Chapter 12 Rotational Invariance and Angular Momentum

12.1 Translations in Two Dimensions

Exercise 12.1.1 Verify that $\hat{a} \cdot \mathbf{P}$ is the generator of infinitesimal translations along a by considering the relation

12.2 Rotations in Two Dimensions

Exercise 12.2.1 Provide the steps linking Eq. (12.2.8) to Eq. (12.2.9). [Hint: Recall the derivation of Eq. (11.2.8) from Eq. (11.2.6).]

Exercise 12.2.2 Using these commutation relations (and your keen hindsight) derive $L_z=X P_y-Y P_x$. At least show that Eqs. (12.2.16) and (12.2.17) are consistent with $L_z=X P_y-Y P_x$.

Exercise 12.2.3 Derive Eq. (12.2.19) by doing a coordinate transformation on Eq. (12.2.10), and also by the direct method mentioned above.

Exercise 12.2.4 Rederive the equivalent of Eq. (12.2.23) keeping terms of order $\varepsilon_x \varepsilon_z^2$. (You may assume $\varepsilon_y=0$.) Use this information to rewrite Eq. (12.2.24) to order $\varepsilon_x \varepsilon_z^2$. By equating coefficients of this term deduce the constraint

This seems to conflict with statement (1) made above, but not really, in view of the identity

Using the identify, verify that the new constraint coming from the $\varepsilon_x \varepsilon_z^2$ term is satisfied given the commutation relations between $P_x$, $P_y$, and $L_z$.

12.3 The Eigenvalue Problem of $L_{z}$

Exercise 12.3.1 Provide the steps linking Eq. (12.3.5) to Eq. (12.3.6).

Exercise 12.3.2 Let us try to deduce the restriction on $l_{z}$ from another angle. Consider a superposition of two allowed $l_{z}$ eigenstates:

By demanding that upon a $2 \pi$ rotation we get the same physical state (not necessarily the same state vector), show that $l_z-l_z^{\prime}=m \hbar$, where $m$ is an integer. By arguing on the grounds of symmetry that the allowed values of $l_z$ must be symmetric about zero, show that these values are either $\ldots, 3 \hbar / 2, \hbar / 2,-\hbar / 2,-3 \hbar / 2, \ldots$ or $\ldots, 2 \hbar, \hbar, 0,-\hbar,-2 \hbar, \ldots$ It is not possible to restrict $l_z$ any further this way.

Exercise 12.3.3 A particle is described by a wave function

Show (by expressing $\cos^2 \phi$ in terms of $\Phi_m$) that

(Hint: Argue that the radial part $e^{-\rho^2 / 2 \Delta^2}$ is irrelevant here.)

Exercise 12.3.4 A particle is described by a wave function

Show that

Exercise 12.3.5 Note that the angular momentum seems to generate a repulsive potential in Eq. (12.3.13). Calculate its gradient and identify it as the centrifugal force.

Exercise 12.3.6 Consider a particle of mass $\mu$ constrained to move on a circle of radius $a$. Show that $H=L_z^2/2\mu a^2$. Solve the eigenvalue problem of $H$ and interpret the degeneracy.

Exercise 12.3.7 (The Isotropic Oscillator) Consider the Hamiltonian

(1) Convince yourself $\left[H, L_z\right]=0$ and reduce the eigenvalue problem of $H$ to the radial differential equation for $R_{E m}(\rho)$.

(2) Examine the equation as $\rho \rightarrow 0$ and show that

(3) Show likewise that up to powers of $\rho$

So assume that $R_{E m}(\rho)=\rho^{|m|} \mathrm{e}^{-\mu \omega \rho^2 / 2 \hbar} U_{E m}(\rho)$.

(4) Switch to dimensionless variables $\varepsilon=E/\hbar\omega$, $y=(\mu \omega / \hbar)^{1/2}\rho$.

(5) Convert the equation for $R$ into an equation for $U$. (I suggest proceeding in two stages: $R=y^{|m|}f$, $f=\mathrm{e}^{-y^2/2}U$.) You should end up with

(6) Argue that a power series for $U$ of the form

will lead to a two-term recursion relation.

(7) Find the relation between $C_{r+2}$ and $C_r$. Argue that the series must terminate at some finite $r$ if the $y \rightarrow \infty$ behavior of the solution is to be acceptable. Show $\varepsilon=r+|m|+1$ leads to termination after $r$ terms. Now argue that $r$ is necessarily even——i.e, $r=2 k$. (Show that if $r$ is odd, the behavior of $R$ as $\rho \rightarrow 0$ is not $\rho^{|m|}$.) So finally you must end up with

Define $n=2 k+|m|$, so that

(8) For a given $n$, what are the allowed values of $|m|$? Given this information show that for a given $n$, the degeneracy is $n+1$. Compare this to what you found in Cartesian coordinates (Exercise 10.2.2).

