Bose Statistics: How a 1924 Paper Changed Counting

Most revolutions in physics come with new equations. Satyendra Nath Bose's came with a new way of counting. In 1924, while a Reader in physics at the University of Dhaka, he derived Planck's radiation law without leaning on classical electrodynamics at all Satyendra Nath Bose, Wikipedia^[1]. The trick was to treat light quanta as truly identical. Our Satyendra Nath Bose sticker sums it up: particle guy, major stats.

How did a paper from Dhaka reach Einstein?

Bose first sent his article to the Philosophical Magazine in Britain, which rejected it; nobody knows why Satyendra Nath Bose, Wikipedia^[1]. He then sent it directly to Albert Einstein. Einstein saw its importance, translated it into German himself and submitted it on Bose's behalf to Zeitschrift für Physik, where it appeared in 1924 under the title “Plancks Gesetz und Lichtquantenhypothese” (Planck's law and the light-quantum hypothesis) Satyendra Nath Bose, Wikipedia^[1] Bose 1924^[2]. The recognition let Bose spend two years working in European laboratories Satyendra Nath Bose, Wikipedia^[1].

What problem did Bose solve?

By 1924 Planck's law had been confirmed for a quarter of a century, but every derivation used a classical step: counting the electromagnetic modes in a cavity as if light were a wave, then adding quantum rules on top Satyendra Nath Bose, Wikipedia^[1]. Bose wanted a derivation that used light quanta from start to finish.

Step 1: divide phase space into cells of size h³

Bose treated light quanta as particles in a six-dimensional phase space of position and momentum, divided into elementary cells of volume h³ Satyendra Nath Bose, Wikipedia^[1]. A quantum of frequency ν has momentum p = hν/c. Counting the cells in a thin shell of momentum, and doubling for the two polarizations of light, gives the number of cells available at frequencies between ν and ν + dν:

A_s = 8πν²V·dν / c³

where A_s is the number of cells in frequency band s, ν is the frequency in Hz, V is the volume in m³ and c is the speed of light in m/s. This is the same mode count that appears in the energy density form of Planck's law, u_ν = (8πhν³/c³)/(exp(hν/k_BT) − 1) Planck's law, Wikipedia^[3], but reached by counting particle states rather than waves.

Step 2: count arrangements without labels

Here is the new idea. Bose did not ask which quantum sits in which cell. He asked only how many quanta sit in each cell. The number of ways to place N_s identical quanta into A_s cells is

W = ∏_s (N_s + A_s − 1)! / (N_s!·(A_s − 1)!)

where the product runs over all frequency bands. Maximizing the logarithm of W at fixed total energy gives Planck's law exactly Bose–Einstein statistics, Wikipedia^[4] Satyendra Nath Bose, Wikipedia^[1].

Why does counting without labels change anything?

Toss two coins. If the coins are distinct, there are four outcomes, HH, HT, TH and TT, and the chance of two heads is one in four. If the coins were identical in the quantum sense, so that HT and TH were the same state, there would be only three outcomes and the chance of two heads would be one in three Satyendra Nath Bose, Wikipedia^[1]. Bose's counting is the second kind. It makes particles more likely to share a state than classical counting predicts Satyendra Nath Bose, Wikipedia^[1]. When Einstein later asked whether he knew he had invented a new kind of statistics, Bose reportedly said he had not realized he was counting differently from Boltzmann Satyendra Nath Bose, Wikipedia^[1].

What did Einstein add?

Einstein extended the method to atoms of an ideal gas in 1924–25 Bose–Einstein statistics, Wikipedia^[4]. The result is the Bose–Einstein distribution, the average number of particles in a state of energy ε_i Bose–Einstein statistics, Wikipedia^[4]:

n̄_i = g_i / (exp((ε_i − μ)/k_BT) − 1)

where g_i is the number of states with energy ε_i, μ is the chemical potential in joules (zero for photons), k_B is the Boltzmann constant and T the temperature in kelvin Bose–Einstein statistics, Wikipedia^[4]. Compared with the classical Boltzmann factor, the −1 in the denominator boosts the occupation of low-energy states.

For atoms, whose number is fixed, the −1 has a striking consequence. Below a critical temperature the excited states can no longer hold all the atoms, and the rest pile into the single lowest state. For an ideal gas of spinless bosons in a box this happens at Bose gas, Wikipedia^[5]

T_c = (h²/(2π m k_B))·(n/ζ(3/2))^2/3

where m is the atomic mass in kg, n is the number density in m⁻³ and ζ(3/2) ≈ 2.612. Our arithmetic for rubidium-87 at n = 10²⁰ m⁻³ gives T_c ≈ 0.40 µK, which is why the condensate needed laser and evaporative cooling.

When was the condensate seen?

On 5 June 1995 Eric Cornell and Carl Wieman produced the first gaseous condensate at JILA in Boulder, in rubidium atoms; later that year Wolfgang Ketterle made one with sodium at MIT Bose–Einstein condensate, Wikipedia^[6]. The three shared the 2001 Nobel Prize in Physics Bose–Einstein condensate, Wikipedia^[6]. The story of that experiment is in our Bose–Einstein condensate explainer and on our Bose–Einstein Condensate 1995 tee and Condensa sticker.

Why is a whole class of particles named after Bose?

Particles with whole-number spin, such as photons, gluons, the W and Z and the Higgs, obey Bose's counting and are called bosons, a name given in his honour Satyendra Nath Bose, Wikipedia^[1]. Composite particles with whole-number spin follow the same rule, which is behind the frictionless flow of superfluid helium Bose–Einstein statistics, Wikipedia^[4]. Particles with half-integer spin obey the opposite rule, at most one per state; see our post on the Pauli exclusion principle. Why spin decides the statistics is the subject of the spin–statistics theorem.

Bose was nominated for the Nobel Prize in Physics by K. Banerjee in 1956, D. S. Kothari in 1959, and S. N. Bagchi and A. K. Dutta in 1962, but never received it Satyendra Nath Bose, Wikipedia^[1].

Quick FAQ

Did Bose make a mistake that turned out right?

A popular story says so. What the sources support is that he counted identical quanta as indistinguishable, and that this is now understood to be correct for bosons Satyendra Nath Bose, Wikipedia^[1] Bose–Einstein statistics, Wikipedia^[4].

Is Bose–Einstein statistics the same as Planck's law?

For photons, with μ = 0, Bose–Einstein statistics gives Planck's law Bose–Einstein statistics, Wikipedia^[4].

Does it apply at room temperature?

Both quantum statistics become classical Maxwell–Boltzmann statistics at high temperature or low density Bose–Einstein statistics, Wikipedia^[4].

References

1. Wikipedia, “Satyendra Nath Bose”. https://en.wikipedia.org/wiki/Satyendra_Nath_Bose
2. S. N. Bose (1924), “Plancks Gesetz und Lichtquantenhypothese,” Zeitschrift für Physik 26, 178–181 (DOI 10.1007/BF01327326; metadata verified via Crossref). https://doi.org/10.1007/BF01327326
3. Wikipedia, “Planck's law”. https://en.wikipedia.org/wiki/Planck%27s_law
4. Wikipedia, “Bose–Einstein statistics”. https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics
5. Wikipedia, “Bose gas”. https://en.wikipedia.org/wiki/Bose_gas
6. Wikipedia, “Bose–Einstein condensate”. https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.