if k is a constant or a function of the positional coordinates of particles. However, because of the rather special nature of spin variables, time inversion, when applied to spin functions, is somewhat more complicated  (pp. 331-333).
Physically, time inversion would be expected to correspond to a transformation
of variables in which the time t is replaced by - t. Thus,
for example, a position coordinate x should remain invariant under time
inversion, while a velocity dx/dt or a momentum
m(dx/dt) should transform into its negative. In quantum
mechanics momenta are represented by operators of the form
Arguments such as this make the following transformation equations for angular momentum operators seem reasonable. (They are also correct  (pp. 329-330).)
From these relations it is possible to show that when time inversion is applied to wave functions, the signs of all angular momentum projection quantum numbers are reversed. For example,
We thus conclude
When the system being considered contains an even number of electrons, θ2 = +1  (p. 332). Under these circumstances, it happens that L, S, and J are all whole numbers, so that zero is a possible value for each of the projection quantum numbers Λ, Σ, Ω, and M. It is relatively easy to show that the phase factor of the wave function having a projection quantum number equal to zero can be chosen such that the function is unchanged when the time inversion operation is carried out. In other words, it is always possible to choose phases such that
When the system being considered contains an odd number of electrons,
The effect of time inversion on all other functions can be obtained as follows. Our choice of phases for the ladder operators S± in eqs (1.13) implies that
where m and n are positive integers and k1 and k2 are positive constants. The transformation properties of components of the angular momenta given in (2.28) and eqs (2.32) and (2.33) allow us to write
Equations (2.34) can easily be shown to be equivalent to
for all values of Σ allowed for given half-integral S.
A set of four equations similar to (2.34) holds for the functions when
J is half-integral, except that both laboratory-fixed
where m and n are positive integers and k3 is a positive constant. This equation and the three analogous equations obtained by using different combinations of (Jx iJy)m and (JX ± iJY)n can be shown to be equivalent to
for all values of Ω and M allowed for given half-integral J.
For integral values of L, S, and J we find, by using (2.31) and equations similar to (2.34) and (2.36), that
Since an expression of the form 〈a|a〉 must always equal
a real, positive number, we conclude that if
We consider an example of the use of time inversion in determining which matrix elements of the dipole moment operator are real and which are not in sect. 3.5.