math and physics play

Angular momentum expectation values

December 14, 2015 phy1520 ,

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Q: [1] pr 3.18

Compute the expectation values for the first and second powers of the angular momentum operators with respect to states \( \ket{lm} \).

A:

We can write the expectation values for the \( L_z \) powers immediately

\begin{equation}\label{eqn:angularMomentumExpectation:20}
\expectation{L_z}
= m \Hbar,
\end{equation}

and

\begin{equation}\label{eqn:angularMomentumExpectation:40}
\expectation{L_z^2} = (m \Hbar)^2.
\end{equation}

For the x and y components first express the operators in terms of the ladder operators.

\begin{equation}\label{eqn:angularMomentumExpectation:60}
\begin{aligned}
L_{+} &= L_x + i L_y \\
L_{-} &= L_x – i L_y.
\end{aligned}
\end{equation}

Rearranging gives

\begin{equation}\label{eqn:angularMomentumExpectation:80}
\begin{aligned}
L_x &= \inv{2} \lr{ L_{+} + L_{-} } \\
L_y &= \inv{2i} \lr{ L_{+} – L_{-} }.
\end{aligned}
\end{equation}

The first order expectations \( \expectation{L_x}, \expectation{L_y} \) are both zero since \( \expectation{L_{+}} = \expectation{L_{-}} \). For the second order expectation values we have

\begin{equation}\label{eqn:angularMomentumExpectation:100}
\begin{aligned}
L_x^2
&= \inv{4} \lr{ L_{+} + L_{-} } \lr{ L_{+} + L_{-} } \\
&= \inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} + L_{+} L_{-} + L_{-} L_{+} } \\
&= \inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} + 2 (L_x^2 + L_y^2) } \\
&= \inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} + 2 (\BL^2 – L_z^2) },
\end{aligned}
\end{equation}

and
\begin{equation}\label{eqn:angularMomentumExpectation:120}
\begin{aligned}
L_y^2
&= -\inv{4} \lr{ L_{+} – L_{-} } \lr{ L_{+} – L_{-} } \\
&= -\inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} – L_{+} L_{-} – L_{-} L_{+} } \\
&= -\inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} – 2 (L_x^2 + L_y^2) } \\
&= -\inv{4} \lr{ L_{+} L_{+} + L_{-} L_{-} – 2 (\BL^2 – L_z^2) }.
\end{aligned}
\end{equation}

Any expectation value \( \bra{lm} L_{+} L_{+} \ket{lm} \) or \( \bra{lm} L_{-} L_{-} \ket{lm} \) will be zero, leaving

\begin{equation}\label{eqn:angularMomentumExpectation:140}
\begin{aligned}
\expectation{L_x^2}
&=
\expectation{L_y^2} \\
&=
\inv{4} \expectation{2 (\BL^2 – L_z^2) } \\
&=
\inv{2} \lr{ \Hbar^2 l(l+1) – (\Hbar m)^2 }.
\end{aligned}
\end{equation}

Observe that we have
\begin{equation}\label{eqn:angularMomentumExpectation:160}
\expectation{L_x^2}
+
\expectation{L_y^2}
+
\expectation{L_z^2}
=
\Hbar^2 l(l+1)
=
\expectation{\BL^2},
\end{equation}

which is the quantum mechanical analogue of the classical scalar equation \( \BL^2 = L_x^2 + L_y^2 + L_z^2 \).

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.

Expectation of spherically symmetric 3D potential derivative

December 14, 2015 phy1520 , , , , , ,

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Q: [1] pr 5.16

For a particle in a spherically symmetric potential \( V(r) \) show that

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:20}
\Abs{\psi(0)}^2 = \frac{m}{2 \pi \Hbar^2} \expectation{ \frac{dV}{dr} },
\end{equation}

for all s-states, ground or excited.

Then show this is the case for the 3D SHO and hydrogen wave functions.

