D O C U M E N T 4 3 3 F E B R U A R Y 1 9 2 5 4 2 9

433. From Erwin Schrödinger

Zurich, 9 Hutten Street, 5 February 1925

Highly esteemed Professor,

I just read your interesting paper on the degeneration of

gases[1]

and stumbled

across a serious misgiving. May I explain it to you?

I was taken aback by the fact that the distribution (11), p. 263, does not exactly

agree with the Boltzmann distribution, which ever since your famous radiation pa-

per of 1917 tends to be regarded as exactly valid in quantum theory as

well.[2]

I will

now consider the first eq. (9). If I understand it correctly, is the probability for

the sth cell to accommodate just r molecules. Namely, this really is valid for that

particular determined by the maximum condition, because noticeably differing

distributions are, of course, vanishingly improbable. Now, if ns is the average num-

ber of molecules per cell “in the region s,” then, according to a familiar formula,

the probability of a molecule number r is

.

If I compare this with the first equation (9), (in that I initially only direct my fo-

cus on the dependence of r), then (9) indicates the exponential dependence correct-

ly, but the r! is missing in the denominator. This is then also connected with the fact

that factor

βs

is not , but (approximately) 1–ns , the approximate value for

at low ns. It seems to me that the lack of 1/r! is not legitimate, but evidently the re-

sult of the Stirling approximations on the previous page. The formula is, as it were,

only right for r = 1; the cases where more molecules belong in a cell are in fact ne-

glected. Formula (4) also leaves the same impression, purely intuitively. If one sup-

plements the denominator r! in (9), then one obtains instead of (11):

and instead of (10):

.

Thus the contradiction to Boltzmann’s principle disappears and, I believe, also all

the deviations from the ideal gas, which essentially depend on this –1 in the denom-

inator of equation (11).

pr s

pr s

ns)re–n-

(

s

r!

--------------------

e–ns e–ns

ns r

r

¦rβse–αs

d

ee

–αs)

–βsdαs(

βse–αsee–αs

= = =

1

e–αsr

r!

-----------

r

¦βs

βsee

–αs;

with ns

e–αs

= = =