Bose and Einstein – A Historic Collaboration

Preface

Einstein and Tagore, two of the greatest personalities of the last century, found music as a common thread linking them when they met twice in the former's home in Germany in 1930.  When I wrote about this in one of my earlier blog posts (see Tagore and Einstein on Music, Aug 10) some friends pointed out that there was a similar connection between Einstein and another great Bengali personality, Satyendranath Bose, this time the linkage being vastly more enduring and of tremendous importance as well in the history of scientific thought.  They suggested that I touch upon this too without delving too deep into the complexities of the subject matter that provided the linkage.  I had taught Bose-Einstein Statistics very admiringly as well as passionately as part of a Statistical Mechanics course in Physics at the postgraduate level for many years.  Despite this background, I am attempting the task here with considerable misgivings, particularly because of the formidable difficulty in communicating the ideas involved in a non-mathematical language.

Background

At the turn of the twentieth century, the world of Physics had faced a crisis on several fronts because of the failure of its long established theories to explain a number of puzzling discoveries.  It speaks for the outstanding genius of Albert Einstein, a largely self-taught and unknown clerk in a German patent office, that these were all resolved through radically new and revolutionary ideas which gave a refreshingly different direction to the march of science.   The fact that one of the edifices of this revolution was the Special Theory of Relativity in 1905, followed ten years later by the even more important General Theory of Relativity, is well known even among non-scientists.  However, the fact that Einstein was also principally responsible for the other great edifice of the revolution, Quantum Theory, later to develop into Quantum Mechanics, is less well known.   The name of Satyendranath Bose (often shortened as S N Bose) is linked intimately with one of the earliest and most important applications of this theory along with that of Einstein himself, the collaborative result going into the history of science as Bose-Einstein statistics.  As we shall see later, the name of Bose is also permanently enshrined in the annals of Physics as boson, the name used to describe any fundamental entity of nature conforming to the Bose-Einstein statistics.

Particles and Radiation

The term 'particle' is often used to describe entities like molecules (even tightly bound collections of molecules), atoms, nuclei of atoms, components of nuclei which are principally neutrons and protons, electrons, neutrinos and a variety of other entities.  They are all characterized by definite rest masses, i.e., the mass they possess when they are at rest relative to a given frame of reference (if they are in motion with reference to such a frame, their masses increase with their speeds in a manner described by Einstein's Special Theory of Relativity).  They also have other characteristics like electric charge, intrinsic spin, magnetic moment, etc.

The term 'radiation' is used to denote energy emitted or absorbed by matter in the form of 'electromagnetic waves' which travel in vacuum at a constant speed of about 300 000 km/sec (this gets reduced in material media like glass or water in inverse proportion to their optical densities).   They are characterized by a 'frequency' and a 'wavelength' such that the product of these two always gives the speed. Visible light of any color, infrared rays, ultraviolet rays, X rays, gamma rays, microwaves and radio waves are all examples of electromagnetic radiation.  Their behavior is governed by Maxwell's Electromagnetic Theory, just as the behavior of particles is governed by Newton's Laws of Motion (this is true only for low speeds, the behavior of particles at speeds close to that of light being governed by the Theory of Relativity).

Scope of Statistical Mechanics

Let us consider a collection of particles enclosed in a container under normal conditions. These particles will be in constant random motion, often bumping against each other and with the walls of the container, thereby changing their speeds and directions after each collision.  Given a set of initial conditions, these changes can be calculated in principle for each particle and its future course worked out by applying Newton's laws of motion.   However, the number of particles that exist even in the tiniest such container will be so enormously large that it is an inconceivably horrendous task to perform such calculations for each particle individually and thereby work out the consequence for the collection as a whole.  Fortunately however, it is the collective behavior and averaged out properties that would be of paramount interest when considering such a huge collection of particles.  The individual behavior hardly matters if the collective behavior can be understood in some way.  This is where statistical techniques come in handy.

  

Statistical Mechanics was developed in the nineteenth century primarily through the efforts of James Clerk Maxwell in England and Ludwig Boltzmann in Germany.  They applied statistical techniques to a collection of particles under normal conditions and some simplifying assumptions, called an ideal gas, to average out the microscopic dynamics of individual particles and work out their macroscopic (large-scale) thermodynamic features.  This has come to be known as Maxwell-Boltzmann (MB) Statistics.  One important consequence of this exercise was that the temperature of the substance was a measure of the average kinetic energy of the microscopic particles.