A:

The text works a problem that looks similar to this by considering the commutator of an operator \( A \), later set to \( A = p_r = -i \Hbar \PDi{r}{} \) the radial momentum operator. First it is noted that

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:40}
0 = \bra{nlm} \antisymmetric{H}{A} \ket{nlm},
\end{equation}

since \( H \) operating to either the right or the left is the energy eigenvalue \( E_n \). Next it appears the author uses an angular momentum factoring of the squared momentum operator. Looking earlier in the text that factoring is found to be

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:60}
\frac{\Bp^2}{2m}
= \inv{2 m r^2} \BL^2 – \frac{\Hbar^2}{2m} \lr{ \PDSq{r}{} + \frac{2}{r} \PD{r}{} }.
\end{equation}

With
\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:80}
R = – \frac{\Hbar^2}{2m} \lr{ \PDSq{r}{} + \frac{2}{r} \PD{r}{} }.
\end{equation}

we have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:100}
\begin{aligned}
0
&= \bra{nlm} \antisymmetric{H}{p_r} \ket{nlm} \\
&= \bra{nlm} \antisymmetric{\frac{\Bp^2}{2m} + V(r)}{p_r} \ket{nlm} \\
&= \bra{nlm} \antisymmetric{\inv{2 m r^2} \BL^2 + R + V(r)}{p_r} \ket{nlm} \\
&= \bra{nlm} \antisymmetric{\frac{-\Hbar^2 l (l+1)}{2 m r^2} + R + V(r)}{p_r} \ket{nlm}.
\end{aligned}
\end{equation}

Let’s consider the commutator of each term separately. First

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:120}
\begin{aligned}
\antisymmetric{V}{p_r} \psi
&=
V p_r \psi
–
p_r V \psi \\
&=
V p_r \psi
–
(p_r V) \psi
–
V p_r \psi \\
&=
–
(p_r V) \psi \\
&=
i \Hbar \PD{r}{V} \psi.
\end{aligned}
\end{equation}

Setting \( V(r) = 1/r^2 \), we also have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:160}
\antisymmetric{\inv{r^2}}{p_r} \psi
=
-\frac{2 i \Hbar}{r^3} \psi.
\end{equation}

Finally
\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:180}
\begin{aligned}
\antisymmetric{\PDSq{r}{} + \frac{2}{r} \PD{r}{} }{ \PD{r}{}}
&=
\lr{ \partial_{rr} + \frac{2}{r} \partial_r } \partial_r
–
\partial_r \lr{ \partial_{rr} + \frac{2}{r} \partial_r } \\
&=
\partial_{rrr} + \frac{2}{r} \partial_{rr}
–
\lr{
\partial_{rrr} -\frac{2}{r^2} \partial_r + \frac{2}{r} \partial_{rr}
} \\
&=
-\frac{2}{r^2} \partial_r,
\end{aligned}
\end{equation}

so
\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:200}
\antisymmetric{R}{p_r}
=-\frac{2}{r^2} \frac{-\Hbar^2}{2m} p_r
=\frac{\Hbar^2}{m r^2} p_r.
\end{equation}

Putting all the pieces back together, we’ve got
\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:220}
\begin{aligned}
0
&= \bra{nlm} \antisymmetric{\frac{-\Hbar^2 l (l+1)}{2 m r^2} + R + V(r)}{p_r} \ket{nlm} \\
&=
i \Hbar
\bra{nlm} \lr{
\frac{\Hbar^2 l (l+1)}{m r^3} – \frac{i\Hbar}{m r^2} p_r +
\PD{r}{V}
}
\ket{nlm}.
\end{aligned}
\end{equation}

Since s-states are those for which \( l = 0 \), this means

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:240}
\begin{aligned}
\expectation{\PD{r}{V}}
&= \frac{i\Hbar}{m } \expectation{ \inv{r^2} p_r } \\
&= \frac{\Hbar^2}{m } \expectation{ \inv{r^2} \PD{r}{} } \\
&= \frac{\Hbar^2}{m } \int_0^\infty dr \int_0^\pi d\theta \int_0^{2 \pi} d\phi r^2 \sin\theta \psi^\conj(r,\theta, \phi) \inv{r^2} \PD{r}{\psi(r,\theta,\phi)}.
\end{aligned}
\end{equation}

