NBS Monograph 115: 2.11. The Time Inversion Operation Theta

2.11. The Time Inversion Operation θ

The time inversion operation θ [17] (pp. 325-348) is of interest here principally in connection with intensity calculations. The intensity of absorption or emission of light by diatomic molecules depends on matrix elements of the components of the dipole moment operator. These matrix elements are usually not calculated explicitly, but are rather treated as parameters, to be determined from a fit to experimental data. It sometimes happens that several such parameters occur in the intensity expressions, which then involve, for example, the squares of sums and differences of these parameters. It is clearly desirable to know which of the parameters are real and which are complex, since the arithmetic of real numbers is not identical to the arithmetic of complex numbers. Time inversion is a useful tool, since it is essentially the operation of taking complex conjugates. Indeed,

(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 dx/dt or a momentum m(dx/dt) should transform into its negative. In quantum mechanics momenta are represented by operators of the form - i h bar (d/dx), which do not contain the time variable at all. However, in contrast to position coordinates, they do contain the pure imaginary number 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).)

eq 2.28 (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

eq 2.30 (2.30)

When the system being considered contains an even number of electrons, θ² = +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

eq 2.31 (2.31)

When the system being considered contains an odd number of electrons, θ² = -1 [17] (p. 332). Under these circumstances. 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

eq 2.32 (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

eq 2.33 (2.33)

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

eq 2.34 (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 $|\Omega JM\rangle$ when J is half-integral, except that both laboratory-fixed (J_X ± iJ_Y) and molecule-fixed (J_x minus plus iJ_y) ladder operators must be used. One of these equations takes the form

eq 2.36 (2.36)

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 (J_x minus plus iJ_y)^m and (J_X ± iJ_Y)ⁿ can be shown to be equivalent to

(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

eq 2.38 (2.38)

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