   

Radiation and the Ultraviolet Catastrophe

It is common experience that when a body is heated it starts emitting radiation, initially in the form of heat (infrared), and later, as the temperature increases, in the form of visible light varying in color from red to blue. At sufficiently high temperatures it is possible to detect emitted radiation over almost the whole of the electromagnetic spectrum, with an energy distribution characteristic of the temperature.  In thermodynamics, one talks of 'black body radiation', the energy emitted by an idealized entity called the black body.   A black body is visualized as a cavity of matter into which all radiation that falls on it is completely absorbed. This is not to be confused in any way with a 'black hole' which is one of the possible terminal stages of a super massive star in its evolutionary process.  Also, the body does not have to look black in appearance.  The brilliantly bright Sun with a surface temperature of about six thousand degrees can in fact be approximated as a functionally black body.

The distribution of energy in different wavelength (or frequency) ranges at different temperatures of a black body was measured accurately and compared with the predictions made on the basis of Maxwell's electromagnetic theory which postulates the radiation as being emitted or absorbed by natural 'oscillators'.  As shown by the British physicists Rayleigh and Jeans, the energy emitted at any frequency should increase very rapidly with the frequency of the radiation.  This means that there should be no upper limit to the energy and the spectrum of radiation should be swamped by the more energetic radiations corresponding to the higher frequencies at any particular temperature.  This classical theoretical prediction of what came to be known as 'ultraviolet catastrophe' deviates sharply from the experimental observations in which the energy distribution curve goes through a smooth peak after an initial rise and decreases thereafter as the wavelength increases.   This irreconcilable difference between theory and observation was one of the major crises in the world of Physics referred to earlier.

Planck's Remedy

Max Planck of Germany came up with a remedy to the black body radiation crisis by deriving a formula to fit the experimental data perfectly (see figure below for a black body temperature of 5000 K) with a simple and elegant but ad-hoc assumption – that the radiation is emitted by the oscillators only in certain discrete 'quantized' steps and not continuously as was assumed earlier. In doing this he considered the possible ways of distributing electromagnetic energy over the different modes of vibrations of the charged oscillators.  He visualized the energy (E) of such a quantum (later to be called a photon) as proportional to the frequency (n) of the emitted radiation.  This is given by the relation E = hn where h is a constant of proportionality, later to be enshrined as the fundamental and universal Planck's constant.   Planck himself had been troubled by the unsatisfactory nature of his assumption, but it had worked wonders.  If one regards the proof of the pudding as lying in its eating, here was such an unqualified success and so fundamental to the future course of what came to be known as Quantum Physics that Plank was awarded the Nobel Prize for Physics.

The Photon Theory

One of the other crises in Physics at that time was the Photoelectric Effect which went squarely against the wave nature of radiation propounded in the electromagnetic theory.  Experiments had shown that, when light of frequency above a threshold value fell on the surfaces of certain materials, electrons were emitted with energies in proportion to the frequency of the incident radiation.  There was no way this could be reconciled with the well established wave nature of radiation.  Einstein came up with a revolutionary solution to the problem by going well beyond Planck's hypothesis and proposing that radiation is not only emitted in quantized form but also propagated and absorbed as quanta.  In other words, radiation could be thought of as existing as discrete packets or 'particles of energy' just as one could think of particles of matter like electrons and protons.  The energy of such a quantum of radiation corresponded to Planck's relation E = hn.

Einstein's simple and successful explanation of the Photoelectric Effect brought him a Nobel Prize for Physics in 1921, greatly overshadowing his vastly more important theories of relativity.

The photon theory opened up a new dilemma.  Though conceptually very different, the electromagnetic theory and photon theory had their respective domains of unquestionable success in explaining well known phenomena – like diffraction, interference, polarization, etc., by the wave theory and Photoelectric Effect, Compton Effect, etc., by the photon theory.   The subsequent discovery of other phenomena like electron diffraction (which is the basis for the revolutionary electron microscope), in which traditional particulate matter shows up a distinctly wave nature under certain conditions, uncovered a deep rooted duality in the very make up of nature, validating both the particle and wave pictures.  This duality is at the heart of the new Quantum Mechanics that was developed later by the efforts of great minds like Heisenberg, Schrodinger, Dirac and a host of others. 

Quantum Properties

Maxwell-Boltzmann statistics is successful in predicting the macroscopic properties of aggregates of particles like atoms or molecules in a gas to which Newton's laws can be applied in principle.  Such particles are distinguishable from each other, at least in principle, the same way that a set of identical looking table tennis balls can be distinguished from each other by leaving an identifying mark on each.  In contrast, two or more identical particles such as electrons, photons or nucleons obeying quantum laws cannot be so distinguished even in principle.  This fundamental distinction between classical and quantum particles first came to light from the path-breaking work of S N Bose as we shall see later and produces radically different results when statistical techniques are applied to the two classes of particles.  We need a quantum mechanical formulation of statistical mechanics applicable to particles that are indistinguishable even in principle. This was contributed by Bose and Einstein for one type of particles and later by Fermi and Dirac for another type.

