(2.27) |

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
[17] (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 d*x*/d*t* or a momentum
*m*(d*x*/d*t*) should transform into its negative. In quantum
mechanics momenta are represented by operators of the form
*i*(d/d*x*),*i*. It is thus
convenient in quantum mechanics to construct a formalism in which time
inversion corresponds to the taking of complex conjugates rather than the
replacing of *t* by *t*.

Arguments such as this make the following transformation equations for angular momentum operators seem reasonable. (They are also correct [17] (pp. 329-330).)

(2.28) |

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,

(2.29) |

We thus conclude

(2.30) |

When the system being considered contains an even number of electrons,
θ^{2} = +1 [17] (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

(2.31) |

When the system being considered contains an odd number of electrons,
^{2} = -1*S* and *J* are
half-integers, so that a value of zero for Σ, Ω, and *M* is
not possible. There are then no spin functions and no rotational functions
which remain unchanged by the time inversion operation [see
(2.30)]. However, it is possible to show that one can
always choose phases consistent with those of Condon and Shortley
[7], such that

(2.32) |

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

(2.33) |

where *m* and *n* are positive integers and *k*_{1} and
*k*_{2} 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

(2.34) |

Equations (2.34) can easily be shown to be equivalent to

(2.35) |

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
*J _{X}* ±

(2.36) |

where *m* and *n* are positive integers and *k*_{3} is a
positive constant. This equation and the three analogous equations obtained by
using different combinations of (*J _{x}*

(2.37) |

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

(2.38) |

Since an expression of the form ⟨*a*|*a*⟩ must always equal
a real, positive number, we conclude that if *a*⟩ =
e^{ia} |*a*′⟩*a*| =
e^{-ia} ⟨*a*′|.

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.