meters in the Bay of Bengal, which brings disastrous flooding to coastal areas.

Heavy rainfall is characteristic of Bangladesh causing it to flood every year. With the exception of the relatively dry western region of Rajshahi, where the annual rainfall is about 1600 mm, most parts of the country receive at least 2300 mm of rainfall per year. Because of its location just south of the foothills of the Himalayas, where monsoon winds turn west and northwest, the region of Sylhet in northeastern Bangladesh receives the greatest average precipitation. From 1977 to 1986, annual rainfall in that region ranged between 3280 and 4780 mm per year. Average daily humidity ranged from March lows of between 55 and 81 % to July highs of between 94 and 100 %, based on readings taken at selected stations nationwide in 1986.

About 80 % of Bangladesh’s rain falls during the monsoon season. The monsoons result from the contrasts between low and high air pressure areas that result from differential heating of land and water. During the hot months of April and May hot air rises over the Indian subcontinent, creating low-pressure areas into which rush cooler, moisture-bearing winds from the Indian Ocean. This is the southwest monsoon, commencing in June and usually lasting through September. Dividing against the Indian landmass, the monsoon flows in two branches, one of which strikes western India. The other travels up the Bay of Bengal and over eastern India and Bangladesh, crossing the plain to the north and northeast before being turned to the west and northwest by the foothills of the Himalayas.

Natural calamities, such as floods, tropical cyclones, tornadoes, and tidal bores—destructive waves or floods caused by flood tides rushing up estuaries—ravage the country, particularly the coastal belt, almost every year. Between 1947 and 1988, 13 severe cyclones hit Bangladesh, causing enormous loss of life and property. In May 1985, for example, a severe cyclonic storm packing 154 kilometer-per-hour winds and waves 4 meters high swept into southeastern and southern Bangladesh, killing more than 11,000 persons, damaging more than 94,000 houses, killing some 135,000 head of livestock, and damaging nearly 400 kilometers of critically needed embankments.

Annual monsoon flooding results in the loss of human life, damage to property and communication systems, and a shortage of drinking water, which leads to the spread of disease. For example, in 1988 two-thirds of Bangladesh’s 64 districts experienced extensive flood damage in the wake of unusually heavy rains that flooded the river systems. Millions were left homeless and without potable water. Half of Dhaka, including the runways at the Zia International Airport—an important transit point for disaster relief supplies—was flooded. About 2 million tons of crops were reported destroyed, and relief work was rendered even more challenging than usual because the flood made transportation of any kind exceedingly difficult. A tornado in April 1989 killed more than 600 people (possibly many more; it may have been the deadliest tornado in world history).

There are no precautions against cyclones and tidal bores except giving advance warning and providing safe public buildings where people may take shelter. Adequate infrastructure and air transport facilities that would ease the sufferings of the affected people had not been established by the late 1980s. Efforts by the government under the Third Five-Year Plan (1985-90) were directed toward accurate and timely forecast capability through agrometeorology, marine meteorology, oceanography, hydrometeorology, and seismology. Necessary expert services, equipment, and training facilities were expected to be developed under the United Nations Development Programme.

River systems

Ganges River Delta, Bangladesh and India

The rivers of Bangladesh mark both the physiography of the nation and the life of the people. About 700 in number, these rivers generally flow south. The larger rivers serve as the main source of water for cultivation and as the principal arteries of commercial transportation. Rivers also provide fish, an important source of protein. Flooding of the rivers during the monsoon season causes enormous hardship and hinders development, but fresh deposits of rich silt replenish the fertile but overworked soil. The rivers also drain excess monsoon rainfall into the Bay of Bengal. Thus, the great river system is at the same time the country’s principal resource and its greatest hazard.

The profusion of rivers can be divided into five major networks. The Jamuna-Brahmaputra is 292 kilometers long and extends from northern Bangladesh to its confluence with the Padma. Originating as the Yarlung Zangbo Jiang in China’s Xizang Autonomous Region (Tibet) and flowing through India’s state of Arunachal Pradesh, where it becomes known as the Brahmaputra (“Son of Brahma”), it receives waters from five major tributaries that total some 740 kilometers in length. At the point where the Brahmaputra meets the Tista River in Bangladesh, it becomes known as the Jamuna. The Jamuna is notorious for its shifting subchannels and for the formation of fertile silt islands (chars). No permanent settlements can exist along its banks.

The second system is the Padma-Ganges, which is divided into two sections: a 258-kilometer segment, the Ganges, which extends from the western border with India to its confluence with the Jamuna some 72 kilometers west of Dhaka, and a 126-kilometer segment, the Padma, which runs from the Ganges-Jamuna confluence to where it joins the Meghna River at Chandpur. The Padma-Ganges is the central part of a deltaic river system with hundreds of rivers and streams—some 2,100 kilometers in length—flowing generally east or west into the Padma.

The third network is the Surma-Meghna River System, which courses from the northeastern border with India to Chandpur, where it joins the Padma. The Surma-Meghna, at 669 kilometers by itself the longest river in Bangladesh, is formed by the union of six lesser rivers. Below the city of Kalipur it is known as the Meghna. When the Padma and Meghna join together, they form the fourth river system—the Padma-Meghna—which flows 145 kilometers to the Bay of Bengal.

This mighty network of four river systems flowing through the Bangladesh Plain drains an area of some 1.5 million square kilometers. The numerous channels of the Padma-Meghna, its distributaries, and smaller parallel rivers that flow into the Bay of Bengal are referred to as the Mouths of the Ganges. Like the Jamuna, the Padma-Meghna and other estuaries on the Bay of Bengal are also known for their many chars.

A fifth river system, unconnected to the other four, is the Karnaphuli. Flowing through the region of Chittagong and the Chittagong Hills, it cuts across the hills and runs rapidly downhill to the west and southwest and then to the sea. The Feni, Karnaphuli, Sangu, and Matamuhari—an aggregate of some 420 kilometers—are the main rivers in the region. The port of Chittagong is situated on the banks of the Karnaphuli. The Karnaphuli Reservoir and Karnaphuli Dam are located in this area. The dam impounds the Karnaphuli River’s waters in the reservoir for the generation of hydroelectric power.

During the annual monsoon period, the rivers of Bangladesh flow at about 140,000 cubic meters per second, but during the dry period they diminish to 7,000 cubic meters per second. Because water is so vital to agriculture, more than 60 % of the net arable land, some 91,000 km², is cultivated in the rainy season despite the possibility of severe flooding, and nearly 40 % of the land is cultivated during the dry winter months. Water resources development has responded to this “dual water regime” by providing flood protection, drainage to prevent overflooding and waterlogging, and irrigation facilities for the expansion of winter cultivation. Major water control projects have been developed by the national government to provide irrigation, flood control, drainage facilities, aids to river navigation and road construction, and hydroelectric power. In addition, thousands of tube wells and electric pumps are used for local irrigation. Despite severe resource constraints, the government of Bangladesh has made it a policy to try to bring additional areas under irrigation without salinity intrusion.

Water resources management, including gravity flow irrigation, flood control, and drainage, were largely the responsibility of the Bangladesh Water Development Board. Other public sector institutions, such as the Bangladesh Krishi Bank, the Bangladesh Rural Development Board, the Bangladesh Bank, and the Bangladesh Agricultural Development Corporation were also responsible for promotion and development of minor irrigation works in the private sector through government credit mechanisms.

