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