Spin (intrinsic angular momentum) is a property applicable to both classical and quantum particles, but with different connotations.  The spin associated with a classical particle is very much like the spinning motion of a top or the rotation of the Earth about an axis embedded within it.  The spin of a quantum particle is somewhat analogous to such rotational motion of macroscopic objects, but is quantized just like its energy.  It can only be an integral or half-integral multiple of a fundamental unit which involves, not surprisingly, the Planck constant h.  Also, the classical analogy fails completely when attributing spin to an entity like the electron which is viewed as just a point mass.

We need to make a distinction between two types of quantum particles – photons i.e., quantized packets or particles of energy that are massless and 'material' particles like electrons, nucleons, mesons, etc., that have a finite rest mass.  Photons and the other quantum particles are different in one other major respect.  Photons can be created or destroyed and their number is not conserved, unlike the other quantum particles. 

The Exclusion Principle

As emerged from the work of Planck and Einstein any quantum particle can have only discrete (quantized) energy states and, as may be expected, they tend to occupy the lowest energy states at any given temperature.  In general, the lower the temperature the lower the energy.  According to classical mechanics, the energy is zero at absolute zero temperature. However, this is not so for quantum particles.  Quantum Mechanics leads to the finding that even at absolute zero temperature a quantum particle should have a non-zero value, called the zero point energy.

The state of a particle can be described by assigning various quantum 'numbers' characteristic of the particle in that state; these numbers correspond to the energy, charge, spin, magnetic moment, etc.  It is now opportune to introduce one of the most fundamental principles of quantum behavior, discovered by Wolfgang Pauli.  According to this, no two particles with identical quantum numbers can exist in the same state if they have half-integral spins.  This has come to be known as the Exclusion Principle.  Particles of half integral spin, necessarily obeying this principle, have come to be known as fermions in honor of the great Italian physicist Enrico Fermi who first elucidated their statistical behavior.

In contrast, particles with integral or zero spin, including photons, are excluded by nature from the exclusion principle and any number of such particles can occupy the same state. Such particles have come to be known as bosons in honor of S N Bose. This fundamental distinction between bosons and fermions has far reaching consequences as we shall see later.

The Bose-Einstein Connection

Along with great luminaries like J C Bose, P C Ray, C V Raman, Meghnad Saha and others, Satyendranath Bose belonged to the golden era of early twentieth century Science most of which flourished in pre-independence Bengal, the cradle of Indian science.  Though brilliant in both Physics and Mathematics, he was a multi-faceted personality, with deep rooted interests in literature, arts and music. Even within the domain of Physics he did notable work in both theoretical and experimental areas, a fact that is not reflected adequately in terms of the number of research papers to his credit. He followed with keen interest the revolutionary new developments of contemporary science in Europe, particularly the works of the legendary Einstein for whom he developed great respect and reverence in characteristic Indian intellectual tradition.   Good at both German and French, he translated and published Einstein's original scientific papers in English and this was how most Indian scientists got to know about Einstein's work.

Bose visualized black body radiation as a gas of photons, similar to a gas of classical particles obeying Maxwell-Boltzmann statistics, and tried to apply similar techniques to derive the immensely successful Planck's radiation formula in a manner radically different from what Planck himself had done.   In counting the energy states of the photons Bose stumbled upon the property of indistinguishability in a fortuitous way.  A highly simplistic analogy may help to clarify this. If two objects are labeled A and B, their combinations AB and BA are treated as different entities in the MB formalism.  Apparently Bose made the 'mistake' of treating them as the same, and this is what basically led him to a derivation of the correct formula for the distribution of photon energies, viz., Planck's formula.  Though it was much later that he realized the revolutionary and true meaning of his methodology, Bose's euphoria was understandable.  He had derived Planck's formula bereft of the ad hoc and unsatisfactory nature of Planck's assumptions and based on Einstein's photon concept. 

Bose's persistent efforts to get his discovery published by any reputed science journal proved futile.   Apparently, he was way ahead of his times with an idea few could even understand during those days.  There must have been a lot of skepticism about something like this coming from an unknown native of a country that was equally unknown in the scientific world.  In despair, Bose hit upon an idea almost as daring as the one he was trying to publicize.  He mustered enough courage to communicate his work to the one person he most admired professionally and upon whose work it was partly based – the great Einstein himself, by then an international celebrity and whom Bose always regarded as a great master.  Here is his covering letter, in his own hand:

For those who may find it difficult to decipher the handwriting, here is the text of the historic letter:

Einstein was highly impressed and his response to Bose's plea was as decisive as it was prompt. He translated Bose's paper into German and sent it for publication with his comment:"...Bose's derivation of Planck's formula appears to me to be an important step forward. The method used here gives also the quantum theory of an ideal gas, as I shall show elsewhere". Bose's paper, translated under the title Plancks Gesetz und Lichtquanten-hypothese, was published in the August 1924 issue of the renowned German journal Zeitschrift fur Physik. 

