Question about Time-Reversal Symmetry and Crystal Field Hamiltonians I have a bit of a confused question, hopefully I can refine it further as people comment. Spatial symmetry is readily apparent in a crystal-field Hamiltonian, that is, one can just examine the potential and see the spatial symmetries. I'm a bit confused about how to check if the system has time-reversal symmetry, particularly in the case of integer J (even number of electrons). Does one just check to see if the Hamiltonian commutes with the time-reversal operator? Does time-reversal necessarily imply that all $\pm$Jz pairs have degenerate eigenvalues?
 A: Now the confusing part about time reversal operator is how to define it. because it is $T=UK$, and $U$ changes from basis to basis, so if I give you a random hamiltonian matrix it would be very hard to determine the U. 
But there is a theorem if your system is non interacting, then there are only two anti unitary symmetry operators you can define, because if you define any other you can write these anti unitary operators as a product of a unitary symmetry and your original anti unitary symmetry.
but here is the interesting part, the irreducible block diagonal representation of your hamiltonian is independent of your unitary symmetry transformations,[for example angular momentum parts of the hydrogen model comes from O(3) symmetry but the energy part is independent from these symmetry and the irreps of your hamiltonian is just the energy parts].
Hence once you find two anti unitary operators one is commuting and other is anti commuting with your hamiltonian, then former is your time reversal and latter is charge conjugation operator. and the other anti unitary operators are irrelevant since they are just product of the original ones and unitary symmetries, since the unitary symmetries are irrelevant , they are irrelevant.
please see: arXiv:1512.08882v1