(9) Write down all the normalized eigenfunctions corresponding to $n=0,1$.

(10) Argue that the $n=0$ function must equal the corresponding one found in Cartesian coordinates. Show that the two $n=2$ solutions are linear combinations of their counterparts in Cartesian coordinates. Verify that the parity of the states is $(-1)^n$ as you found in Cartesian coordinates.

Exercise 12.3.8 Consider a particle of charge $q$ in a vector potential

(1) Show that the magnetic field is $\mathbf{B}=B \mathbf{k}$.

(2) Show that a classical particle in this potential will move in circles at an angular frequency $\omega_0=q B / \mu c$.

(3) Consider the Hamiltonian for the corresponding quantum problem:

Show that $Q=\left(c P_x+q Y B / 2\right) / q B$ and $P=\left(P_y-q X B / 2 c\right)$ are canonical. Write $H$ in terms of $P$ and $Q$ and show that allowed levels are $E=(n+1 / 2) \hbar \omega_0$.

(4) Expand $H$ out in terms of the original variables and show

where $H\left(\omega_0 / 2, \mu\right)$ is the Hamiltonian for an isotropic two-dimensional harmonic oscillator of mass $\mu$ and frequency $\omega_0 / 2$. Argue that the same basis that diagonalized $H\left(\omega_0 / 2, \mu\right)$ will diagonalize $H$. By thinking in terms of this basis, show that the allowed levels for $H$ are $E=\left(k+\dfrac{1}{2}|m|-\dfrac{1}{2} m+\dfrac{1}{2}\right) \hbar \omega_0$, where $k$ is any integer and $m$ is the angular momentum. Convince yourself that you get the same levels from this formula as from the earlier one $\left[E=(n+1 / 2) \hbar \omega_0\right]$. We shall return to this problem in Chapter 21.

12.4 Angular Momentum in Three Dimensions

Exercise 12.4.1 (1) Verify that Eqs. (12.4.9) and Eq. (12.4.8) are equivalent, given the definition of $\varepsilon_{i j k}$.

(2) Let $U_1, U_2$, and $U_3$ be three energy eigenfunctions of a single particle in some potential. Construct the wave function $\psi_A\left(x_1, x_2, x_3\right)$ of three fermions in this potential, one of which is in $U_1$, one in $U_2$, and one in $U_3$, using the $\varepsilon_{i j k}$ tensor.

Exercise 12.4.2 (1) Verify Eq. (12.4.2) by first constructing the $3 \times 3$ matrices corresponding to $R\left(\varepsilon_x \mathbf{i}\right)$ and $R\left(\varepsilon_y \mathbf{j}\right)$, to order $\varepsilon$.

(2) Provide the steps connecting Eqs. (12.4.3) and (12.4.4a).

(3) Verify that $L_x$ and $L_y$ defined in Eq. (12.4.1) satisfy Eq. (12.4.4a). The proof for other commutators follows by cyclic permutation.

Exercise 12.4.3 We would like to show that $\hat{\theta} \cdot \mathbf{L}$ generates rotations about the axis parallel to $\hat{\theta}$. Let $\delta\boldsymbol{\theta}$ be an infinitesimal rotation parallel to $\boldsymbol{\theta}$.

(1) Show that when a vector $\mathbf{r}$ is rotated by an angle $\delta \boldsymbol{\theta}$, it changes to $\mathbf{r}+\delta \boldsymbol{\theta} \times \mathbf{r}$. (It might help to start with $\mathbf{r} \perp \delta \boldsymbol{\theta}$ and then generalize.)

(2) We therefore demand that (to first order, as usual)

Comparing to $U[R(\delta \boldsymbol{\theta})]=I-(i \delta \theta / \hbar) L_{\hat{\theta}}$, show that $L_{\hat{\theta}}=\hat{\theta} \cdot \mathbf{L}$.

Exercise 12.4.4 Recall that $\mathbf{V}$ is a vector operator if its components $V_i$ transform as

(1) For an infinitesimal rotation $\delta \boldsymbol{\theta}$, show, on the basis of the previous exercise, that

(2) Feed in $U[R]=1-(i / \hbar) \delta \boldsymbol{\theta} \cdot \mathbf{L}$ into the left-hand side of Eq. (12.4.13) and deduce that

12.5 The Eigenvalue Problem of $L^{2}$ and $L_{z}$

Exercise 12.5.1 Consider a vector field $\Psi(x, y)$ in two dimensions. From Fig. 12.1 it follows that under an infinitesimal rotation $\varepsilon_z \mathbf{k}$,

Show that (to order $\varepsilon_z$)

so that

where $I^{(2)}$ is a $2 \times 2$ identity matrix with respect to the vector components, $I^{(1)}$ is the identity operator with respect to the argument $(x, y)$ of $\Psi(x, y)$. This example only illustrates the fact that $J_z=L_z+S_z$ if the wave function is not a scalar. An example of half-integral eigenvalues will be provided when we consider spin in a later chapter. (In the present example, $S_z$ has eigenvalues $\pm \hbar$.)

