4 2 0 D O C . 4 2 7 Q U A N T U M T H E O R Y O F I D E A L G A S
equation of state, negative entropy values would have to occur for states that are
realizable in principle. That is why we must reject the classical equation of state in
principle, and regard it as a limiting law, similar to Wien’s radiation formula.
§3. Dimensional Consideration. Method Used in the Following
From (1) it follows that ρ is dimensionless. From this we can draw conclusions
about the structure of the function ρ, if we assume that ρ contains no dimensional
constant other than Planck’s constant h. One then deduces in the familiar way that
ρ must be of the form
, (2)
[8]
where ψ is an unknown universal function of two dimensionless variables. The
function ψ is hereby subject to the condition
, (3)
where it is assumed by definition that
(4)
More cannot be concluded from dimensional considerations. Function ψ of two
variables can, however, without proposing somehow questionable hypotheses, be
defined to the extent that only one function of a single variable remains undeter-
mined. This can be done in two ways independent of each other, by concluding
from the two statements that:
1. The entropy of a gas does not change under infinitely slow adiabatic
compression.
2. In an ideal gas there is a stationary state, in which the sought-after velocity
distribution prevails even in the presence of a conservative static external force
field.
These two statements should be valid, neglecting the effect of collisions between
the molecules. However, owing to this neglect of the collisions in principle, two
nonprovable preconditions are involved that are very natural and whose correctness
is made the more probable in that they both lead to the same result and, in the lim-
iting case of a vanishing influence by quanta, they lead to Maxwell’s distribution.
[p. 20]
ρ ψ¨
L
κT
------ ,

V

---·
-
©
2 /3
h2
--------------------¸ -
© ¹
¨ ¸
¨ ¸
§ ·
=
V
h3
----
-³ρ
N =
pd
1
pd
2
dp3
L
L dL +
³
2π(2m)3 /4L1 /2dL = =
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