Time reversal symmetry and real symmetric Hamiltonian matrix

Question

In the literature (like those in quantum chaos), it seems that time-reversal symmetry implies that the Hamiltonian of the system is a real symmetric one, instead of just being complex Hermitian.

Is there any rigorous justification about this link?

Answer 1

A time reversal operator is an anti-unitary operator, which can be expressed as:

$\mathcal{T}=UK$

where $K$ denotes complex conjugate and $U$ is a unitary operator. In case of spinless particles, $U$ is chosen to be Identity. Thus $\mathcal{T}=K$.

If the system has $\mathcal{T}$-reversal symmetry:

$$ KH\psi=HK\psi $$

which leads to:

$$ H^*\psi^*=H\psi^* $$ meaning that the Hamiltonian must be real symmetric.