Every earlier derivation of Planck’s law borrowed its prefactor from classical electrodynamics and quantised only the energy. Bose borrowed nothing. He counted cells, counted them for indistinguishable quanta, and the entire law came out — prefactor included. The counting is redone here from scratch, and the identical counting with distinguishable quanta is shown to give Wien’s law instead.
In 1924 Bose was Reader in Physics at the University of Dacca, four years into a department that barely existed. He wrote a four-page derivation of Planck’s radiation law and sent it to Philosophical Magazine, which declined it. On 4 June 1924 he posted it instead to Einstein, in Berlin, with a covering note: “I have ventured to send you the accompanying article for your perusal and opinion.”
Einstein translated it into German himself and placed it in Zeitschrift für Physik 26, 178, where it appeared in August 1924 with a translator’s note in his own name: in his opinion, Bose’s derivation signified an important advance. Einstein then wrote the sequel, carrying the same counting onto massive particles, and got condensation out of it.
The popular account has Bose making a counting mistake during a lecture and arriving at the right answer by luck. Neither source used here supports it, and the paper itself reads as a deliberate argument about indistinguishability rather than a slip. It is left out for that reason, not because it is unflattering.
Everything turns on one combinatorial choice. If the quanta are distinguishable, each of the $N$ of them independently picks a cell, giving $g^{N}$ arrangements. If they are not — if swapping two quanta produces no new arrangement — the count is instead the number of ways to write $N$ as an ordered sum of $g$ non-negative parts:
| N | g | Bose | Boltzmann |
|---|---|---|---|
| 2 | 2 | 3 | 4 |
| 3 | 2 | 4 | 8 |
| 2 | 3 | 6 | 9 |
| 3 | 3 | 10 | 27 |
| 5 | 4 | 56 | 1024 |
These are not small corrections to one another. At $N=g=3$ it is 10 against 27. The two counts agree only when $N=1$, where there is nothing to permute.
Maximising $\log W$ under fixed energy gives the occupancy of a cell at $x = h\nu/kT$. The Bose count gives $\langle n\rangle = 1/(e^{x}-1)$; the Boltzmann count gives $e^{-x}$. Multiply either by the same mode density and the same $h\nu$:
$$u(\nu)\,d\nu = \frac{8\pi h\nu^{3}}{c^{3}}\cdot\frac{1}{e^{x}-1}\,d\nu \quad\text{(Planck)} \qquad \frac{8\pi h\nu^{3}}{c^{3}}\cdot e^{-x}\,d\nu \quad\text{(Wien)}$$ Same prefactor, same energy quantum. Planck’s law and Wien’s law differ by nothing except whether the quanta are counted as distinguishable.
And Bose did not take the $8\pi\nu^{2}/c^{3}$ from Maxwell’s equations, which is the part that matters for attribution. He obtained it by counting phase-space cells of volume $h^{3}$, with a factor 2 for polarisation. The derivation is closed: no classical electrodynamics enters it anywhere.
Expanding at small $x$: $1/(e^{x}-1) = 1/x - 1/2 + O(x)$, so $u \to 8\pi\nu^{2}kT/c^{3}$ — Rayleigh–Jeans, including its ultraviolet divergence. At large $x$ the $-1$ stops mattering and the formula becomes Wien’s. Wien’s law is the dilute corner of Planck’s, not a rival to it: at $x=20$ the two agree to better than $10^{-8}$, and at $x=0.1$ the Bose occupancy is more than nine times the Boltzmann one.
Integrating over all frequencies needs $\int_{0}^{\infty} x^{3}/(e^{x}-1)\,dx$, which is $\Gamma(4)\,\zeta(4)$. Both factors are exact:
So $u_{\text{total}} \propto T^{4}$. The fourth power is a consequence of the counting and of $\zeta(4)=\pi^{4}/90$ — Euler’s, and derived in ch-euler — not an extra physical assumption. The displacement law is the root of $3(1-e^{-x})=x$, which is $x^{*} = 2.8214393721$ to ten places.
Applied to massive particles, the same counting produces a threshold at $n\lambda^{3} = \zeta(3/2) = 2.6123753\ldots$ — again a pure number out of the combinatorics, with no new physics inserted. Below that temperature a macroscopic fraction of the particles occupies one state. It took until 1995 to make one in a laboratory, and the 2001 Nobel Prize in Physics went to Cornell, Ketterle and Wieman for doing so.
Why quanta are indistinguishable is not derived on this page and was not derived by Bose. He assumed it and the answer came out right, which is evidence and not a proof. The modern account — that the state space is a symmetrised tensor product, and that the choice between symmetric and antisymmetric is tied to spin by the spin–statistics theorem — came later and needs relativistic quantum field theory. Nothing in the counting above explains why photons take the symmetric option and electrons do not.
The maximisation step in Part II is also stated rather than performed here: the script checks the resulting occupancies and their consequences, not the Lagrange-multiplier argument that produces them. And the chapter makes no claim about the prefactor beyond Bose’s own — that it can be had by cell-counting, which is checkable, rather than that cell-counting is the only route to it.
Bose 1924 S. N. Bose, “Plancks Gesetz und Lichtquantenhypothese”, Zeitschrift für Physik 26, 178 (August 1924). Sent to Einstein 4 June 1924 after Philosophical Magazine declined it; translated into German by Einstein, who appended a note calling the derivation an important advance.
Einstein 1924–25 A. Einstein, “Quantentheorie des einatomigen idealen Gases” — the same counting carried onto massive particles, and the condensation prediction.
Ghose P. Ghose, “Counting in the Dark”, Inference — on what was actually new in the derivation, in particular the prefactor.
in-corpus ch-euler ($\zeta(4)=\pi^{4}/90$, derived) · ch-chandrasekhar (the other statistics, and what it holds up) · ch-dirac · ch-boltzmann (the same counting move applied to boiling water)
verification book7/ch-bose-verify.py —
eight blocks, standard library only. Block [8] is a control that fails on purpose
to show the failure path works. Every number on this page is printed by it.
No priority is claimed for anything on this page. The combinatorics, the two occupancies, Planck’s law, the Stefan–Boltzmann integral, the Wien root and $\zeta(3/2)$ are all classical. They are derived here rather than looked up, which is this corpus’s practice, and that is a statement about method and not about priority.