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
then (using Tp =- p)
One can easily see that the eigenvalue equation
has no solutions for any complex eigenvalue l= reiq.
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
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:
Under time reversal we get
and applying parity we have
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.