Question on Time Reversal and Degeneracy

[RedX]

Under time reversal T, the momentum operator changes sign but the position operator remains the same. So if you have a Hamiltonian of the form H(X,P)=P^2 + V(X) , then it's invariant under time reversal since momentum is squared. This means H and T commute, so that if a state has eigenvalue E of the operator H, then T operated on the state also has the same eigenvalue E under the operator H. However, since T^2=-1 and not +1, this implies that T operated on the state is not the same state. In other words, whenever the Hamiltonian is of the form P^2+V(X), every state is twice degenerate?

For the electrons, this kind of makes sense because spin isn't specified by the Hamiltonian of the form P^2+V(X), and spin can offer a degeneracy of two in this case. But what about for spinless particles?

Also is it sloppy to say that if the Hamiltonian doesn't depend on time, then it is invariant under time reversal? What if the Hamilotnian were H=P^2+P? Here the Hamiltonian seems to me to not depend on time, but the way time flows?

 

[RedX]

Never mind. Wikipedia explains it:

http://en.wikipedia.org/wiki/T-symmetry#Kramers.27_theorem

So it turns out that T^2 can be +1 or -1, the latter for spin 1/2, the former for spin 0. So there is no degeneracy if T^2=1.

Still, it's a little weird. If you charge conjugate twice, you should get the same state. If you do parity twice, you may or may not get the same state depending on your phase convention? And if you do time reversal twice, you definitely don't get the same state (for a spin 1/2 particle).

I suppose this is not too weird since you have to rotate an electron around two circles to get it back the same. But one would normally expect that if you reversed time twice, nothing would happen, and you'd be back where you were. Same with parity.

 

[denian]

I would like to clarify and provide some insights on the topic of time reversal and degeneracy. First of all, it is important to understand that time reversal is a mathematical operation and does not necessarily have a physical interpretation. It is a tool used in quantum mechanics to understand the behavior of systems under different conditions.

Now, let's address the statement that the Hamiltonian is invariant under time reversal if it is of the form P^2 + V(X). This is not entirely accurate. The Hamiltonian is only invariant under time reversal if it commutes with the time reversal operator T. In the case of P^2 + V(X), T and H commute, which means that T can be applied to the state without changing its energy eigenvalue. However, this does not imply that every state is twice degenerate. Degeneracy is a property of the energy levels and can be affected by other factors such as symmetries and interactions.

Regarding the statement about spinless particles, it is true that spin is not specified by the Hamiltonian P^2 + V(X). However, this does not mean that spinless particles cannot have degeneracy. Degeneracy can arise from other factors such as spatial symmetries or conservation laws.

Lastly, it is not entirely accurate to say that a Hamiltonian is invariant under time reversal if it does not depend on time. The Hamiltonian may depend on other variables such as position or momentum, which can still be affected by time reversal. In the case of H = P^2 + P, the Hamiltonian does depend on time and its behavior under time reversal can be more complex.

In conclusion, time reversal and degeneracy are complex topics and require a thorough understanding of quantum mechanics. It is important to be precise and clear when discussing these concepts to avoid misconceptions.