Area and boundaries

Area:
total: 144,000 km²
county comparison to the world: 101
land: 133,910 km²
water: 10,090 km²

Area comparative:

• Australia comparative: 1.5 times larger than Tasmania
• Canada comparative: twice the size of New Brunswick
• United Kingdom comparative: larger than England
• United States comparative: slightly smaller than Iowa

Land boundaries:
total: 4,246 km
border countries: Myanmar 193 km, India 4,053 km

Coastline: 580 km

Maritime claims:
contiguous zone: 18 nautical miles (33 km)
continental shelf: up to the outer limits of the continental margin
exclusive economic zone: 200 nautical miles (370 km)
territorial sea: 12 nautical miles (22 km)

Elevation extremes:
lowest point: Indian Ocean 0 m
highest point: In the Mowdok range at 1052m (at N 21°47’12″ E 92°36’36″), NOT Keokradong (883 m not 1,230m) or Tajingdong, 985m not 1,280m as sometimes reported)

Resources and land use

Natural resources: natural gas, arable land, timber, coal

Land use:
Arable land: 55.39%
Permanent crops: 3.08%
other: 41.53% (2005)

Irrigated land: 47,250 km² (2003)

Total renewable water resources: 1,210.6 cu km (1999)

Freshwater withdrawal (domestic/industrial/agricultural):
total: 79.4 cu km/yr (3%/1%/96%)
per capita: 560 cu m/yr (2000)

Environmental concerns

Natural hazards: Cyclones; much of the country routinely swamped with water during the summer monsoon season; droughts

Environment – current issues: many people are landless and forced to live on and cultivate flood-prone land; limited access to potable water; water-borne diseases prevalent; water pollution especially of fishing areas results from the use of commercial pesticides; ground water contaminated by naturally occurring arsenic; intermittent water shortages because of falling water tables in the northern and central parts of the country; soil degradation and erosion; deforestation; severe overpopulation

Environment – international agreements:
party to: Biodiversity, Climate Change, Climate Change-Kyoto Protocol, Desertification, Endangered Species, Environmental Modification, Hazardous Wastes, Law of the Sea, Ozone Layer Protection, Ship Pollution, Wetlands

signed, but not ratified: none of the selected agreements Geography-note

Most of the country is situated on deltas of large rivers flowing from the Himalayas; the Ganges unites with the Jamuna (main channel of the Brahmaputra) and later joins the Meghna to eventually empty into the Bay of Bangal

Why Time is Absolute, and Relative, But Never Universal

by Vincent SauvéAbstract

The author elaborates upon a materialist view of the absolute and relative nature of time in a simple and (hopefully) clear manner for what has always been a complicated subject. Albert Einstein’s relativity of time is defended against those who think of time as a universal absolute that extends beyond its proper frame–the inertial frame. The inertial frame is clarified and elaborated upon to show why the scale of time is naturally, through Newton’s Laws of Motion, in congruence to this frame. The conclusion of this paper is that the supposed paradoxes or absurdities of Einstein’s relativity theory are really the result of a false conception of time.

Introduction

Many argue against Einstein’s relativity theory because they cannot conceive how time can be anything other than universal (1). Great confusion and error abounds both by those who defend Einstein’s relativity and those who attack it; so much so that it is easy to be discouraged from trying to clarify matters, yet I believe it is important to make whatever attempts possible.

The concept of time

First of all, a clear understanding of the concept of time (2) is necessary. Time is reckoned by noting the intervals that occur by the motion of material things. Historically, this has meant how many times the sun is at its highest point in the sky (days), the moon at the same phase (month), and the passing of the seasons (year). Recognition of the passage of time is always in relation to something material. The more uniform the material motion the more accurate we can be in our measurements and divisions of time. I also agree with this view of the measurement of time, by Einstein:

The measurement of time is effected by means of clocks. A clock is a thing which automatically passes in succession through a (practically) equal series of events (period). The number of periods (clock-time) elapsed serves as a measure of time. The meaning of this definition is at once clear if the event occurs in the immediate vicinity of the clock in space; for all observers then observe the same clock-time simultaneously with the event (by means of the eye) independently of their position. Until the theory of relativity was propounded it was assumed that the conception of simultaneity had an absolute objective meaning also for events separated in space.
This assumption was demolished by the discovery of the law of propagation of light. For if the velocity of light in empty space is to be a quantity that is independent of the choice (or, respectively, of the state of motion) of the inertial system to which it is referred, no absolute meaning can be assigned to the conception of the simultaneity of events that occur at points separated by a distance in space. Rather, a special time must be allocated to every inertial system. If no co-ordinate system (inertial system) is used as a basis of reference there is no sense in asserting that events at different points in space occur simultaneously. It is in consequence of this that space and time are welded together into a uniform four-dimensional continuum. (Einstein, 1992)

It is also important to know that in the evolution of concepts, the word space first of all applies to matter in the sense of that which extends spatially. L. Feuerbach makes an interesting and succinct statement about space and time in an argument against the philosophical idealism of Hegel:

“In reality, exactly the opposite holds good, …it is not things that presuppose space and time, but space and time that presupposes things, for space or extension presupposes something that extends, and time, movement, for time is indeed only a concept derived from movement, presupposes something that moves. Everything is spatial and temporal….” (Lenin, 1981)

Newton’s view of time can also be characterized as idealistic, or in any case, incomplete. Newton, apparently not giving sufficient significance to how the absolute is contained within the relative, instead came up with “Absolute, true and mathematical time, of itself, and by its own nature, flows uniformly on, without regard to anything external. ” (Capek, 1973, p. 393)So, if we can agree with the materialistic view that time is solidly linked to the motions of matter, is there any sense in speaking of universal motion (time)? How about universal rest which is the other side of the coin to universal motion? A favorite quote of mine on this subject is the following:

All rest, all equilibrium, is only relative, only has meaning in relation to one or another definite form of motion….
A motionless state of matter therefore proves to be one of the most empty and nonsensical of ideas–a “delirious fantasy” of the purest water. In order to arrive at such an idea, it is necessary to conceive as absolute rest the relative mechanical equilibrium in which a body on earth may find itself, and then to extend this absolute rest over the whole universe….This conception is nonsensical, because it transfers as absolute to the entire universe a state which by its nature is relative and which therefore can never be simultaneously applied except to a part of matter. (Engels, 1976)

Let us note how the following statement by Einstein merges into this, specifically, the absolute and relative nature of motion (and light):

The law of the constant velocity of light in empty space, which has been confirmed by the development of electro-dynamics and optics, and the equal legitimacy of all inertial systems (special principle of relativity), which was proved in a particularly incisive manner by Michelson’s famous experiment, between them made it necessary, to begin with, that the concept of time should be made relative, each inertial system being given its own special time. …According to the special theory of relativity, spatial co-ordinates and time still have an absolute character in so far as they are directly measurable by stationary clocks and bodies. But they are relative in so far as they depend on the state of motion of the selected inertial system. (Einstein, 1934)

Light, like everything else that moves, is both absolute (denoted by c) and relative. Every material thing can also he said to have an absolute character, providing we choose the appropriate reference frame to consider with. So that Einstein, when he says in his second postulate that “Any ray of light moves in the ‘resting’ coordinate system with the definite velocity c, which is independent of whether the ray was emitted by a resting or by a moving body” he is saying, that when time is measured within such system, that there is a constant to this system. This is a specific absolute, not a universal absolute. A “moving” system is the same as a “resting” system from the perspective of those doing the measuring within the system. “It is essential to have time defined by means of clocks at rest in the resting system, and the time now defined being appropriate to the resting system we call ‘the time of the resting system.’ ” (Einstein, 1905)That motion, and hence light and time, is both absolute and relative is only natural, it is the nature of things. Moreover, what is referred to as universal time is merely Greenwich time calculated from midnight at the Greenwich (England) meridian, an arbitrary designation that is quite useful in our modern world, but unknown a few centuries ago.