Since s-states are spherically symmetric, this is
\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:260}
\expectation{\PD{r}{V}}
= \frac{4 \pi \Hbar^2}{m } \int_0^\infty dr \psi^\conj \PD{r}{\psi}.
\end{equation}

That integral is

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:280}
\int_0^\infty dr \psi^\conj \PD{r}{\psi}
=
\evalrange{\Abs{\psi}^2}{0}{\infty} – \int_0^\infty dr \PD{r}{\psi^\conj} \psi.
\end{equation}

With the hydrogen atom, our radial wave functions are real valued. It’s reasonable to assume that we can do the same for other real-valued spherical potentials. If that is the case, we have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:300}
2 \int_0^\infty dr \psi^\conj \PD{r}{\psi}
=
\Abs{\psi(0)}^2,
\end{equation}

and

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:320}
\boxed{
\expectation{\PD{r}{V}}
= \frac{2 \pi \Hbar^2}{m } \Abs{\psi(0)}^2,
}
\end{equation}

which completes this part of the problem.

A: show this is the case for the 3D SHO and hydrogen wave functions

For a hydrogen like atom, in atomic units, we have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:360}
\begin{aligned}
\expectation{
\PD{r}{V}
}
&=
\expectation{
\PD{r}{} \lr{ -\frac{Z e^2}{r} }
} \\
&=
Z e^2
\expectation
{
\inv{r^2}
} \\
&=
Z e^2 \frac{Z^2}{n^3 a_0^2 \lr{ l + 1/2 }} \\
&=
\frac{\Hbar^2}{m a_0} \frac{2 Z^3}{n^3 a_0^2} \\
&=
\frac{2 \Hbar^2 Z^3}{m n^3 a_0^3}.
\end{aligned}
\end{equation}

On the other hand for \( n = 1 \), we have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:380}
\begin{aligned}
\frac{2 \pi \Hbar^2}{m} \Abs{R_{10}(0)}^2 \Abs{Y_{00}}^2
&=
\frac{2 \pi \Hbar^2}{m} \frac{Z^3}{a_0^3} 4 \inv{4 \pi} \\
&=
\frac{2 \Hbar^2 Z^3}{m a_0^3},
\end{aligned}
\end{equation}

and for \( n = 2 \), we have

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:400}
\begin{aligned}
\frac{2 \pi \Hbar^2}{m} \Abs{R_{20}(0)}^2 \Abs{Y_{00}}^2
&=
\frac{2 \pi \Hbar^2}{m} \frac{Z^3}{8 a_0^3} 4 \inv{4 \pi} \\
&=
\frac{\Hbar^2 Z^3}{4 m a_0^3}.
\end{aligned}
\end{equation}

These both match the potential derivative expectation when evaluated for the s-orbital (\( l = 0 \)).

For the 3D SHO I verified the ground state case in the Mathematica notebook sakuraiProblem5.16bSHO.nb

There it was found that

\begin{equation}\label{eqn:symmetricPotentialDerivativeExpectation:420}
\expectation{\PD{r}{V}}
= \frac{2 \pi \Hbar^2}{m } \Abs{\psi(0)}^2
= 2 \sqrt{\frac{m \omega ^3 \Hbar}{ \pi }}
\end{equation}

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.

Quadratic Zeeman effect

December 13, 2015 phy1520 , , , ,

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Q: [1] pr. 5.18

Work out the quadratic Zeeman effect for the ground state hydrogen atom due to the usually neglected \( e^2 \BA^2/2 m_e c^2 \) term in the Hamiltonian.

