Time reversal symmetry and Bloch states

[Joker93]

Hello!

The time reversal operator, $\hat{\Theta}$ transforms a Bloch state as follows:
$\hat{\Theta} \psi_{nk}=\psi^*_{nk}$.
How does one proceed to prove the condition that $\psi_{nk}$ and $\psi^*_{nk}$ must satisfy in order for our system to be time reversal invariant?

Thanks in advance!

 

[DrDu]

First, i would like to point out that your expression is only correct for scalar functions, but not for spinor valued functions, i.e. electrons.
A Bloch function is proportional to exp(ikr) which becomes exp(-ikr) on time reversal. Hence time reversal converts a state with k into one with -k.
In the scalar case, it will be possible to write down an invariant wavefunction if the wavefunctions with k and -k are energetically degenerate. As you then can form $\psi_k+\psi_{-k}$ and $i\psi_k-i\psi_{-k}$.