Einstein, the genius that he was, understood the implication of Bose seminal work even better than the originator himself.  While Bose had applied a new statistical technique for the understanding of a photon gas, Einstein realized that it could be generalized and applied even to material particles that had an integral spin; in other words, to all bosons. He published this work a few months later.  The resulting theoretical edifice has come to be known as Bose-Einstein statistics and is one of the cornerstones of all of Quantum Physics, on par with Fermi-Dirac statistics.

One of Bose's great ambitions was realized when he managed to spend two years in France and Germany, meeting and working with some towering personalities of the time, including Marie Curie, Paul Langevin, Louis de Broglie, Lise Meitner, Wolfgang Pauli and Werner Heisenberg.  Of course the most memorable of these was his association in Berlin with the master himself, Albert Einstein. 

Birds of the same feather flocking together...

Nature's exclusion of bosons from Pauli's exclusion principle results in some startling consequences when applied to material particles as we shall now examine.

In view of the fact that helium atoms have zero spin, they also would behave like a photon gas. Bose-Einstein statistics can then be applied to study their behavior. In the case of photons, the total number of particles actually decreases as we decrease the total energy of the system (equivalently, as we lower the temperature). However, if we apply BE statistics to a gas of helium atoms in an enclosure, we must obviously keep the number of atoms fixed as the temperature varies. So the precise formula for the number of atoms in a state of specified energy would be different from what Bose used in the case of photons. As the temperature is lowered, more and more particles pile up on each other and crowd together into the (same) lowest energy state (For the gas of photons, they just disappear from the system). At an extremely low temperature, but one which is still just above absolute zero, the number of particles in the lowest state becomes enormously large.  Practically all the particles should be found at this energy level.  Here we have bosons behaving like birds of the same feather all flocking together!  The state of matter corresponding to such a situation is called a bose condensate.  The prediction of such an extraordinary phase of matter remained largely a theoretical curiosity until the end of the twentieth century. 

The phenomenon of superconductivity dramatically illustrates the differences between systems of quantum mechanical particles that obey BE statistics instead of Fermi-Dirac statistics. At room temperature, electrons, which have one-half spin, are distributed among their possible energy states according to FD statistics. At very low temperatures, the electrons pair up to form spin-zero electron pairs which behave as bosons, and promptly condense into the same ground state. A large energy gap between this ground state and the first excited state ensures that any electrical current is “frozen in.” This causes the current to flow through without resistance; this is one of the defining properties of superconducting materials.

 ...and singing together

To achieve Bose-Einstein condensation, a boson gas must be cooled to considerably less than one millionth of a degree above absolute zero. After some historic experimental breakthroughs by several research teams this was finally achieved in 1995 by two American physicists, Eric Cornell and Carl Weiman. A pure bose condensate of about 2000 rubidium atoms at an unbelievably low temperature of 20 nK, i.e. two billionths of a degree above absolute zero, was achieved.  The atoms were shown to lose their individual identities and behave as though they were a single “super atom” for a full ten seconds. The atoms’ physical properties, such as their motions, became identical to one another.

Another American physicist Wolfgang Ketterle worked independently of Cornell and Wieman, and reported large condensates of sodium atoms. Ketterle was also able to extract a beam of coherent matter (much like a laser beam which is coherent radiation) from the condensate, thus achieving the first ever atom laser. When a gas consisting of uncoordinated atoms turns into a Bose-Einstein condensate, it is very much like various instruments of an orchestra all joining to produce the same tone perfectly synchronously.  Here we have bosons not only behaving like birds that are flocking together, but also singing together...the same note and all in unison!

Postscript

A large number of scientists have received the Nobel Prize for highly significant and path breaking work based on or connected with Bose-Einstein statistics.  The list includes Cornell, Weiman and Ketterley for their experimental work.  Paradoxically, the only name missing from such a list is that of Satyendranath Bose himself.  What an irony that his work was not considered worthy of such recognition! A similar fate befell the other great Bose of Bengal – Jagadish Chandra Bose, for his discovery of radio communication (he should have at least shared the prize that was given to Marconi).  It is of considerable comfort to all Indians that C V Raman didn't share such a fate!

Unfortunately, Nobel prizes are not awarded posthumously.  If I could travel back in time and set right some of the anomalies and injustices in the award of Nobel prizes for scientific achievement, I would give a Nobel Prize in Physics jointly to Bose and Einstein for their historic contributions.  Would I like to take away Einstein's award for the Photoelectric Effect? No, not all!  On the contrary, I would give him a third one for his work on Relativity, the one that he most deserves!