A:

The first order energy shift is

For a z-oriented magnetic field we can use

\begin{equation}\label{eqn:quadraticZeeman:20}
\BA = \frac{B}{2} \setlr{ -y, x, 0 },
\end{equation}

so the perturbation potential is

\begin{equation}\label{eqn:quadraticZeeman:40}
\begin{aligned}
V
&= \frac{e^2 \BA^2}{2 m_e c^2} \\
&= \frac{e^2 \BB^2 (x^2 + y^2)}{8 m_e c^2} \\
&= \frac{ e^2 \BB^2 r^2 \sin^2\theta }{8 m_e c^2}
\end{aligned}
\end{equation}

The ground state wave function is

\begin{equation}\label{eqn:quadraticZeeman:60}
\begin{aligned}
\psi_0
&= \braket{\Bx}{0} \\
&= \inv{\sqrt{\pi a_0^3}} e^{-r/a_0},
\end{aligned}
\end{equation}

so the energy shift is

\begin{equation}\label{eqn:quadraticZeeman:80}
\begin{aligned}
\Delta
&= \bra{0} V \ket{0} \\
&= \inv{ \pi a_0^3 } 2 \pi \frac{ e^2 \BB^2 }{8 m_e c^2} \int_0^\infty r^2 \sin\theta e^{-2r/a_0} r^2 \sin^2\theta dr d\theta \\
&=
\frac{ e^2 \BB^2 }{4 a_0^3 m_e c^2}
\int_0^\infty r^4 e^{-2r/a_0} dr \int_0^\pi \sin^3\theta d\theta \\
&= –
\frac{ e^2 \BB^2 }{4 a_0^3 m_e c^2}
\frac{4!}{(2/a_0)^{4+1} } \evalrange{\lr{u – \frac{u^3}{3}}}{1}{-1} \\
&=
\frac{ e^2 a_0^2 \BB^2 }{4 m_e c^2}.
\end{aligned}
\end{equation}

If this energy shift is written in terms of a diamagnetic susceptibility \( \chi \) defined by

\begin{equation}\label{eqn:quadraticZeeman:100}
\Delta = -\inv{2} \chi \BB^2,
\end{equation}

the diamagnetic susceptibility is

\begin{equation}\label{eqn:quadraticZeeman:120}
\chi = -\frac{ e^2 a_0^2 }{2 m_e c^2}.
\end{equation}

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.

L_y perturbation

December 13, 2015 phy1520 , , , ,

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Q: \( L_y \) perturbation. [1] pr. 5.17

Find the first non-zero energy shift for the perturbed Hamiltonian

\begin{equation}\label{eqn:LyPerturbation:20}
H = A \BL^2 + B L_z + C L_y = H_0 + V.
\end{equation}

A:

The energy eigenvalues for state \( \ket{l, m} \) prior to perturbation are

\begin{equation}\label{eqn:LyPerturbation:40}
A \Hbar^2 l(l+1) + B \Hbar m.
\end{equation}

The first order energy shift is zero

\begin{equation}\label{eqn:LyPerturbation:60}
\begin{aligned}
\Delta^1
&=
\bra{l, m} C L_y \ket{l, m} \\
&=
\frac{C}{2 i}
\bra{l, m} \lr{ L_{+} – L_{-} } \ket{l, m} \\
&=
0,
\end{aligned}
\end{equation}

so we need the second order shift. Assuming no degeneracy to start, the perturbed state is

\begin{equation}\label{eqn:LyPerturbation:80}
\ket{l, m}’ = \sum’ \frac{\ket{l’, m’} \bra{l’, m’}}{E_{l,m} – E_{l’, m’}} V \ket{l, m},
\end{equation}

and the next order energy shift is
\begin{equation}\label{eqn:LyPerturbation:100}
\begin{aligned}
\Delta^2
&=
\bra{l m} V
\sum’ \frac{\ket{l’, m’} \bra{l’, m’}}{E_{l,m} – E_{l’, m’}} V \ket{l, m} \\
&=
\sum’ \frac{\bra{l, m} V \ket{l’, m’} \bra{l’, m’}}{E_{l,m} – E_{l’, m’}} V \ket{l, m} \\
&=
\sum’ \frac{ \Abs{ \bra{l’, m’} V \ket{l, m} }^2 }{E_{l,m} – E_{l’, m’}} \\
&=
\sum_{m’ \ne m} \frac{ \Abs{ \bra{l, m’} V \ket{l, m} }^2 }{E_{l,m} – E_{l, m’}} \\
&=
\sum_{m’ \ne m} \frac{ \Abs{ \bra{l, m’} V \ket{l, m} }^2 }{
\lr{ A \Hbar^2 l(l+1) + B \Hbar m }
-\lr{ A \Hbar^2 l(l+1) + B \Hbar m’ }
} \\
&=
\inv{B \Hbar} \sum_{m’ \ne m} \frac{ \Abs{ \bra{l, m’} V \ket{l, m} }^2 }{
m – m’
}.
\end{aligned}
\end{equation}