Yet why should other sentient beings on other worlds care about our arbitrary standard unit of time, presuming that they may become aware of us someday? Their day and night, and hence, their divisions of time will certainly be different than ours.

But, is there not a universal commonality–if we accept the cosmological principle, as I believe we should–that the laws of nature should be the same everywhere in the universe? Specifically, what if there are two identical atomic clocks, one on planet A, and one on planet B, with a day/night cycle very much different from each other. In this case, one can claim a limited type of universal time, by having each observer of the clocks count a certain number of atomic vibrations or cycles. The clock keeper from planet A can agree with the clock keeper on planet B that, for example, one hundred atomic cycles will be the accepted universal standard of time. But obviously this is a limited form of universal time in that there is to be no relative motion between the two clocks. When relative motion is introduced the signals sent will change. Actually, the signal itself does not change, but how it is received does.

An example of signaling the pace of time using mechanics, and then extending the situation to electrodynamics, will illustrate the similarities and problems:

Consider these two clocks to be no longer on these planets, but now in deep interstellar space at a constant distance from each other. These clocks have a gun attached to each. They also have a target attached safely nearby. After every standard unit of time passes, clock A triggers a gunshot aimed at the target on clock B. The observer (or machine/computer) at clock B records the impact and notes the constant arrival of a bullet (assuming the cartridges have exact loads of powder and that other factors remain the same). Now then, if one of the clocks, say clock A, is to produce an acceleration away from clock B, the observer at clock B would correctly conclude that clock A is accelerating away from clock B by his recording of the time of arrival of bullet impacts. Or, he incorrectly concludes that clock A is no longer constant, that its time is slowing (time dilation).

The same is true when light is used as a means of communication rather than a bullet.

…for every reference system in which the laws of mechanics are valid, the laws of electrodynamics and optics are also valid. (Einstein, 1905)

But, note that we do not see a change in velocity of photons from objects that approach or move away from us because that light first interacts with the atoms that are in our atmosphere and/or in our lens, etc., which are at rest relative to us (3). Though, measurements will reveal the increase or decrease in energy through the interacting medium. And unlike the situation with light, the clock observer in the above mentioned example can conclude which interpretation is correct by measuring the velocity of the bullet.The inertial reference frame

To further elucidate matters it is also very important that we clearly understand the significance of the inertial reference frame (or system), what it is and what it isn’t.

An inertial system is not equivalent to that of the “fixed stars”, as is the Machian notion, although a system (or body) can be inertial with respect to the average velocity of the background stars.

The problem of formulating physical laws for every CS [coordinate system] was solved by the so-called general relativity theory; the previous theory, applying only to inertial systems, is called the special relativity theory. (Einstein and Infeld, pp. 212-3)

An inertial system is one in which

…the laws of classical mechanics are valid for the observer inside the elevator [in free-fall]. All bodies behave in the way expected by the law of inertia. (Einstein and Infeld, p. 215)

An inertial system does not need to be a room in free-fall. It can also be a satellite in orbit or a spacecraft with its thrusters off, whether it is accelerating relative to the earth or not. Very high velocities can also be imagined for an inertial system by simply imagining our inertial coordinate system (we’ll use Einstein’s example of an elevator) free-falling straight down to a very massive, very dense object such as a Neutron star (or orbiting this star at close range). Also, it is not a subjective matter having to do with a human observer. Replace the human observer by a machine that measures the motions within the inertial system and the results will be identical.What is the universal fact to be witnessed by all observers who are part of an inertial frame of which there are an innumerable number (as well as relative velocities) in our universe? It is that all things that are at rest in that system will stay that way and that things that are set in motion will continue that motion in a straight line with a constant speed unless acted upon by external forces (Newton’s first law of motion). An inertial frame is a frame of reference in which bodies are not accelerated from the perspective of within the system. Yet, the whole inertial frame of reference will be accelerated relative to something else. The important point here is that gravity accelerates all material things (including light) equally together, so that an inertial observer will notice that all things move equally and together within his system (or rather, stay in place relative to him), even though his entire system will be accelerating relative to something else. His space-time region has the characteristic of being flat (Euclidean) and isotropic. Any experiment, for example, whether he measures the space (extension, distance), and the time of impact of a bullet fired from a gun, or a beam of light from one wall of his “elevator” to the other wall, will have constant results regardless of direction. For him, (when considering only that which is part of his inertial system) his sense of motion or rest is an absolute one, and therefore, also his definition of time.

The ultimate scale of time is therefore based on our concept of universal laws of nature. This was already recognized last century, long before the advent of modern ultra-precise time-keeping, in particular by Thomson and Tait in their treatise Natural Philosophy (1890). In discussing the law of inertia they argued that it could be stated in the form: the times during which any particular body not compelled by force to alter the speeds of its motions passes through equal spaces are equal; and in this form, they said, the law expresses our convention for measuring time. It is easily seen that this implies a unique time-scale except for the arbitrary choice of time unit and time zero. (Whitrow, 1973, p. 404).

The laws of motion inform us that the unique time-scale should naturally be firmly welded to the inertial reference frame.

The first law of motion, by stating under what circumstances the velocity of a moving body remains constant, supplies us with a method of defining equal intervals of time. (Maxwell, 1952)

(A brief description of regions where spacetime curvature results will be useful before I continue. Spacetime curvature is experienced when the coordinate system experiences the effects of gravity, or, equivalently, acceleration in the sense of what results from a rocket thruster. It is very important to know the difference between free-fall acceleration, and rocket produced acceleration. These two kinds of acceleration are not the same. For example, an observer in an elevator with a rocket engine attached but turned off and left to free-fall will experience Euclidean spacetime geometry in the motions of his little enclosed world. But as soon as he fires his rocket thruster spacetime curvature results; the strength of which depends on the thrust of his engines. All things set in motion (relative to the elevators coordinate system, of course) now will have curved paths, whether it is an object tossed in the elevator compartment or a light beam initiated from the compartment. Furthermore, the velocity of this light will be altered, depending if the light is directed “uphill” or “downhill” relative to the artificially produced gravitational field.)Inertial regions are also known as Galilean regions:

…there are finite regions, where, with respect to a suitably chosen space of reference, material particles move freely without acceleration, and in which the laws of the special theory of relativity,… hold with remarkable accuracy. Such regions we shall call ‘Galilean regions.’ (Einstein, 1974, pp. 58-9)
A system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a ‘Galilean system of co-ordinates’. (O.E.D., 1991)

These Galilean (inertial) regions should not he considered to be very large:

We can therefore always regard an infinitesimally small region of the space-time continuum as Galilean. For such an infinitely small region there will be an inertial system … relatively to which we are to regard the laws of the special theory of relativity as valid…. Space-time regions of finite extent are, in general, not Galilean, so that a gravitational field cannot be done away with by any choice of co-ordinates in a finite region. (Einstein, 1974, pp. 63-4)

I cannot think of a brief way to sufficiently elucidate why these Galilean regions are small, although the reader will find an excellent illustration of why this is so in Taylor and Wheeler’s Spacetime Physics, pp. 5-10.Conclusion

Regardless of Einstein’s faults; his many errors, and ambiguous statements in the expounding of his theory of relativity, (and translation problems?) it is not correct to fault him for introducing the relativity of simultaneity (the relativity of time) as a solution for problems in physics as many authors have done. The absolute and relative nature of time (motion/light) is difficult to grasp for anybody, particularly when our textbooks and our professors are not perfectly clear (often times because they don’t grasp it well). I have to agree with the following comment from the authors of Spacetime Physics, even if I don’t agree with everything they wrote in their textbook:

The problem of understanding relativity is no longer one of learning but one of intuition–a practiced way of seeing. When seen with this intuition, a remarkable number of otherwise incomprehensible experimental results are revealed to be perfectly natural.