Exercise 12.5.2 (1) Verify that the $2 \times 2$ matrices $J_x^{(1 / 2)}$, $J_y^{(1 / 2)}$, and $J_z^{(1 / 2)}$ obey the commutation rule $\left[J_x^{(1 / 2)}, J_y^{(1 / 2)}\right]=\mathrm{i} \hbar J_z^{(1 / 2)}$.

(2) Do the same for the $3 \times 3$ matrices $J_i^{(1)}$.

(3) Construct the $4 \times 4$ matrices and verify that

Exercise 12.5.3 (1) Show that $\left\langle J_x\right\rangle=\left\langle J_y\right\rangle=0$ in a state $|j m\rangle$.

(2) Show that in these states

(use symmetry arguments to relate $\left\langle J_x^2\right\rangle$ to $\left\langle J_y^2\right\rangle$).

(3) Check that $\Delta J_x \cdot \Delta J_y$ from part (2) satisfies the inequality imposed by the uncertainty principle [Eq. (9.2.9)].

(4) Show that the uncertainty bound is saturated in the state $|j, \pm j\rangle$.

Exercise 12.5.4 (1) Argue that the eigenvalues of $J_x^{(j)}$ and $J_y^{(j)}$ are the same as those of $J_z^{(j)}$, namely, $j \hbar,(j-1) \hbar, \ldots, (-j\hbar)$. Generalize the result to $\hat{\theta} \cdot \mathbf{J}^{(j)}$.

(2) Show that

where $J \equiv \hat{\theta} \cdot \mathbf{J}^{(j)}$. (Hint: In the case $J=J_z$ what happens when both sides are applied to an arbitrary eigenket $|j m\rangle$? What about an arbitrary superpositions of such kets?)

(3) It follows from (2) that $J^{2 j+1}$ is a linear combination of $J^0, J^1, \ldots, J^{2 j}$. Argue that the same goes for $J^{2 j+k}$, $k=1,2, \ldots$.

Exercise 12.5.5 Using results from the previous exercise and Eq. (12.5.23), show that

(1) $D^{(1 / 2)}[R]=\exp \left(-\mathrm{i} \hat{\theta} \cdot \mathbf{J}^{(1 / 2)} / \hbar\right)=\cos (\theta / 2) I^{(1 / 2)}-(2\mathrm{i} / \hbar) \sin (\theta / 2) \hat{\theta} \cdot \mathbf{J}^{(1 / 2)}$

(2) $D^{(1)}[R]=\exp \left(-\mathrm{i} \theta_x J_x^{(1)} / \hbar\right)=\left(\cos \theta_x-1\right)\left(\frac{J_x^{(1)}}{\hbar}\right)^2-\mathrm{i} \sin \theta_x\left(\frac{J_x^{(1)}}{\hbar}\right)+I^{(1)}$

Exercise 12.5.6 Consider the family of states $|j j\rangle, \ldots,|j m\rangle, \ldots,|j,-j\rangle$. One refers to them as states of the same magnitude but different orientation of angular momentum. If ones takes this remark literally, i.e., in the classical sense, one is led to believe that one may rotate these into each other, as is the case for classical states with these properties. Consider, for instance, the family $|1,1\rangle,|1,0\rangle,|1,-1\rangle$. It may seem, for example, that the state with zero angular momentum along the $z$ axis, $|1,0\rangle$, may be obtained by rotating $|1,1\rangle$ by some suitable ($\frac{1}{2} \pi$?) angle about the $x$ axis. Using $D^{(1)}\left[R\left(\theta_x \mathbf{i}\right)\right]$ from part (2) in the last exercise show that

The error stems from the fact that classical reasoning should be applied to $\langle\mathbf{J}\rangle$, which responds to rotations like an ordinary vector, and not directly to $|j m\rangle$, which is a vector in Hilbert space. Verify that $\langle\mathbf{J}\rangle$ responds to rotations like its classical counterpart, by showing that $\langle\mathbf{J}\rangle$ in the state $D^{(1)}\left[R\left(\theta_x \mathbf{i}\right)\right]|1,1\rangle$ is $\hbar\left[-\sin \theta_x \mathbf{j}+\cos \theta_x \mathbf{k}\right]$.

