76 
QUANTUM THEORY 
to j 8 m, and its whole energy accordingly changes from me 2 to £ me 2 . Equat 
ing this to the change of energy of the radiation 
E x — E 2 = (fi — 1 ) me 2 . 
Conservation of momentum gives 
(E l — E 2 + me 2 ) 2 — (E x — E 2 cos 9) 2 — E 2 2 sin 2 9 = (¡3 2 mc) 2 (c 2 — u 2 — v 2 ) = m 2 c 4 
which reduces to 
so that, if A x , A 2 are the wave-lengths of the incident and scattered quanta, 
(52*2) gives 
The constant can be interpreted as the wave-length of a quantum contain 
ing as much energy as goes to constitute an electron. 
Radiation scattered by electrons at rest is thus of longer wave-length 
by an amount dependent on the angle of scattering but not on the incident 
wave-length. This reddening is known as the Compton Effect. It is not 
possible to obtain any considerable quantity of free electrons to experi 
ment on in the laboratory; but it appears that when X rays of high fre 
quency are used the more weakly bound electrons of ordinary matter can 
be regarded as sufficiently free to scatter according to this formula. It 
has thus been possible to prove it experimentally. 
The corresponding formula for scattering by a moving electron can 
be obtained by applying the Lorentz transformation. The wave-length is 
not always increased. If the electrons are moving with speeds such as to 
be in thermodynamical equilibrium with the radiation there must be just 
as much shortening of wave-lengths as lengthening. A proof of Planck’s 
Law can be obtained by following up this condition in detail*. 
53. To find the amount of scattering, by free electrons, we consider 
first the classical formulae. 
E x E 2 0 
— cos 9 = Bm.u, 
c c 
sin 9 = Bm . v. 
c 
Hence 
Now 
— 2 E X E 2 (1 — cos 9) + 2 mc 2 (E x — E 2 ) = 0 (52-2). 
E/c = hv/c = h/X, 
(52-31) 
or inserting numerical values 
A 2 — A x = 0-0242 Â x (1 — cos 6) 
(52-32). 
* Pauli, Zeits. für Physik, 18, p. 272.