Physics 401  Homework #9
1)
Time evolution of expectation values (three points each).
If an observable (A) has a
quantum mechanical operator (
A
ˆ
) which does not depend on time, then the timerate
change of the expectation value of that observable in any arbitrary state is given by
A
H
i
dt
A
d
ˆ
,
ˆ
a) Apply the result shown above to the case where the observable (A) is the
momentum. Let the Hamiltonian for the system be that of a particle moving in an
arbitrary onedimensional potential function V(x) . What famous law of classical physics
does this result correspond to?
b) If the expectation value for an observable is constant in time, we say that that
observable is conserved. Suppose we have a free particle (V(x) = constant). Is momentum
conserved? Is energy conserved?
c)) Suppose we have a nonfree particle (V(x)
≠
constant). Is momentum
conserved? Is energy conserved?
2)
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 Fall '11
 HALL
 Work, Quantum Physics, orbital angular momentum, Finite square, canonical commutation relations

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