I think of those who have not yet learned this practiced way of seeing as having three-dimensional thinking. This three-dimensional thinking is wide-spread and has given us the interpretation of increased mass with increased velocity (4), the contraction of lengths as a dynamical phenomenon rather than a kinematic phenomenon (5), and asymmetrical time dilation (6). One needs to develop four-dimensional thinking to realize that (just as absolute motion is contained within relative motion) there are no paradoxes when absolute time is understood to be contained within relative time. It is my hope that this paper will help.

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ReferencesCapek, M., 1973, author of the article “Time” in Dictionary of the History of ideas: Studies of Selected Pivotal Ideas, Philip P. Wiener, editor in chief, (Charles Scribner’s Sons, New York). pp. 389-398.

Einstein, A., 1905, “On the Electrodynamics of Moving Bodies, ” translated from Annalen der Physik by Arthur Miller in the Appendix of his book Albert Einstein’s Special Theory of Relativity, (Addison-Wesley Publishing Company, Inc., Advanced Book Program, Reading, MA, 1981).

Einstein, A., 1934, Essays In Science, (The Wisdom Library, A division of Philosophical Library, New York, NY) pp. 48-9. This is an abridged edition and authorized English translation from the volume ‘Mein Weltbild’ (The World as I See It).

Einstein, A., and Infeld, L., 1938, The Evolution of Physics, (A Clarion Book, published by Simon and Schuster, New York, NY).

Einstein, A., 1974, The Meaning of Relativity, (Princeton University Press, Princeton, NJ, fifth edition).

Einstein, A., 1992, Fadiman, C., general editor, “Albert Einstein On Space-Time,” The Treasury of the Encyclopedia Britannica, (Viking Penguin, a division of Penguin Books USA Inc., New York, NY), pp. 371-383.

Engels, F., 1976, Anti-Duhring (Peking, 1976), pp. 74-5.

Lenin, V. I., 1981, “Philosophical Notebooks,” Collected Works, Vol. 38,
p. 70 (Moscow, 1981).

Maxwell, J. C., 1952, Matter and Motion, (Dover Publications Inc., New York, NY), p. 31.

Oxford English Dictionary (O.E.D.), 1991, under Galilean transformations see 1920, R.W. Lawson’s translation of Einstein’s Relativity for the quote.

Taylor, E. F., and Wheeler, J. A., 1966, Spacetime Physics, (W.H. Freeman and Company, New York).

Whitrow, G. J., 1973, author of the article “Time and Measurement” in Dictionary of the History of Ideas: Studies of Selected Pivotal Ideas, Philip P. Wiener, editor in chief, (Charles Scribner’s Sons, New York), pp. 398-406.

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Notes(1) I was actually moved to write this paper by reading the latest example of this view of universal time in Wen-Xiu Li’s paper, “On the Relativity of Lengths and Times,” Apeiron Vol. 2, No. 1, January 1995, pp. 16-9.

(2) It makes no sense to conceive of the concept of time as arising in any other way than through our observations of matter in motion. Time is not something a God brought forth with the imaginary 6,000-10,000 year old creation of the universe that is the biblical, or the equally imaginary 10-20 billion year age Big Bang model of the universe. Legitimate science, as opposed to the “science” that panders to the Western religious tradition of a God who created us and/or everything else, does not presume that our universe has an age when it was created (from nothing, in many popular speculations). We should recognize (even if it is difficult to comprehend) that our universe is infinite in time and space. The question of how our universe got here is not a legitimate question for the scientist any more than the question is for the creationist who nevertheless seldom worries about the same question in regards to his Creator. All points of view, religious, and scientific, inevitably involve an infinite in time; either an infinite God or an infinite universe, and sometimes both. To argue, as some will, that God created time with his (or her) Creation, leaves open the question: Did time exist for God? If not, then the thoughts of past, present, and future don’t exist for this God either. Likewise, this God gave no time to think about his Creation. For a look into how the Big Bang cosmology has become the religious sophisticates version of creation, see my “Is Big Bang cosmology good science, or ‘creation science’?”, 1994, in: Challenges in Modem Physics, Proceedings of the Pacific Division AAAS Meeting, San Francisco State University, June 20-24. Email me for a copy.

(3) For articles on the phenomenon of light “extinguishing” see “The Unavailability of ‘Old’ Light,” by John B. Schaefer, Am. J. Phys., Vol. 57, No. 3, March 1989, p. 200; “Experimental Evidence for the Second Postulate of Special Relativity,” by J.C. Fox, in Am. J. Phys., Vol. 30, No. 4, April 1962, pp. 297-300; “Speed of Light,” by Carl E. Ockert, in Am. J. Phys., Vol. 36, No. 2, February 1968, pp. 158-161.

(4) Please see Lev Okun’s “The Concept of Mass,” Physics Today, June 1989, pp. 31-6, excerpts:

In the modern language of relativity theory there is only one mass, the Newtonian mass m, which does not vary with velocity; hence the famous formula E=mc² has to be taken with a large grain of salt. … The notion of the dependence of mass on velocity was introduced by Lorentz in 1899 and then developed by him and others in the years preceding Einstein’s formulation of special relativity in 1905, as well as in later years. The basis of this notion is again the application of the nonrelativistic formula p=mv in the relativistic region, where (as we know now) this formula is not valid.

Also see Okun’s reply in the Letters column, “Putting to Rest Mass Misconceptions,” Physics Today, May 1990, pp. 13-4, 115, 117. And, Carl G. Adler’s paper “Does mass really depend on velocity, dad?” Am. J. Phys., August 1987.

(5) We see the phenomenon of contraction and time dilation as real (because they will be “observed”) yet, illusionary. It is a problem of kinematics, not dynamics. See page 21 in Abraham Pais’ book ‘Subtle is the Lord…’: The Science and the Life of Albert Einstein, (Oxford University Press, 1982).

(6) See Mendel Sachs, 1985, “On Einstein’s Later View of the Twin Paradox,” Foundations of Physics, Vol. 15, No. 9, pp. 977-980. His abstract reads: “It is shown that Einstein abandoned his earlier view that there are material consequences, such as asymmetrical aging, implied by the space-time transformation of relativity theory.”

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(Mathematical formulation of quantum mechanics) বলতে কোয়ান্টাম বলবিজ্ঞানের একটি কঠোর বিধিবদ্ধ বর্ণনা প্রদানকারী গাণিতিক সূত্রসমূহের সমষ্টিকে বোঝায়। বিংশ শতাব্দীর প্রথম দিককার বছরগুলিতে এই গাণিতিক সূত্রগুলি লিপিবদ্ধ হয়েছিল। এই সূত্রগুলির প্রকৃতি পূর্বতন সূত্রগুলির চেয়ে আলাদা। এগুলিতে বিমূর্ত গাণিতিক সংগঠন, যেমন অসীম-মাত্রাবিশিষ্ট হিলবার্ট জগৎসমূহ এবং ঐ জগৎগুলির উপর প্রযুক্ত অপারেটর ব্যবহার করা হয়েছে। এই সংগঠনগুলির অনেকগুলিই ফাংশনাল বিশ্লেষণ নামক বিশুদ্ধ গণিতের একটি গবেষণা ক্ষেত্র থেকে নেয়া হয়েছিল। এই বিশেষায়িত ক্ষেত্রটি কোয়ান্টাম বলবিজ্ঞানের সমসাময়িককালে উদ্ভাবন করা হয় এবং কোয়ান্টাম বলবিজ্ঞানের গাণিতিক ব্যাখ্যার চাহিদা ক্ষেত্রটির উন্নয়নে প্রভাব রেখেছিল। সংক্ষেপে, ভৌত পর্যবেক্ষণসম্ভব রাশি যেমন শক্তি ও ভরবেগের মানগুলি দশা জগতের ফাংশনসমূহের মান হিসেবে আর গণ্য করা হচ্ছিল না। বরং এগুলিকে হিলবার্ট জগতে রৈখিক অপারেটরসমূহের আইগেনমান হিসেবে গণ্য করা হল।

