3 7 8 D O C . 3 8 5 Q U A N T U M T H E O R Y O F I D E A L G A S I I

How a material particle, or a system of material particles, can be assigned a (sca-

lar) field of waves has been demonstrated by Mr. E. de Broglie in a very noteworthy

paper.*)

A material particle of mass m is first assigned a frequency ν according to

the equation

(35)

Now let the particle be at rest with reference to a Galilean system K′, in which we

imagine a synchronous oscillation of the frequency ν0 everywhere. Relative to a

system K, with reference to which K′ is moving with mass m at the velocity v along

the (positive) X axis, there then exists a wavelike process of the type

Frequency ν and phase velocity V of this process are hence given by

(36)

(37)

ν is then—as Mr. De Broglie has shown—at the same time the group velocity of

this wave. It is furthermore interesting that, according to (35) and (36), the parti-

cle’s energy is exactly equal to hν, in concordance with the fundamental

relation of quantum theory.

One then sees that to such a gas a scalar wave field can be assigned, and I have

become persuaded by calculation that [14] is the mean square of the fluctuation

*)

Louis de Broglie. Thèses. Paris (Edit. Musson & Co.), 1924. There is a very remarkable

geometrical interpretation of the Bohr-Sommerfeld quantum rule in this dissertation as

well.[13]

mc2

hν0. =

2πν0------------------¸

t

v

c2

----x - –

1

v2¸

c2

----¹ -–

©

¨

¨ ¸

§ ·

. sin¨

ν

ν0

1

v2

c2

---- -–

------------------ =

V

c2

v

---- -=

[p. 10]

mc2

1

v2

c2

---- -–

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

1

Zν

-----