Seminary of the Southwest Quantum Mechanic Practice Problems

2. If y happens to be an eigenfunction of an operator A with the eigenvalue a, evaluatethe expectation value (A).
5. Consider a quantum particle of mass m that is completely free to travel in one-
dimension, V(x) = 0.
(a) Write out the full expression for the time independent Schrodinger
equation.
(b) Consider the two cases where A=0 and then where B=0. Determine if these
wavefunctions are (separately) eigenfunctions of the momentum operator and,
if so, what the eigenvalues are.
(c) Are there any restrictions on the total energy for this particle?
11. Write the normalized wavefunction for a particle in two-dimensional box of sides a
and b. The momentum operator in two-dimensions is
P =-in (i+1)=)
=
a2
дх2
a2
j
ду2.
Calculate the uncertainty in momentum for this operator.
12. Two operators, A and B commute. Prove that if |4> is an eigenfunction of A with
eigenvalue a, then B|Y> is also an eigenfunction of A with eigenvalue a.
8. The Schrodinger equation for a particle of mass m constrained to move on a circle
of radius a is
h2d2W(0)
21 dez
ΕΨ(8) oses 2
Where I = ma? is the moment of inertia and e is the angle that describes the position of
the particle around the ring. Show by direct substitution that the solutions to this
equation are
(O) = Aeine
Where n = 1(218)1/2
Argue that the appropriate boundary conditions is Y() = Y(8 +
21) and use this condition to show that
n?h?
E =
21
n = 0, +1, +2, +3