The sum over \( l’ \) was eliminated because \( V \) only changes the \( m \) of any state \( \ket{l,m} \), so the matrix element \( \bra{l’,m’} V \ket{l, m} \) must includes a \( \delta_{l’, l} \) factor.
Since we are now summing over \( m’ \ne m \), some of the matrix elements in the numerator should now be non-zero, unlike the case when the zero first order energy shift was calculated above.

\begin{equation}\label{eqn:LyPerturbation:120}
\begin{aligned}
\bra{l, m’} C L_y \ket{l, m}
&=
\frac{C}{2 i}
\bra{l, m’} \lr{ L_{+} – L_{-} } \ket{l, m} \\
&=
\frac{C}{2 i}
\bra{l, m’}
\lr{
L_{+}
\ket{l, m}
– L_{-}
\ket{l, m}
} \\
&=
\frac{C \Hbar}{2 i}
\bra{l, m’}
\lr{
\sqrt{(l – m)(l + m + 1)} \ket{l, m + 1}
–
\sqrt{(l + m)(l – m + 1)} \ket{l, m – 1}
} \\
&=
\frac{C \Hbar}{2 i}
\lr{
\sqrt{(l – m)(l + m + 1)} \delta_{m’, m + 1}
–
\sqrt{(l + m)(l – m + 1)} \delta_{m’, m – 1}
}.
\end{aligned}
\end{equation}

After squaring and summing, the cross terms will be zero since they involve products of delta functions with different indices. That leaves

\begin{equation}\label{eqn:LyPerturbation:140}
\begin{aligned}
\Delta^2
&=
\frac{C^2 \Hbar}{4 B} \sum_{m’ \ne m} \frac{
(l – m)(l + m + 1) \delta_{m’, m + 1}
–
(l + m)(l – m + 1) \delta_{m’, m – 1}
}{
m – m’
} \\
&=
\frac{C^2 \Hbar}{4 B}
\lr{
\frac{ (l – m)(l + m + 1) }{ m – (m+1) }
–
\frac{ (l + m)(l – m + 1) }{ m – (m-1)}
} \\
&=
\frac{C^2 \Hbar}{4 B}
\lr{
–
(l^2 – m^2 + l – m)
–
(l^2 – m^2 + l + m)
} \\
&=
-\frac{C^2 \Hbar}{2 B} (l^2 – m^2 + l ),
\end{aligned}
\end{equation}

so to first order the energy shift is

\begin{equation}\label{eqn:LyPerturbation:160}
\boxed{
A \Hbar^2 l(l+1) + B \Hbar m \rightarrow
\Hbar l(l+1)
\lr{
A \Hbar
-\frac{C^2}{2 B}
}
+ B \Hbar m
+\frac{C^2 m^2 \Hbar}{2 B} .
}
\end{equation}

Exact perturbation equation

If we wanted to solve the Hamiltonian exactly, we’ve have to diagonalize the \( 2 m + 1 \) dimensional Hamiltonian

\begin{equation}\label{eqn:LyPerturbation:180}
\bra{l, m’} H \ket{l, m}
=
\lr{ A \Hbar^2 l(l+1) + B \Hbar m } \delta_{m’, m}
+
\frac{C \Hbar}{2 i}
\lr{
\sqrt{(l – m)(l + m + 1)} \delta_{m’, m + 1}
–
\sqrt{(l + m)(l – m + 1)} \delta_{m’, m – 1}
}.
\end{equation}

