PHY357S: Lecture 11

Time Reversal (tÆ-t)

A wavefunction cannot be in an eigenstate of T, because T is an antiunitary operator which changes functions to their complex conjugate. e.g. if

y = e^{i(^p·x^-Et)}

then (using Tp =- p)

Ty = e^{i(-^p·x^+Et)}

= y*

One can easily see that the eigenvalue equation

Ty = ly

has no solutions for any complex eigenvalue l= re^iq.

This does not mean T may not be a symmetry of the Hamiltonian. It just means the wavefunctions cannot be eigenstates.

Probabilities and expectation values are unchanged if T is a symmetry. e.g. For a free particle wave function

T(y*y) = y y* = y*y

This may not be true for interactions if complex matrices are involved.

Time reversal can be tested in several ways, for example, consider a two body reaction involving initial state spinless particles a & b, and final state spinless particles c & d:

a(p_a) + b(p_b) Æ c(p_c) + d(p_d)

Under time reversal we get

c(-p_c) + d(-p_d) Æ a(-p_a) + b(-p_b)

and applying parity we have

c(p_c) + d(p_d) Æ a(p_a) + b(p_b)

So PT symmetry would imply that the rate for the reaction a+b Æc+d should be the same as the rate for the reaction c+d Æa+b. This is the principal of detailed balance.