Problems 391a. Determine the eigenvalues and eigenvectors ofHˆ.b. Suppose the system is prepared with an initial state vector|Ψ(0)〉=(
1
0
)
.
Determine the state vector|Ψ(t)〉at timet.c. Determine the expectation values of the operatorsSˆx,Sˆy, andSˆzin the
time-dependent state computed in part b.d. Suppose, instead, the system is prepared with an initial state vector|Ψ(0)〉=
1
√
2
(
1
1
)
.
Determine the state vector|Ψ(t)〉at timet.e. Using the time-dependent state computed in part d, determine the fol-
lowing expectation values:〈Ψ(t)|Sˆx|Ψ(t)〉,〈Ψ(t)|Sˆy|Ψ(t)〉,〈Ψ(t)|Sˆz|Ψ(t)〉.f. For the time-dependent state in part d, determine the uncertainties ∆Sx,
∆Sy, and ∆Sz, where ∆Sα =√
〈Ψ(t)|Sˆα^2 |Ψ(t)〉−〈Ψ(t)|Sˆα|Ψ(t)〉^2 , for
α=x,y,z.9.3. Consider a free particle in a one-dimensional box that extends fromx=
−L/2 tox=L/2. Assuming periodic boundary conditions, determine the
eigenvalues and eigenfunctions of the Hamiltonian for this problem. Repeat
for infinite walls atx=−L/2 andx=L/2.
9.4. A rigid homonuclear diatomic molecule rotates in thexyplane about an axis
through its center-of-mass. Letmbe the mass of each atom in the molecule,
and letRbe its bond length. Show that the molecule has a discrete set of
energy levels (energy eigenvalues) and determine the corresponding eigen-
functions.
9.5. Given only the commutator relation between ˆxand ˆp, [ˆx,pˆ] =i ̄hIˆand the
fact that ˆp→−i ̄h(d/dx) when projected into the coordinate basis, show that
the inner product relation
〈x|p〉=1
√
2 π ̄heipx/ ̄hfollows.9.6. Using raising and lowering operators, calculate the expectationvalue〈n|ˆx^4 |n〉
and general matrix element〈n′|xˆ^4 |n〉for a one-dimensional harmonic oscilla-
tor.