The Identity of Elementary Particles and Einstein’s Discovery of Quantum Statistics

In 1924, Albert Einstein received an amazing very short paper from India by Satyendra Nath Bose. Einstein must have been pleased to read the title, “Planck’s Law and the Hypothesis of Light Quanta.” It was more attention to Einstein’s 1905 work than anyone had paid in nearly twenty years. The paper began by claiming that the “phase space” (a combination of 3-dimensional coordinate space and 3-dimensional momentum space) should be divided into small volumes of h3, the cube of Planck’s constant. By counting the number of possible distributions of light quanta over these cells, Bose claimed he could calculate the entropy and all other thermodynamic properties of the radiation.

Bose easily derived the inverse exponential function. Einstein too had derived this. Maxwell and Boltzmann derived it, without the additional -1, by analogy from the Gaussian exponential tail of probability and the theory of errors.

1 / (e – hν / kT -1)

(Planck had simply guessed this expression from Wien’s law by adding the term – 1 in the denominator of Wien’s a / e – bν / T).

All previous derivations of the Planck law, including Einstein’s of 1916-17 (which Bose called “remarkably elegant”), used classical electromagnetic theory to derive the density of radiation, the number of “modes” or “degrees of freedom” of the radiation field,

ρνdν = (8πν2dν / c3) E

But now Bose showed he could get this quantity with a simple statistical mechanical argument remarkably like that Maxwell used to derive his distribution of molecular velocities. Where Maxwell said that the three directions of velocities for particles are independent of one another, but of course equal to the total momentum,

px2 + py2 + pz2 = p2 ,

Bose just used Einstein’s relation for the momentum of a photon,

p = hν / c,

and he wrote

px2 + py2 + pz2 = h2ν2 / c2 .

This led him to calculate a frequency interval in phase space as

 dx dy dz dpx dpy dpz = 4πV ( hν / c )3 ( h dν / c ) = 4π ( h3 ν2 / c3 ) V dν,

which he simply divided by h3, multiplied by 2 to account for two polarization degrees of freedom, and he had derived the number of cells belonging to dν,

ρνdν = (8πν2dν / c3) E ,

without using classical radiation laws, a correspondence principle, or even Wien’s law. His derivation was purely statistical mechanical, based only on the number of cells in phase space and the number of ways N photons can be distributed among them.

Einstein immediately translated the Bose paper into German and had it published in Zeitschrift für Physik, without even telling Bose. More importantly, Einstein then went on to discuss a new quantum statistics that predicted low-temperature condensation of any particles with integer values of the spin. So called Bose-Einstein statistics were quickly shown by Dirac to lead to the quantum statistics of half-integer spin particles now called Fermi-Dirac statistics. Fermions are half-integer spin particles that obey Pauli’s exclusion principle so a maximum of two particles, with opposite spins, can be found in the fundamental h3 volume of phase space identified by Bose.

Einstein’s 1916 work on transition probabilities predicted the stimulated emission of radiation that brought us lasers (light amplification by the stimulated emission of radiation). Now his work on quantum statistics brought us the Bose-Einstein condensation. Either work would have made their discoverer a giant in physics, but these are more often attributed to Bose, just as Einstein’s quantum discoveries before the Copenhagen Interpretation are mostly forgotten by historians and today’s textbooks, or attributed to others.

This may have been Einstein’s last positive contribution to quantum physics. Some judge his next efforts as purely negative attempts to discredit quantum mechanics, by graphically illustrating quantum phenomena that seem logically impossible or at least in violation of fundamental theories like his relativity. But information philosophy hopes to provide explanations for Einstein’s paradoxes that depend on the immaterial nature of information.

The phenomena of nonlocalitynonseparability, and entanglement may not be made intuitive by our explanations, but they can be made understandable. And they can be visualized in a way that Einstein and Schrödinger might have liked, even if they would still find the phenomena impossible to believe. We hope even the layperson will see our animations as providing them an understanding of what quantum mechanics is doing in the microscopic world. The animations present standard quantum physics as Einstein saw it, though Schrödinger never accepted the “collapse” of the wave function and the existence of particles.