Exercise 12.5.7 (Euler Angles) Rather than parametrize an arbitrary rotation by the angle $\boldsymbol{\theta}$, which describes a single rotation by $\theta$ about an axis parallel to $\boldsymbol{\theta}$, we may parametrize it by three angles, $\gamma, \beta$, and $\alpha$ called Euler angles, which define three successive rotations:

(1) Construct $D^{(1)}[R(\alpha, \beta, \gamma)]$ explicitly as a product of three $3 \times 3$ matrices. (Use the result from Exercise 12.5.5 with $J_x \rightarrow J_y$.)

(2) Let it act on $|1,1\rangle$ and show that $\langle\mathbf{J}\rangle$ in the resulting state is

(3) Show that for no value of $\alpha, \beta$, and $\gamma$ can one rotate $|1,1\rangle$ into just $|1,0\rangle$.

(4) Show that one can always rotate any $|1, m\rangle$ into a linear combination that involves $\left|1, m^{\prime}\right\rangle$, i.e.,

for some $\alpha, \beta, \gamma$ and any $m, m^{\prime}$. \noindent (5) To see that one can occasionally rotate $|j m\rangle$ into $\left|j m^{\prime}\right\rangle$, verify that a $180^{\circ}$ rotation about the $y$ axis applied to $|1,1\rangle$ turns it into $|1,-1\rangle$.

Exercise 12.5.8 Verify that

Exercise 12.5.9 Show that $L^2$ above is Hermitian in the sense

The same goes for $L_z$, which is insensitive to $\theta$ and is Hermitian with respect to the $\phi$ integration.

Exercise 12.5.10 Write the differential equation corresponding to

in the coordinate basis, using the $L^2$ operator given in Eq. (12.5.36). We already know $\beta=m \hbar$ from the analysis of $-\mathrm{i} \hbar(\partial / \partial \phi)$. So assume that the simultaneous eigenfunctions have the form

and show that $P_\alpha^m$ satisfies the equation

We need to show that

(1) $\dfrac{\alpha}{\hbar^2}=l(l+1)$, $l=0,1,2,\ldots$

(2) $|m| \leqslant l$

We will consider only part (1) and that too for the case $m=0$. By rewriting the equation in terms of $u=\cos \theta$, show that $P_\alpha^0$ satisfies

Convince yourself that a power series solution

will lead to a two-term recursion relation. Show that $\left(C_{n+2} / C_n\right) \rightarrow 1$ as $n \rightarrow \infty$. Thus the series diverges when $|u| \rightarrow 1$ ($\theta \rightarrow 0$ or $\pi$). Show that if $\alpha / \hbar^2=(l)(l+1)$; $l=0,1,2,\ldots$, the series will terminate and be either an even or odd function of $u$. The functions $P_\alpha^0(u)=P_{l(l+1) \hbar^2}^0(u) \equiv P_l^0(u) \equiv P_l(u)$ are just the Legendre polynomials up to a scale factor. Determine $P_0$, $P_1$, and $P_2$ and compare (ignoring overall scales) with the $Y_{l}^{0}$ functions.

Exercise 12.5.11 Derive $Y_1^1$ starting from Eq.(12.5.28) and normalize it yourself. [Remember the $(-1)^l$ factor from Eq. (12.5.32).] Lower it to get $Y_1^0$ and $Y_1^{-1}$ and compare it with Eq. (12.5.39).

Exercise 12.5.12 Since $L^2$ and $L_z$ commute with $\Pi$, they should share a basis with it. Verify that under parity $Y_l^m \rightarrow (-1)^l Y_l^m$. (First show that $\theta \rightarrow \pi -\theta$, $\phi \rightarrow \phi+\pi$ under parity. Prove the result for $Y_l^l$. Verify that $L_{-}$does not alter the parity, thereby proving the result for all $Y_l^m$.)

Exercise 12.5.13 Consider a particle in a state described by

where $N$ is a normalization factor.

(1) Show, by rewriting the $Y_1^{ \pm 1.0}$ functions in terms of $x, y, z$, and $r$, that

(2) Using this result, show that for a particle described by $\psi$ above, $P\left(l_z=0\right)=2 / 3$; $P\left(l_z=+\hbar\right)=1 / 6=P\left(l_z=-\hbar\right)$.

Exercise 12.5.14 Consider a rotation $\theta_x \mathbf{i}$. Under this

Therefore we must have

Let us verify this prediction for a special case

which must go into

(1) Expand $\psi$ in terms of $Y_1^1, Y_1^0, Y_1^{-1}$.

\noindent (2) Use the matrix $\mathrm{e}^{-\mathrm{i} \theta_x L_x / \hbar}$ to find the fate of $\psi$ under this rotation.\footnote{Use Exercise 12.5.15.} Check your result against that anticipated above. [Hint: (1) $\psi \sim Y_1^0$, which corresponds to

(2) Use Eq. (12.5.42).]

12.6 Solution of Rotationally Invariant Problems

Exercise 12.6.1 A particle is described by the wave function

(1) What is the angular momentum content of the state?

(2) