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40 OPTICS & PHOTONICS NEWS APRIL 2013 and Bose-Einstein Statistics

Barry R. Masters

Satyendra Nath Bose is an often overlooked hero in the field of and quantum statistics.

ver since researchers found evidence for the existence of the Higgs boson in 2012, the particle has made frequent appearances in scientific and E mainstream media. Many people recognize Peter Higgs—one half of the particle’s namesake, but fewer realize that the “boson” half of the moniker refers to Indian Satyendra Nath Bose. Although not as well known as his contemporaries, S.N. Bose was arguably one of the most important scientists of the 20th century. His revolutionary way of viewing photon behavior and photon statistics—later known as Bose statistics—changed the field in a dramatic way. played an integral role in publishing his work and also extended Bose’s new statistics to atoms and predicted the Bose- Einstein condensate (BEC). Wikimedia Commons

APRIL 2013 OPTICS & PHOTONICS NEWS 41 1047-6938/13/04/40/8-$15.00 ©OSA In 1918, Bose and published a joint paper on the in the prestigious English journal, Philosophical Magazine.

Several years ago, with the support of an the lectures and laboratory demonstrations of OSA Fellow Travel Award (OPN, May 2004), his physics teacher , who I lectured in the various Indian Institutes of developed novel instruments to produce and Technology and the two Bose Institutes in detect millimeter waves. On his final examina- Calcutta, one of which was named after S.N. tions, Bose achieved the highest ranking among Bose (the second was named after Jagadish the B.Sc. students and repeated this feat on his Chandra Bose). The trip inspired me to take a M.Sc. examinations. closer look at his work and his life. This article In 1914, at the age of 20 and while still an honors the seminal work of Bose, his statistics M.Sc. student, Bose married Ushabati Ghosh, and the BEC. the 11-year-old daughter of a prominent physician in Calcutta. Of their nine children, Early life and education two died in infancy; the surviving children Bose was born on 1 January 1894 in Calcutta, consisted of five daughters and two sons. the capital of British India. At the age of five, he enrolled in the Normal School in North Early professional life Calcutta and completed his final years of pri- In 1917, Bose became a lecturer in physics and mary education at the Hindu School in Cal- applied mathematics at the University College cutta. When he was eight, Bose demonstrated of Science, Calcutta University. In 1918, Bose extraordinary intellect and mathematical and Meghnad Saha published a joint paper on talent. He attributed to his upbringing his the kinetic theory of gases in the prestigious urbane, liberal and open-minded character; English journal, Philosophical Magazine. his deep curiosity and love of the arts; and Subsequently they translated Einstein’s his perfectionism. papers into English so that Indian students Following his studies at the Hindu School, who did not know German could read them. Bose studied at the Presidency College in This work, The , which Calcutta from 1909 to 1915. Although he contained papers by Einstein and Hermann suffered from weak vision, he loved to read; Minkowski, was published in 1919 by Cal- his favorite poets were Tagore and Tennyson. cutta University. His motivation to study science was similar to Bose worked at the University of Dhaka in that of many of his fellow students: The process East Bengal from its founding in 1921 to 1924. of science could facilitate the advancement of His initial post was as a reader (pre-profes- development and prosperity in his country. sorial position) in the physics department. “We wanted to put scientific knowledge to use He taught courses on and through technology for the benefit of the people electromagnetic theory to the M.Sc. students. or to contribute to science by intensive study,” For Bose, Dhaka was isolated from the Euro- Bose said. pean centers and thus many of the modern Presidency College was the best place to developments in relativity theory and quantum study science in Calcutta, and the competition theory. But his colleague saved him from seclu- to get in was intense; about 2,000 applicants sion. Debendra Mohan Bose returned from competed for 32 slots. Bose overcame the odds his doctoral studies in Germany in 1919 and and earned himself a spot. He was impressed gave Bose a wonderful gift: ’s book with his science teachers and especially with Thermodynamik und Wärmestrahlung.

42 OPTICS & PHOTONICS NEWS APRIL 2013 Bose’s 1924 publications Statistical Bose read Planck’s papers on the distribution of Prior to Bose’s Statistics energy in blackbody radiation in order to teach this material to his class. He was disturbed by 1877 Planck’s derivation of the law with its ad hoc develops a method of assumptions. In 1924, Saha stayed with Bose calculating probability by counting states in Dhaka and pointed out the 1923 papers of and applied it to an ideal gas at equilibrium; he found the equilibrium state, which is the , Einstein and Paul Ehrenfest most probable state, is given by the Maxwell- Boltzmann and their relation to Einstein’s 1917 paper. Boltzmann distribution. This led him to develop Bose statistics—a new method to count states of indistinguishable 1900 particles—and applied it to his derivation of Max Planck builds upon Boltzmann’s prior Planck’s radiation law. work that related entropy and probability Bose wrote two papers in which he inde- and develops the Boltzmann principle or pendently derived and further developed the equation: S = klnW, where S is entropy, k is Boltzmann’s constant, and W is the probability previous work of Planck: “Planck’s law and the of a distribution of elements into cells. Planck light-quantum hypothesis,” and “Thermal equi- published his law of blackbody radiation based Planck librium in the radiation fi eld in the presence of on Maxwell’s classical electromagnetic theory and the ad hoc assumption that the energy « of the oscillators or matter.” He considered the papers as a single resonators was quantized. Some years later, Planck accepted that unit and submitted them to the Philosophical it was not possible to derive his law only from Maxwell’s electro- Magazine. He did not receive a reply from the magnetic theory and Maxwell-Boltzmann statistics. prestigious English journal; therefore, on 4 June 1924, he sent the fi rst paper to Einstein, along 1905 with a letter: Albert Einstein analyzes the energy fluc- “If you think the paper worth publication, tuations of radiation that obeyed Wien’s law and concludes that the radiation consists I shall be grateful if you arrange for its of particles of discrete light-quanta with publication in Zeitschrift für Physik. energy, « = hn, where h is Planck’s constant Though a complete stranger to you, I do and n is the frequency of the oscillator. Einstein not feel any hesitation in making such a request. Because we are all your pupils 1910 though profi ting only by your teachings Peter Debye publishes his derivation of through your writings.” Planck’s law; instead of following Planck, who The two had corresponded before, when used the classical relation between energy density of the blackbody radiation and the av- Bose had asked Einstein if he could translate erage energy of a resonator, Debye calculates his paper on generalized relatively into English. the probability of a given state of the radiation (Einstein acceded.) Bose’s fi rst paper stated that, using the state without invoking resonators. Debye in the thermal equilibrium of massless par- ticles (photons), the number of particles are not conserved due to the absorption and emission 1911, 1914 of photons. These particles could be described From a historical perspective, the little- known work of Ladislas in a new way—with Bose statistics, which Natanson is significant. used the counting of cells in one-particle phase He shows that both Planck space instead of the previous method of totaling and Debye have made the the number of standing waves in a cavity in tacit assumption of the in- distinguishability of quanta Onnes order to obtain the number of states. in their derivations. Both Only one year before, Arthur Compton Paul Ehrenfest and Kamerlingh Onnes had published his seminal paper on the reach the same conclusion. Ehrenfest Compton effect. Bose interpreted Z, the Wikimedia commons

APRIL 2013 OPTICS & PHOTONICS NEWS 43 S.N. Bose number of states in the frequency interval logical fl aws. He stated that Einstein’s 1917 viewing a between n and n + dn as the number of cells derivation used “Wien’s law based on classical photo of Albert Einstein that contain a given number of photons with theory and the Bohr correspondence principle, (1953). frequency n, and not the number of particles, which assumes that the quantum theory agrees

AIP Emilio Segre as was previously done by Boltzmann. with the classical theory in certain limiting Visual Archives, Gift of cases.” Bose went on to say that “a new form of Kameshwar Wali and On 14 June, he sent Einstein his second Etienne Eisenmann paper: is required to be compat- ible with quantum theory; these logical fl aws “I hope my fi rst paper has reached your can be solved by using statistical mechanics hands. The result to which I have arrived without any assumption of mechanism of seems rather important (to me at any rate). elementary processes on which the energy You will see that I have dealt with the exchange depends.” problem of thermal equilibrium between Bose followed Einstein’s light-quantum radiation and matter in a different way, hypothesis and considered blackbody radiation and have arrived at a different law for the as light-quanta in a volume. He then continued probability of elementary processes, which in several steps, the most signifi cant of which seems to have simplicity in its favour.” was how to calculate the number of states of a Both papers put forward the idea that the light-quantum in the frequency range of n and fundamental assumptions of quantum theory n + dn. Bose extended Planck’s “fi rst quantiza- are not compatible with the laws of classical tion” of the material oscillators to the radiation electrodynamics. According to Bose, all previ- fi eld itself (photons). With the knowledge of the ous derivations of the relation between energy Compton effect, Bose assumed that the photon n density sn and the average energy of a resonator, of frequency “has momentum of magnitude E, use classical assumptions and resulted in hn/c in direction of its .”

44 OPTICS & PHOTONICS NEWS APRIL 2013 While Einstein had both of Bose’s papers published in Zeitschrift für Physik, his appended comments on each were very different.

Bose divided the total phase space volume removed it from his paper and replaced it with h3 into cells of volume and multiplied the the sentence about polarization. resulting formula by a factor of two. His Bose then stated: “The total number of cells paper explained that the factor of two was must be regarded as the number of possible required “in order to take into account the arrangements of a quantum in a given volume.” fact of polarization.” He then showed that the factor 8πn2/c3 in While Einstein had both of Bose’s papers Planck’s law is the total number of quantum published in Zeitschrift für Physik, his states of the radiation. Next, he calculated appended comments on each were very the thermodynamic probability of a state. different. On Bose’s fi rst paper, Einstein Furthermore, he stated, “… since each cell is to was enthusiastic and viewed the work as a be counted as a single … conse- signifi cant contribution. But Einstein’s long quently the cells can be partitioned into distinct appended comments on the second paper classes characterized solely by their occupation were very different: numbers.” Finally, he followed the standard techniques to derive Planck’s law. Bose never “Your principle is not compatible with the cited the previous works of Natanson nor the following two conditions: (1) The absorp- works of Ehrenfest and Onnes. tion coeffi cient is independent of the radia- tion density. (2) The behavior of a resonator in a radiation fi eld should follow from the statistical laws as a limiting case.” Bosons and Bose was disappointed with Einstein’s Fermions comments on his second publication and on 27 n 1926, Wolfgang Pauli published January, 1925, he wrote his rebuttal to Einstein: I a paper stating his exclusion prin- ciple, which said that no two identical “I have written down my ideas in the form particles with half-integer may of a paper [his third] which I send under occupy the same quantum state separate cover … I have tried to look at the simultaneously. This was followed radiation fi eld from a new standpoint and by ’s publication on the Fermi have sought to separate the propagation of statistics of particles that obey the quantum of energy from the propagation of exclusion principle. In the same year, connected the Bose and Fermi statistics of par- electromagnetic infl uence…” ticles with the symmetry properties of their wave functions Bose later stated that, in his paper, he gave and named the particles “bosons” and “fermions.” Bosons (e.g., photons) have integral a quantum mechanical explanation for the spin and obey Bose-Einstein statistics. factor of two, but that Einstein removed it and Moreover, two or more particles can oc- replaced it with the polarization argument. cupy the same quantum state. Fermions While the original Bose paper in English is (e.g., electrons), on the other hand, have missing from the archives, it is likely that Bose half-integer spin and obey Fermi-Dirac statistics (discovered in 1926). More- postulated the light-quanta had an intrinsic over, two or more Fermions cannot spin with only the values of ±h/2π. occupy the same quantum state due to Dirac He felt that his postulated idea of photon the Pauli . spin was lost when Einstein presumably

APRIL 2013 OPTICS & PHOTONICS NEWS 45 Velocity- Bose believed that Einstein misunderstood separation occurs; one part ‘condenses,’ the rest distribution the content of his paper. This appeared to remains a ‘saturated ideal gas.’ ” data of a gas of rubidium atoms, dampen his desire to pursue his research. The concept of a BEC changed from an idea confirming the While in Germany, Bose did not publish any to reality in 1938 when Fritz London made the discovery of a new papers and even after he returned to India, he proposal that the 1928 discovery of superfl uid- phase of matter, the Bose-Einstein did not produce any original work in theoreti- ity in helium-4 was due to a BEC. condensate. cal physics for the next 12 years. A modern description states that, at high temperatures, the particles in a Boson gas are NIST/JILA/CU-Boulder Bose-Einstein condensate distributed into energy levels given by Bose- One week after he received Bose’s fi rst paper, Einstein statistics. Below a critical temperature, Einstein extended Bose’s statistics and applied a macroscopic number of particles condense in them to a monoatomic ideal gas whose numbers the ground state and form a BEC. Thus, we can are conserved. He submitted three papers on observe quantum behavior on a macroscopic this work to the Prussian Academy of Science scale and investigate many of the fundamental in Berlin on 10 July 1924, 8 January 1925 and properties of matter. The fi rst BEC of a gas was 29 January 1925. He did this without Bose’s made in 1995 by Eric Cornell and Carl Wieman; consent and without asking him to collaborate. they and Wolfgang Ketterle shared the 2001 Furthermore, in two of the papers, Einstein for their work. incorrectly cited Bose’s statistics as having been devised by D. Bose, referring to another Europe and late life physicist, Debendra Mohan Bose. From 1924 to 1926, Bose lived in Europe. The paper Einstein presented to the Prussian He spent the fi rst year in Paris and then Academy on 8 January 1925 showed that, below went to Berlin, where he was hoping to a critical temperature, there was a collection of work with Einstein, since he knew that identical atoms, some of which would make a Einstein had published a subsequent paper transition to the fi rst quantum state—a state that extended his work and predicted the without kinetic energy. Einstein wrote: “A Bose-Einstein condensation.

46 OPTICS & PHOTONICS NEWS APRIL 2013 One week after he received Bose’s first paper, Einstein extended Bose’s statistics and applied them to a monoatomic ideal gas whose numbers are conserved.

When they finally met in Berlin, Einstein was working of India established the Satyendra Nath Bose National on a new subject: the unification of the electromagnetic Center for Basic Sciences in Calcutta in 1986. Although and the gravitational fields. This change of research they didn’t always agree, Bose’s work with Einstein precluded them from working together; however, Einstein changed the landscape of new physics in the 20th century. gave Bose a letter of introduction, which proved helpful Nearly 40 years after his death, his additions to scientific for Bose to meet many German . knowledge continue to benefit the community and will With letters of recommendation from Albert Einstein, do so for generations to come. OPN and Herman Mark, Bose became a profes- sor and the head of the department of physics when he Barry R. Masters ([email protected]) is a Fellow of AAAS, OSA and SPIE. He is a visiting scientist with the department of returned to Dhaka, despite the fact that he did not have a biological engineering, Massachusetts Institute of Technology, doctoral degree. (Incidentally, D.M. Bose had previously Cambridge, Mass., U.S.A. turned down an offer for the same position.) Bose’s mis- sion was to introduce the latest developments in physics and to his university. REFERENCES & RESOURCES In 1945, Bose moved to Calcutta University, where he c S.N. Bose. “Plancks Gesetz und Lichtquantenhypothese,” remained until 1956 when he became a professor emeritus Zeitschrift für Physik 26, 178 (1924). in his retirement. He then taught physics at Santiniketan, c S.N. Bose. “Wärmegleichgewicht im Strahlungsfeld bei An- the school founded by , a well- wesenheit von Materie,” Zeitschrift für Physik 27, 384 (1924). known Renaissance man, in 1921. Bose and Tagore were c A. Einstein. “Quantentheorie des einatomigen idealen gases,” friends, and Tagore dedicated his book Our Universe to Sitzungsberichte der Preussischen Akademie der Wissen- Bose. However, Bose faced severe opposition from the schaften XXII, 261 (1924). other teachers in the arts who feared that the teaching of c A. Einstein. “Quantentheorie des einatomigen idealen gases. Zweite Abhandlung,” Sitzungsberichte der Preussischen science would move the schools’ focus away from their Akademie der Wissenschafte I, 3 (1925). subjects; therefore, after two years at Santiniketan, Bose c S.N. Bose. S.N. Bose, The Man and His Work, Life, Part I, again returned to Calcutta as a national professor. Collected Scientific Papers. S.N. Bose National Centre for Later in his career, Bose received many awards Basic Sciences, Calcutta, India (1994). and honors for his contributions to science education c S.N. Bose. S.N. Bose, The Man and His Work, Life, Part II, Lectures and Addresses, Miscellaneous Pieces. S.N. Bose and research in India. He was elected president of the National Centre for Basic Sciences, Calcutta, India (1994). Indian Physical Society in 1945 and held the position c S. Dasgupta. Jagadis Chandra Bose and the Indian Response until 1948. He also served as president of the National to Western Science, Oxford University Press, New Delhi, India Institute of Sciences of India from 1948 to 1950. In 1958, (1999). he was made a Fellow of the Royal Society. From 1954 c J.R. Anglin and W. Ketterle. “Bose-Einstein condensation of atomic gases,” Nature 416, 211-18 (2002). to 1959, he was made a member of the upper chamber c E.A. Cornell and C.E. Wieman. “Nobel lecture: Bose-Einstein of the national parliament. condensation in a dilute gas; the first 70 years and some th Bose died one month after his 80 birthday on 4 Feb- recent experiments,” Rev. Mod. Phys. 74, 875-93 (2002). ruary 1974. As a tribute to his life’s work, the government c A. Einstein. Facsimile of Einstein’s handwritten 1925 paper on BEC, Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung. www.lorentz.leidenuniv.nl/history/Einstein_ar- ONLINE EXTRA chive/Einstein_1925_manuscript/index_1.html To view the complete set of letters between S.N. Bose c W. Ketterle. When atoms behave as waves: Bose-Einstein and Albert Einstein, see the online version of this article condensation and the atom laser, Nobel Lecture, 8 Decem- at www.osa-opn.org. ber 2001. www.nobelprize.org/nobel_prizes/physics/laure- ates/2001/ketterle-lecture.pdf

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