Satyendra Nath Bose

(1874-1974)

Quantum Statistics for Photons (1924)

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

*h*^{3}, the cube of Planck's constant. By simply 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, including the famous Planck law.

*ρ*_{ν}dν = (8πhν^{2}/c^{3}) / (e^{ - hν / kT } -1)

Maxwell and Boltzmann had derived their distribution law for material particles by analogy with the Gaussian exponential tail of probability in the theory of errors. The number of gas particles with velocity between v and v + dv is

*N ( v ) dv = (4 / α*^{2} √π) v^{2} e^{ - v2 / α2 } dv.

Max Planck had simply guessed his expression from Wien's law for high frequency radiation

*ρ*_{ν}dν = aν^{3} e^{ - bν/T}.

and from Lord Rayleigh's expression for low-frequency (long-wavelength) radiation.

*ρ*_{ν}dν = ν^{2}T.

Planck simply added the term - 1 in the denominator of Wien's expression *(1 / e*^{ - bν / T}). Planck later acknowledged that his breakthrough was luckily finding a mathematical expression that interpolates between Wien's Law at high frequencies and the Rayleigh-Jeans Law at low frequencies. He called it a *glückliche Interpolationformel*.

Einstein said Planck's work was "monstrous," but obviously in perfect agreement with experiment. How had Planck done it?

All Einstein's *derivations* of the Planck law, including that of 1916-17 (which Bose called "remarkably elegant"), used classical electromagnetic theory to derive the density of radiation as the number of "modes" or "degrees of freedom" of the radiation field,

*ρ*_{ν}dν = (8πν^{2}dν / c^{3}) dE

In 1906, Einstein had criticized Planck's use of this classical expression in deriving his "quantum" radiation law. He called it a contradiction...

"this assumption...contradicts the theoretical basis from which [this expression] was developed."

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,

*p*_{x2} + p_{y2} + p_{z2} = p^{2} ,

Bose just used Einstein's 1917 relation for the momentum of a photon, its energy *hν* divided by its velocity *c*,

*p = hν / c*,

and he wrote

*p*_{x2} + p_{y2} + p_{z2} = h^{2}ν^{2} / c^{2} .

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

*∫ dx dy dz dp*_{x} dp_{y} dp_{z} = 4πV ( hν / c )^{2} ( h dν / c ) = 4π ( h^{3} ν^{2} / c^{3} ) V dν,

which he simply divided by *h*^{3}, multiplied by 2 to account for two polarization degrees of freedom, and he had derived the number of cells belonging to dν,

*ρ*_{ν}dν = (8πν^{2}dν / c^{3}) 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 *p*cells.

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 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 *h*^{3} volume of phase space identified by Bose.

Einstein's 1916 work on transition probabilities predicted the stimulated emission of radiation that brought us lasers (**l**ight **a**mplification by the **s**timulated **e**mission of **r**adiation). 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 work with Bose is often seen as Einstein's last *positive* contribution to quantum physics. Some judge his later 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 in some ways the phenomena of nonlocality (seen as early as 1905 but made clear at the Solvay conference in 1927), and nonseparability, and entanglement (which were introduced by the 1935 Einstein-Podolsky-Rosen paper) are as amazing as anything Einstein ever did.

As we shall see, just like his visions of light quanta and ontological chance that were denied or ignored for so long, the “founders” of quantum mechanics told us to not even try to visualize the “mysteries” produced by Einstein’s last insights. They told us not to try to understand what is happening in “quantum reality.” Despite them, Einstein’s visions of entanglement and “spooky action at a distance” have been confirmed by the latest experiments. He didn’t like them, but he saw them first.

They can only be understood by trying to see what it is that Einstein saw so long ago. Information philosophy will try to illustrate his vision.

They 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.

References

Planck's Law and the Hypothesis of Light Quanta
Normal |

Teacher |

Scholar