68 
QUANTUM THEORY 
When there is perfect quantisation we can divide the whole domain of the 
coordinates into unit cells such that each unit cell contains just one 
quantum orbit. This may be done by taking Sg x to correspond to a complete 
cycle of q x and Sp x to correspond to an increase of 1 in the associated 
quantum number (preferably chosen so that the integral number corre 
sponds to the middle of the cell). Thus 
We have already made the hypothesis that each quantum orbit has 
the same (unit) weight. We shall now regard this as a particular case of 
the more general hypothesis that each cell of volume h 3 has equal (unit) 
weight; so that for large cells 
When there is no quantisation (as in non-periodic motion) the states and 
the weight are spread through the cell. When there is imperfect periodicity 
the weight concentrates towards the quantum orbits in it. For perfect 
periodicity it is w'holly concentrated in the quantum orbits. 
For an electron moving in an electric field the rectangular coordinates 
x, y, z and the associated momenta mu, mv, mw satisfy Hamilton’s 
equations and may be taken as the variables q x ... p 3 . Thus by (48T) 
This is the same as (45-6) but it is not restricted to the particular type of 
system there considered. In consequence of our more general assumption 
(48-4) applies generally both to bound and to free electrons, except that 
where there is periodicity the cell must be large enough to average out the 
ridginess in the distribution of weight induced by the quantisation. 
49. It is necessary to show that the volume of a cell is invariant, that 
is to say, that it does not depend on the particular choice of coordinates 
q x ... p 3 , provided that they satisfy Hamilton’s equations. If it were not 
invariant the weights based on it would be ambiguous, and so also would 
be the quantisation. For readers familiar with the tensor calculus the 
following proof is probably the simplest. 
We write g 4 , g 5 , g 6 for p x , p 2 , p 3 so that Hamilton’s equations (42-3) 
= (% + i ) h - K - h) h > b y ( 42>4 ) 
= h. 
Hence for a unit cell 
V = h* 
(48-2). 
q = V/h* 
(48-3). 
and hence 
(48-4). 
become 
dq x dH dq t dll