কোয়ান্টাম বলবিজ্ঞানের এই সূত্রায়ন আজও ব্যবহৃত হয়। এই বর্ণনার কেন্দ্রে রয়েছে “কোয়ান্টাম অবস্থা” এবং “কোয়ান্টাম পর্যবেক্ষণসম্ভব” এই দুইটি ধারণা। পারমাণবিক মাপের ব্যবস্থাগুলির ক্ষেত্রে এই ধারণা দুটি অতীতের ভৌত বাস্তবতার মডেলগুলিতে ব্যবহৃত ধারণাগুলি অপেক্ষা অত্যন্ত ভিন্ন প্রকৃতির। এই গাণিতিক সূত্রায়ন অনুসারে যদিও অনেক রাশি পরীক্ষণের মাধ্যমে পরিমাপ করা সম্ভব, তা সত্ত্বেও এতে একই সাথে পরিমাপযোগ্য মানের ব্যাপারে একটি নির্দিষ্ট তাত্ত্বিক সীমা রয়েছে। এই সীমাবদ্ধতাটি সর্বপ্রথম হাইজেনবের্গ একটি চিন্তা পরীক্ষণের মাধ্যমে ব্যাখ্যা করেন। নতুন সূত্রায়নে গাণিতিকভাবে এই সীমাকে কোয়ান্টাম পর্যবেক্ষণসম্ভবগুলির অবিনিময়যোগ্যতার মাধ্যমে উপস্থাপন করা হয়েছে।

একটি আলাদা তত্ত্ব হিসেবে কোয়ান্টাম বলবিজ্ঞানের আবির্ভাবের পূর্বে পদার্থবিজ্ঞানে যে গণিত ব্যবহৃত হত, তা ছিল মূলত অন্তরক জ্যামিতি এবং আংশিক অন্তরক সমীকরণসমূহ। এছাড়া পরিসংখ্যানিক বলবিজ্ঞানে সম্ভাবনা তত্ত্ব ব্যবহৃত হত। অন্তরক জ্যামিতি ও আংশিক অন্তরক সমীকরণে জ্যামিতিক স্বজ্ঞা নিঃসন্দেহে একটি প্রধান ভূমিকা পালন করত। আপেক্ষিকতার তত্ত্বগুলি সম্পূর্ণভাবে জ্যামিতিক ধারণার আশ্রয় নিয়েই সূত্রায়িত করা হয়েছিল। মোটামুটি ১৮৯৫ ও ১৯১৫ সালের মধ্যবর্তী সময়ে কোয়ান্টাম বলবিজ্ঞানের ব্যাপারগুলি নিয়ে বিজ্ঞানীরা চিন্তাভাবনা শুরু করেন। ১৯২৫ সালে কোয়ান্টাম তত্ত্বের আবির্ভাবের ১০-১৫ বছর আগেও পদার্থবিজ্ঞানীরা চিরায়ত বলবিজ্ঞানের মধ্যে থেকেই, বিশেষত একই ধরনের গাণিতিক সংঘটন ব্যবহার করে, ঘটনাগুলি ব্যাখ্যার চেষ্টা চালান। এই প্রচেষ্টাগুলির মধ্যে সবচেয়ে পরিশীলিত উদাহরণ হল সমারফেল্ড-উইলসন-ইশিওয়ারা কোয়ান্টায়ন নিয়ম। এই নিয়মটি সম্পূর্ণভাবে চিরায়ত দশা জগতে সূত্রায়িত করা হয়েছিল।

Overview

The word quantum is Latin for “how great” or “how much.” In quantum mechanics, it refers to a discrete unit that quantum theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves have discrete energy packets (called quanta) that behave in a manner similar to particles led to the branch of physics that deals with atomic and subatomic systems which we today call quantum mechanics. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. The foundations of quantum mechanics were established during the first half of the twentieth century by Werner Heisenberg, Max Planck, Louis de Broglie, Albert Einstein, Niels Bohr, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied[1].

Quantum mechanics is essential to understand the behavior of systems at atomic length scales and smaller. For example, if classical mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus, making stable atoms impossible. However, in the natural world the electrons normally remain in an unknown orbital path around the nucleus, defying classical electromagnetism.

Quantum mechanics was initially developed to provide a better explanation of the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom was developed as an explanation for the electron’s staying in its orbital, which could not be explained by Newton’s laws of motion and by Maxwell’s laws of classical electromagnetism.

In the formalism of quantum mechanics, the state of a system at a given time is described by a complex wave function (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space. This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows one to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one can never make simultaneous predictions of conjugate variables, such as position and momentum, with arbitrary accuracy. For instance, electrons may be considered to be located somewhere within a region of space, but with their exact positions being unknown. Contours of constant probability, often referred to as “clouds” may be drawn around the nucleus of an atom to conceptualize where the electron might be located with the most probability. Heisenberg’s uncertainty principle quantifies the inability to precisely locate the particle.

The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave–particle duality in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain “macroscopic quantum systems” such as superconductors and superfluids.

Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (I) the quantization (discretization) of certain physical quantities, (II) wave-particle duality, (III) the uncertainty principle, and (IV) quantum entanglement. Each of these phenomena is described in detail in subsequent sections.

History

The history of quantum mechanics[2] began essentially with the 1838 discovery of cathode rays by Michael Faraday, the 1859 statement of the black body radiation problem by Gustav Kirchhoff, the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete, and the 1900 quantum hypothesis by Max Planck that any energy is radiated and absorbed in quantities divisible by discrete ‘energy elements’, E, such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy, as defined by the following formula:

E = h \nu = \hbar \omega\,

where h is Planck’s Action Constant. Although Planck insisted[3] that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the physical reality of the radiation itself, in 1905, to explain the photoelectric effect (1839), i.e. that shining light on certain materials can function to eject electrons from the material, Albert Einstein[4] postulated, as based on Planck’s quantum hypothesis, that light itself consists of individual quanta, which later came to be called photons (1926). From Einstein’s simple postulation was borne a flurry of debating, theorizing and testing, and thus, the entire field of quantum physics.

Relativity and quantum mechanics

The modern world of physics is founded on the two tested and demonstrably sound theories of general relativity and quantum mechanics —theories which appear to contradict one another. The defining postulates of both Einstein’s theory of relativity and quantum theory are indisputably supported by rigorous and repeated empirical evidence. However, while they do not directly contradict each other theoretically (at least with regard to primary claims), they are resistant to being incorporated within one cohesive model.

Einstein himself is well known for rejecting some of the claims of quantum mechanics. While clearly inventive in this field, he did not accept the more philosophical consequences and interpretations of quantum mechanics, such as the lack of deterministic causality and the assertion that a single subatomic particle can occupy numerous areas of space at one time. He also was the first to notice some of the apparently exotic consequences of entanglement and used them to formulate the Einstein-Podolsky-Rosen paradox, in the hope of showing that quantum mechanics had unacceptable implications. This was 1935, but in 1964 it was shown by John Bell (see Bell inequality) that Einstein’s assumption was correct, but had to be completed by hidden variables and thus based on wrong philosophical assumptions. According to the paper of J. Bell and the Copenhagen interpretation (the common interpretation of quantum mechanics by physicists for decades), and contrary to Einstein’s ideas, quantum mechanics was

* neither a “realistic” theory (since quantum measurements do not state pre-existing properties, but rather they prepare properties)

* nor a local theory (essentially not, because the state vector \scriptstyle |\psi\rangle determines simultaneously the probability amplitudes at all sites, |\psi\rangle\to\psi(\mathbf r), \forall \mathbf r).