This Hamiltonian matrix has a very regular structure

\begin{equation}\label{eqn:LyPerturbation:200}
\begin{aligned}
H &=
(A l(l+1) \Hbar^2 – B \Hbar (l+1)) I \\
&+ B \Hbar
\begin{bmatrix}
1 & & & & \\
& 2 & & & \\
& & 3 & & \\
& & & \ddots & \\
& & & & 2 l + 1
\end{bmatrix} \\
&+
\frac{C \Hbar}{i}
\begin{bmatrix}
0 & -\sqrt{(2l-1)(1)} & & & \\
\sqrt{(2l-1)(1)} & 0 & -\sqrt{(2l-2)(2)}& & \\
& \sqrt{(2l-2)(2)} & & & \\
& & \ddots & & \\
& & & 0 & – \sqrt{(1)(2l-1)} \\
& & & \sqrt{(1)(2l-1)} & 0
\end{bmatrix}
\end{aligned}
\end{equation}

Solving for the eigenvalues of this Hamiltonian for increasing \( l \) in Mathematica (sakuraiProblem5.17a.nb), it appears that the eigenvalues are

\begin{equation}\label{eqn:LyPerturbation:220}
\lambda_m = A \Hbar^2 (l)(l+1) + \Hbar m B \sqrt{ 1 + \frac{4 C^2}{B^2} },
\end{equation}

so to first order in \( C^2 \), these are

\begin{equation}\label{eqn:LyPerturbation:221}
\lambda_m = A \Hbar^2 (l)(l+1) + \Hbar m B \lr{ 1 + \frac{2 C^2}{B^2} }.
\end{equation}

We have a \( C^2 \Hbar/B \) term in both the perturbative energy shift, and the first order expansion of the exact solution. Comparing this for the \( l = 5 \) case, the coefficients of \( C^2 \Hbar/B \) in the perturbative solution are all negative \( -17.5, -17., -16.5, -16., -15.5, -15., -14.5, -14., -13.5, -13., -12.5 \), whereas the coefficient of \( C^2 \Hbar/B \) in the first order expansion of the exact solution are \( 2 m \), ranging from \( [-10, 10] \).

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.

L_z and L^2 eigenvalues and probabilities for a given wave function

December 13, 2015 phy1520 , , , , ,

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Q: [1] 3.17

Given a wave function

\begin{equation}\label{eqn:LsquaredLzProblem:20}
\psi(r,\theta, \phi) = f(r) \lr{ x + y + 3 z },
\end{equation}

  • (a) Determine if this wave function is an eigenfunction of \( \BL^2 \), and the value of \( l \) if it is an eigenfunction.

  • (b) Determine the probabilities for the particle to be found in any given \( \ket{l, m} \) state,
  • (c) If it is known that \( \psi \) is an energy eigenfunction with energy \( E \) indicate how we can find \( V(r) \).

A: (a)

Using
\begin{equation}\label{eqn:LsquaredLzProblem:40}
\BL^2
=
-\Hbar^2 \lr{ \inv{\sin^2\theta} \partial_{\phi\phi} + \inv{\sin\theta} \partial_\theta \lr{ \sin\theta \partial_\theta} },
\end{equation}

and

\begin{equation}\label{eqn:LsquaredLzProblem:60}
\begin{aligned}
x &= r \sin\theta \cos\phi \\
y &= r \sin\theta \sin\phi \\
z &= r \cos\theta
\end{aligned}
\end{equation}

it’s a quick computation to show that

\begin{equation}\label{eqn:LsquaredLzProblem:80}
\BL^2 \psi = 2 \Hbar^2 \psi = 1(1 + 1) \Hbar^2 \psi,
\end{equation}

so this function is an eigenket of \( \BL^2 \) with an eigenvalue of \( 2 \Hbar^2 \), which corresponds to \( l = 1 \), a p-orbital state.

