A solved problem for time reversal symmetry

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Source: Problem 4-10,'Modern Quantum Mechanics'; J.J. Sakurai

problem: suppose a spinless particle is bound to a fixed center by a potential V(x) so asymmetrical that no energy level is degenerate. Using time-reversal invariance prove <L>=o

for any energy eigenstate. (This is known as quenching of orbital angular momentum.) If the wave function of such a non degenerate eigenstate is expanded as

what kind of phase restriction do we obtain on ?

Solution: Under reversal time and , [H,K]=0 Assume where is the eigenket of Hamiltonian. So, is also a eigenket of H with the same eigenvalue.

If we do not have any degeneracy, so

that is a real number.

Also

with comparison two different form of function, we get