The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner, although the two particles can be an arbitrary distance apart; however, this effect does not violate causality, since no transfer of information happens. These experiments are the basis of some of the most topical applications of the theory, quantum cryptography, which works well, although at small distances of typically \scriptstyle \le 1000 km, being on the market since 2004.

There do exist quantum theories which incorporate special relativity—for example, quantum electrodynamics (QED), which is currently the most accurately tested physical theory [5]—and these lie at the very heart of modern particle physics. Gravity is negligible in many areas of particle physics, so that unification between general relativity and quantum mechanics is not an urgent issue in those applications. However, the lack of a correct theory of quantum gravity is an important issue in cosmology.

Attempts at a unified theory

Inconsistencies arise when one tries to join the quantum laws with general relativity, a more elaborate description of spacetime which incorporates gravitation. Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics. Many prominent physicists, including Stephen Hawking, have labored in the attempt to discover a “Grand Unification Theory” that combines not only different models of subatomic physics, but also derives the universe’s four forces—the strong force, electromagnetism, weak force, and gravity— from a single force or phenomenon.

Quantum mechanics and classical physics

Predictions of quantum mechanics have been verified experimentally to a very high degree of accuracy. Thus, the current logic of correspondence principle between classical and quantum mechanics is that all objects obey laws of quantum mechanics, and classical mechanics is just a quantum mechanics of large systems (or a statistical quantum mechanics of a large collection of particles). Laws of classical mechanics thus follow from laws of quantum mechanics at the limit of large systems or large quantum numbers.

The main differences between classical and quantum theories have already been mentioned above in the remarks on the Einstein-Podolsky-Rosen paradox. Essentially the difference boils down to the statement that quantum mechanics is coherent (addition of amplitudes), whereas classical theories are incoherent (addition of intensities). Thus, such quantities as coherence lengths and coherence times come into play. For microscopic bodies the extension of the system is certainly much smaller than the coherence length; for macroscopic bodies one expects that it should be the other way round.

This is in accordance with the following observations:

Many “macroscopic” properties of “classic” systems are direct consequences of quantum behavior of its parts. For example, stability of bulk matter (which consists of atoms and molecules which would quickly collapse under electric forces alone), rigidity of this matter, mechanical, thermal, chemical, optical and magnetic properties of this matter—they are all results of interaction of electric charges under the rules of quantum mechanics.

While the seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles, the laws of classical “Newtonian” physics still remain accurate in predicting the behavior of surrounding (“large”) objects—of the order of the size of large molecules and bigger—at velocities much smaller than the velocity of light.

Theory

There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg)[6] and wave mechanics (invented by Erwin Schrödinger).

In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or “observables”. Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Oftentimes these results are skewed by many causes, such as dense probability clouds or quantum state nuclear attraction. Much of the time, these small anomalies are attributed to different causes such as quantum dislocation. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. When the probability amplitudes of four or more quantum nodes are similar, it is called a quantum parallelism. There are, however, certain states that are associated with a definite value of a particular observable. These are known as “eigenstates” of the observable (“eigen” can be roughly translated from German as inherent or as a characteristic). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time; rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for (a) the state of something having an uncertainty relation and (b) a state that has a definite value. The latter is called the “eigenstate” of the property being measured.

For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wave function. The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. If one performs a position measurement on such a wavefunction, the result x will be obtained with almost 100% probability. In other words, the position of the free particle will almost be known. This is called an eigenstate of position (mathematically more precise: a generalized eigenstate (eigendistribution) ). If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. It can be shown that the wavelength is equal to h/p, where h is Planck’s constant and p is the momentum of the eigenstate. If the particle is in an eigenstate of momentum then its position is completely blurred out.

Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse. It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x.

Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton’s second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such “static” wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig. 1). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric).

The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random.

The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a “measurement” has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of “wavefunction collapse”; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics.

Mathematical formulation
In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac[7] and John von Neumann[8], the possible states of a quantum mechanical system are represented by unit vectors (called “state vectors”) residing in a complex separable Hilbert space (variously called the “state space” or the “associated Hilbert space” of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator’s spectrum is discrete, the observable can only attain those discrete eigenvalues.

The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator – which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg’s uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute.

The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states.

It turns out that analytic solutions of Schrödinger’s equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the “semi-classical equation of motion” approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos.

An alternative formulation of quantum mechanics is Feynman’s path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics.

Interactions with other scientific theories

The fundamental rules of quantum mechanics are very deep. They assert that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers. In other words, classic mechanics is simply a quantum mechanics of large systems. This “high energy” limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit.
Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the “superposition of states” and “wavefunction collapse”, give rise to the reality we perceive?

When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.

The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical \scriptstyle -\frac{e^2}{4 \pi\ \epsilon_0\ } \frac{1}{r} Coulomb potential. This “semi-classical” approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.

Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory, by the physicists Abdus Salam, Sheldon Glashow and Steven Weinberg.

It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.

Example

The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. For the 1-dimensional case in the x direction, the time-independent Schrödinger equation can be written as[9]:

- \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.

The general solutions are:

\psi = A e^{ikx} + B e ^{-ikx} \;\;\;\;\;\; E = \frac{\hbar^2 k^2}{2m}

or

\psi = C \sin kx + D \cos kx \; (by Euler’s formula)

The presence of the walls of the box restricts the acceptable solutions of the wavefunction. At each wall:

\psi = 0 \; \mathrm{at} \;\; x = 0,\; x = L

Consider x = 0

* sin 0 = 0, cos 0 = 1. To satisfy \scriptstyle \psi = 0 \; the cos term has to be removed. Hence D = 0

Now consider: \scriptstyle \psi = C \sin kx\;

* at x = L, \scriptstyle \psi = C \sin kL =0\;
* If C = 0 then \scriptstyle \psi =0 \; for all x. This would conflict with the Born interpretation
* therefore sin kL = 0 must be satisfied, yielding the condition

kL = n \pi \;\;\;\; n = 1,2,3,4,5,… \;

In this situation, n must be an integer showing the quantization of the energy levels.

Applications

Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behaviour of the subatomic particles that make up all forms of matter—electrons, protons, neutrons, photons and others—can often only be satisfactorily described using quantum mechanics. Quantum mechanics has strongly influenced string theory, a candidate for a theory of everything (see reductionism) and the multiverse hypothesis. It is also related to statistical mechanics.

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. (Relativistic) quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much. Most of the calculations performed in computational chemistry rely on quantum mechanics.

Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics.

Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

In many devices, even the simple light switch, quantum tunneling is vital, as otherwise the electrons in the electric current could not penetrate the potential barrier made up, in the case of the light switch, of a layer of oxide. Flash memory chips found in USB drives also use quantum tunneling to erase their memory cells.

Philosophical consequences

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. Even fundamental issues such as Max Born’s basic rules concerning probability amplitudes and probability distributions took decades to be appreciated.

The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of quantum mechanics most widely accepted amongst physicists. According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and does not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement (this dislike is the source of his famous quote, “God does not play dice with the universe.”). Einstein held that there should be a local hidden variable theory underlying quantum mechanics and that, consequently, the present theory was incomplete. He produced a series of objections to the theory, the most famous of which has become known as the EPR paradox. John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local realistic theories. Experiments have been performed confirming the accuracy of quantum mechanics, thus demonstrating that the physical world cannot be described by local realistic theories.[citation needed] The Bohr-Einstein debates provide a vibrant critique of the Copenhagen Interpretation from an epistemological point of view.