(b)

Recall that the angular representation of \( L_z \) is

\begin{equation}\label{eqn:LsquaredLzProblem:100}
L_z = -i \Hbar \PD{\phi},
\end{equation}

so we have

\begin{equation}\label{eqn:LsquaredLzProblem:120}
\begin{aligned}
L_z x &= i \Hbar y \\
L_z y &= – i \Hbar x \\
L_z z &= 0,
\end{aligned}
\end{equation}

The \( L_z \) action on \( \psi \) is

\begin{equation}\label{eqn:LsquaredLzProblem:140}
L_z \psi = -i \Hbar r f(r) \lr{ – y + x }.
\end{equation}

This wave function is not an eigenket of \( L_z \). Expressed in terms of the \( L_z \) basis states \( e^{i m \phi} \), this wave function is

\begin{equation}\label{eqn:LsquaredLzProblem:160}
\begin{aligned}
\psi
&= r f(r) \lr{ \sin\theta \lr{ \cos\phi + \sin\phi} + \cos\theta } \\
&= r f(r) \lr{ \frac{\sin\theta}{2} \lr{ e^{i \phi} \lr{ 1 + \inv{i}} + e^{-i\phi} \lr{ 1 – \inv{i} } } + \cos\theta } \\
&= r f(r) \lr{
\frac{(1-i)\sin\theta}{2} e^{1 i \phi}
+
\frac{(1+i)\sin\theta}{2} e^{- 1 i \phi}
+ \cos\theta e^{0 i \phi}
}
\end{aligned}
\end{equation}

Assuming that \( \psi \) is normalized, the probabilities for measuring \( m = 1,-1,0 \) respectively are

\begin{equation}\label{eqn:LsquaredLzProblem:180}
\begin{aligned}
P_{\pm 1}
&= 2 \pi \rho \Abs{\frac{1\mp i}{2}}^2 \int_0^\pi \sin\theta d\theta \sin^2 \theta \\
&= -2 \pi \rho \int_1^{-1} du (1-u^2) \\
&= 2 \pi \rho \evalrange{ \lr{ u – \frac{u^3}{3} } }{-1}{1} \\
&= 2 \pi \rho \lr{ 2 – \frac{2}{3}} \\
&= \frac{ 8 \pi \rho}{3},
\end{aligned}
\end{equation}

and

\begin{equation}\label{eqn:LsquaredLzProblem:200}
P_{0} = 2 \pi \rho \int_0^\pi \sin\theta \cos\theta = 0,
\end{equation}

where

\begin{equation}\label{eqn:LsquaredLzProblem:220}
\rho = \int_0^\infty r^4 \Abs{f(r)}^2 dr.
\end{equation}

Because the probabilities must sum to 1, this means the \( m = \pm 1 \) states are equiprobable with \( P_{\pm} = 1/2 \), fixing \( \rho = 3/16\pi \), even without knowing \( f(r) \).

(c)

The operator \( r^2 \Bp^2 \) can be decomposed into a \( \BL^2 \) component and some other portions, from which we can write

\begin{equation}\label{eqn:LsquaredLzProblem:240}
\begin{aligned}
H \psi
&= \lr{ \frac{\Bp^2}{2m} + V(r) } \psi \\
&=
\lr{
– \frac{\Hbar^2}{2m} \lr{ \partial_{rr} + \frac{2}{r} \partial_r – \inv{\Hbar^2 r^2} \BL^2 } + V(r) } \psi.
\end{aligned}
\end{equation}

(See: [1] eq. 6.21)

In this case where \( \BL^2 \psi = 2 \Hbar^2 \psi \) we can rearrange for \( V(r) \)

\begin{equation}\label{eqn:LsquaredLzProblem:260}
\begin{aligned}
V(r)
&= E + \inv{\psi} \frac{\Hbar^2}{2m} \lr{ \partial_{rr} + \frac{2}{r} \partial_r – \frac{2}{r^2} } \psi \\
&= E + \inv{f(r)} \frac{\Hbar^2}{2m} \lr{ \partial_{rr} + \frac{2}{r} \partial_r – \frac{2}{r^2} } f(r).
\end{aligned}
\end{equation}

References

[1] Jun John Sakurai and Jim J Napolitano. Modern quantum mechanics. Pearson Higher Ed, 2014.