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