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a “multiverse” composed of mostly independent parallel universes. This is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. (Such a superposition of consistent state combinations of different systems is called an entangled state.) While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe, i.e. the consistent state contribution to the mentioned superposition, we inhabit. Everett’s interpretation is perfectly consistent with John Bell’s experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, the parallel universes will never be accessible to us. This inaccessibility can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away towards the other end of the universe; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again, together with the system that was measured originally. This is completely impractical, but even if one could theoretically do this, it would destroy any evidence that the original measurement took place (including the physicist’s memory).

Notes

1. ^ Compare the list of conferences presented here [[1]].
2. ^ J. Mehra and H. Rechenberg, The historical development of quantum theory, Springer-Verlag, 1982.
3. ^ e.g. T.S. Kuhn, Black-body theory and the quantum discontinuity 1894-1912, Clarendon Press, Oxford, 1978.
4. ^ A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt (On a heuristic point of view concerning the production and transformation of light), Annalen der Physik 17 (1905) 132-148 (reprinted in The collected papers of Albert Einstein, John Stachel, editor, Princeton University Press, 1989, Vol. 2, pp. 149-166, in German; see also Einstein’s early work on the quantum hypothesis, ibid. pp. 134-148).
5. ^ Life on the lattice: The most accurate theory we have
6. ^ Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in 1932 for the creation of quantum mechanics, the role of Max Born has been obfuscated. A 2005 biography of Born details his role as the creator of the matrix formulation of quantum mechanics. This was recognized in a paper by Heisenberg, in 1940, honoring Max Planck. See: Nancy Thorndike Greenspan, “The End of the Certain World: The Life and Science of Max Born” (Basic Books, 2005), pp. 124 – 128, and 285 – 286.
7. ^ P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, 1930.
8. ^ J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 (English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955).
9. ^ Derivation of particle in a box, chemistry.tidalswan.com

References

For the lay public:

* Feynman, Richard P.. QED: The Strange Theory of Light and Matter. Princeton University Press. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing many insights for the expert.
* Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and Multiple Universes. Buffalo NY: Prometheus Books. Includes cosmological and philosophical considerations.

More technical:

* Marvin Chester, 1987. Primer of Quantum Mechanics. John Wiley. ISBN 0-486-42878-8
* Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press. ISBN 0-691-08131-X
* Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. The beginning chapters make up a very clear and comprehensible introduction.
* Hugh Everett, 1957, “Relative State Formulation of Quantum Mechanics,” Reviews of Modern Physics 29: 454-62.
* Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The Feynman Lectures on Physics. 1-3. Addison-Wesley.
* Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7. OCLC 40251748. A standard undergraduate text.
* Max Jammer, 1966. The Conceptual Development of Quantum Mechanics. McGraw Hill.
* Hagen Kleinert, 2004. Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3rd ed. Singapore: World Scientific. Draft of 4th edition.
* Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon Press. ISBN 0-08-203204-1
* George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0-486-43517-2.
* Albert Messiah, 1966. Quantum Mechanics (Vol. I), English translation from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf. chpt. IV, section III.
* Omnès, Roland (1999). Understanding Quantum Mechanics. Princeton University Press. ISBN 0-691-00435-8. OCLC 39849482.
* Scerri, Eric R., 2006. The Periodic Table: Its Story and Its Significance. Oxford University Press. Considers the extent to which chemistry and the periodic system have been reduced to quantum mechanics. ISBN 0-19-530573-6
* Transnational College of Lex (1996). What is Quantum Mechanics? A Physics Adventure. Language Research Foundation, Boston. ISBN 0-9643504-1-6. OCLC 34661512.
* von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press.
* Hermann Weyl, 1950. The Theory of Groups and Quantum Mechanics, Dover Publications.

The Postulates of Special Relativity
Inertial Coordinate Systems
From the first postulate, it follows that there is no coordinate system which is in absolute rest. All motion with constant speed is relative and any coordinate system moving with constant speed (relative to the “fixed stars”) is called an inertial coordinate system (or inertial frame [of reference]).

Two inertial frames A and B are moving with constant speed relative to each other. An observer at rest in A will say that objects at rest in B are moving with respect to A. On the other hand, an observer at rest in B will say that it is the objects at rest in A that are moving with respect to B. Motion is relative!

Actually, Einstein was not influenced so much by the Michelson-Morley experiment at the time when he wrote down The Postulates of Special Relativity as he was by his so-called “Gedankenexperimenten” (imaginary “experiments” in his head) and by Ernst Mach and his principle, Mach’s principle1, as well as by Poincaré and his book La Science et l’Hypothèse.

1Mach’s principle: The inertial forces experienced by a body in nonuniform motion are determined by the quantity and distribution of matter in the Universe.

The Postulates of Special Relativity

On June 30, 1905 Einstein formulated the two postulates of special relativity:

1. The Principle of Relativity
The laws of physics are the same in all inertial frames of reference.

2. The Constancy of Speed of Light in Vacuum
The speed of light in vacuum has the same value c in all inertial frames of reference.

The speed of light in vacuum c (299792458 m/s) is so enormous that we do not notice a delay between the transmission and reception of electromagnetic waves under normal circumstances.

The speed of light in vacuum is actually the only speed that is absolute and the same for all observers as was stated in the second postulate.

In 1808, a scientist and teacher John Dalton developed the atomic model of matter that we base all modern chemistry upon. Three of the main ideas of modern atomic theory are:

An element is composed of tiny particles called atoms. All atoms of a certain element have similar chemical properties. Atoms of different elements have different properties.
In an ordinary chemical reaction, no atom of any element disappears or is changed into an atom of another element.
Compounds are formed when atoms of two or more elements combine. In a given compound, the relative numbers of atoms of each kind are definite and constant.
Based on Dalton’s propositions the atom can be defined as the smallest particle of an element that defines its chemical properties.

Inside the atom there are three very important particles: electrons, protons and neutrons.

In the very middle of the atom is the nucleus which contains protons and neutrons. Each element is identified by the number of protons in its nucleus. Carbon, for example, always has six protons. The number of neutrons in an element’s nucleus can vary, however. Carbon can have from two to 16 neutrons, forming different isotopes of carbon. Of these, only the carbon nucleii with six, seven or eight neutrons are naturally occurring. The proton has a mass nearly equal to an ordinary hydrogen atom. It carries a positive charge, which is equal and opposite to the negative charge of the electron. The neutron is an uncharged particle with a mass slightly greater than a proton. In a neutral atom, the number of protons in the nucleus is equal to the number of electrons outside the nucleus. Compared to the mass of protons and neutrons electrons are very small. See the table below for a comparison in properties of the three subatomic particles.

PROPERTIES OF SUBATOMIC PARTICLES Particle Location Relative Charge Mass (amu)* Mass (kg)
proton nucleus + 1 1.00728 1.673 x 10 -27kg
neutron nucleus 0 1.00867 1.675 x 10 -27kg
electron outside the nucleus – 1 0.00055 9.109 x 10 -31kg

* amu = atomic mass units

Sources: Masterton & Hurley, Chemistry Principles and Reactions, Third Edition, (1997, 1993, 1989 Saunders College Publishing)

The fusion process shown in the animation involves deuterium (hydrogen-2) and tritium (hydrogen-3) fusing to give helium and a neutron which is ejected from the atom at a high speed. The energy released in this reaction goes into the kinetic energy of the helium and the neutron. Fusion occurs in a star because the massive gravity of the star creates pressures so intense light elements such hydrogen are fused together to make deuterium. In this process, a positron and a neutrino are released (due to the conversion of a proton into a neutron). Fusion can also be achieved using extreme temperatures which also occur in stars.

