XVII. I] THE NEW QUANTUM MECHANICS 
225 
alone should enter, Heisenberg took as his starting point the 
frequency relation of Kramers’ dispersion theory. 
The chief problem to be solved by the quantum theory 
consists in determining the frequencies and amplitudes of the 
emitted radiation and the energy levels of the atomic system. 
It is known that the frequencies of spectral lines can be expressed 
according to the quantum theory in the form : 
v[n, n 
a) = =^{W(n) - W(n 
a)} 
17:1 
where the expression v(n, n — a) implies that the frequency is 
a function of the two quantum numbers, n and n — a. 
In the classical theory the corresponding expression may be 
written 
v[n, a) — a . v(n) — a 
1 dW 
h dn 
17: 2 
The fact that we have a certain difference in the first formulation 
and a differential in the second is of special significance, and 
suggests that the same modification of classical results may be 
necessary in other instances. 
Again, classical frequencies conform to the combination 
relation 
v{n, a) + v(n, ft) = v(n, a + (5) . . 17 : 3 
while on the quantum theory the relation is assumed as 
v(n, n — a) + v(n — a, n — a — ft) — v{n, n — a — fi) 17 : 4 
In addition to the frequencies it is necessary for the description 
of the radiation to know the amplitudes. 
The amplitudes may be denoted by complex three-dimensional 
vectors A, which define intensity, polarization and phase. 
Heisenberg considers some assigned quantity x{t) represented by 
the totality of terms A(n, n — a)e iio{ - n • M ~ a)i , this form being 
suggested by the expression for the co-ordinates of any particle 
in a classical multiply-periodic system.* In the new theory he 
writes : 
GO 
and proceeds to inquire what will be the value of x 2 (t). On the 
quantum theory he considers the simplest and most natural 
assumption to be 
0 = + co 
x 2 (t) = £B(n, n - fi)e iM[n > . . . 17 : 6 
/3 = — 00 
* For a clear account of such systems see J. H. van Vleck, Quantum 
Principles and Line Spectra (1926).