All life on Earth is dependant on nuclear fusion. It is the energy source that warms our planet and gives us light. Here on Earth scientists have only been able to achieve fusion reactions lasting a few seconds but around the world there has been research into using nuclear fusion as an energy source. If successful, nuclear fusion could possibly result in a limitless supply of clean energy.

Einstein’s letter to FDR regarding the possibility of the creation of a nuclear bomb.
Scientists in the 1930s, using machines that could break apart the nuclear cores of atoms, confirmed Einstein’s formula E=mc² . The release of energy in a nuclear transformation was so great that it could cause a detectable change in the mass of the nucleus. But the study of nuclei — in those years the fastest growing area of physics — had scant effect on Einstein. Nuclear physicists were gathering into ever-larger teams of scientists and technicians, heavily funded by governments and foundations, engaged in experiments using massive devices. Such work was alien to Einstein’s habit of abstract thought, done alone or with a mathematical assistant. In return, experimental nuclear physicists in the 1930s had little need for Einstein’s theories.

In August 1939 nuclear physicists came to Einstein, not for scientific but for political help. The fission of the uranium nucleus had recently been discovered. A long-time friend, Leo Szilard, and other physicists realized that uranium might be used for enormously devastating bombs. They had reason to fear that Nazi Germany might construct such weapons. Einstein, reacting to the danger from Hitler’s aggression, had already abandoned his strict pacifism. He now signed a letter that was delivered to the American president, Franklin D. Roosevelt, warning him to take action. This letter, and a second Einstein-Szilard letter of March 1940, joined efforts by other scientists to prod the United States government into preparing for nuclear warfare. Einstein played no other role in the nuclear bomb project. As a German who had supported left-wing causes, he was denied security clearance for such sensitive work. But during the war he did perform useful service as a consultant for the United States Navy’s Bureau of Ordnance.

After the Japanese surrendered under nuclear bombardment, Einstein was often in the public eye. In May 1946 he became chairman of the newly formed Emergency Committee of Atomic Scientists, joining their drive for international and civilian control of nuclear energy. He recorded fund-raising radio messages for the group, and wrote a widely read article on their work. Einstein’s appeals for nuclear disarmament had an influence among both scientists and the general public. He also spoke out in opposition to German rearmament, defended conscientious objectors against military service, and criticized the Cold War policies of the United States. An early and firm supporter of the United Nations, he was convinced that the solution to international conflict was world law, world government, and a strong world police force. “I am opposed to the use of force under any circumstances, except when confronted by an enemy who pursues the destruction of life as an end in itself.”
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Like many in the 1950s who supported liberal causes, Einstein was suspected of disloyalty. He publicly opposed such McCarthyism. Asked how intellectuals should respond, he declared, “I can only see the revolutionary way of non-cooperation.”

Although his activity was limited by advancing age and ill health, Einstein made clear his commitment to civil liberties. He attacked racial prejudice and supported the black civil rights movement. He called for a homeland in Palestine for the Jewish people, in which the rights of Arabs would also be respected. Meanwhile, he supported the creation of a Jewish university in the United States (the future Brandeis University). When the House Committee on Un-American Activities maligned teachers and other intellectuals, Einstein publicly advised the people under attack not to cooperate, but to follow the principle of civil disobedience. He was equally uncompromising when he refused any association with Germany. He even rejected honors from his native land — he could not forgive the murder of Jews by Germans.

In 1952 Einstein was offered the position of President of Israel, a chiefly honorific post. Old and sick, but at peace in his Princeton home and office, he turned down the invitation. His interest in public affairs, however, continued. In 1955 he joined Bertrand Russell in urging scientists toward mediation between East and West and limitation of nuclear armament. Meanwhile he was writing a speech for the anniversary of Israel’s independence. An incomplete draft of the speech was found at his bedside after he died.

“The abolition of war will demand distasteful limitations to national sovereignty. But what perhaps impedes understanding of the situation more than anything else is that the term mankind feels vague and abstract. People… can scarcely bring themselves to grasp that they, individually, and those whom they love are in imminent danger of perishing agonizingly. And so they hope that perhaps war may be allowed to continue… this hope is illusory.”

“The feeling for what ought and ought not to be grows and dies like a tree, and no fertilizer of any kind will do much good. What the individual can do is give a fine example, and have the courage to firmly uphold ethical convictions in a society of cynics. I have for a long time tried to conduct myself this way, with varying success.”

“Here, then, is the problem which we present to you, stark and dreadful and inescapable: Shall we put an end to the human race or shall mankind renounce war? People will not face this alternative because it is so difficult to abolish war.”

RU Professor discloses New alternative thesis to Einstein’s Relativity
proposed
RAJSHAHI, June 4:-Dr Osman Gani Talukder, Associate Professor, Department of
Applied Physics and Electronics, Rajshahi University proposed a new
hypothesis on “Alternative Approach to Relativity” through his massive
research work in the said specific line of physics.
He disclosed the matter in a press conference held here at a Chinese
Restaurant. Addressing the press conference Dr Talukder said, “according to
Einstein’s Relativity Theory, no particle can achieve the speed of light.
This is a consequence of Einstein’s Relativity Theory that when the speed of
a particle approaches that of light, its mass becomes infinite which is
absurd. It should be pointed here that it was experimentally found that a
neutrino has a finite mass but it moves at the speed of light in
contradiction with Einstein’s Theory”. He further said on the contrary,
according to the alternative approach, there is no such restriction on any
particle’s speed being equal to or greater than that of light. When the
speed of a particle approaches that of light its mass becomes half which is
reasonable and consistent with the experimental evidence.
Further, in the Theory of Relativity, Einstein introduces a new concept of
mass, called “Relativity Mass” which increases with the increase of speed,
he said, adding whereas, in the alternative approach, a new concept of mass,
called “Potential Mass” which is equivalent to the potential energy of
matter, has been introduced. The Potential Mass of a body decreases with the
increase of speed. The concept originates through the integration of the
important fundamental law of classical mechanics (the Energy Conservation
Law) with the most famous consequence of the postulates of relativity (the
mass-energy equivalent). An important implication is that the unification of
some of the fundamental laws of Quantum mechanics is possible through it. In
other words, the generalisation of some fundamental laws of physics can be
done through this concept, he added.
He said, “Einstein, at the last stage of his life tried of unify the four
natural forces of the Universe. But he failed to do so. This alternative
approach to the relativity can pave the way towards completing his
unfinished work”.
Dr Talukder said that the concept gave clear and conspicuous idea about
energy. It also has signified that it may be possible to make necessary
technological development so that we can use energy from a source in a
controlled way.
As a result, energy can be extracted from the enormous energy sources like
atom bombs and can be utilised for the betterment of mankind. In this
context he pointed out that such controlled release of energy from atom
bombs is possible, at least in principle, according to the concept but is
not at all possible according to Einstein’s special Relativity Theory. “Now,
let us think about how much energy is stored in a modern atom bomb.
Generally, the explosion of a present bomb converts about one kilogram of
matter into energy instantly”, he said adding if this amount of energy is
used in a controlled way, it can provide the required amount of electrical
energy (which is 3000 megawatts) in Bangladesh for about a year. Dr Osman
Gani Talukder is now also a part time teacher and researcher in a university
at Perth in Australia and full time teacher in a college of the same
country.
ASA Motakabbir, Assistant Professor, Applied Physics and Electronics
Department, RU and some of the family members of Dr Gani were present at the
Press conference. ( The Bangladesh Observer )
Source URL:
http://www.bangladesh-web.com/news/jun/05/nv4n588.htm#A10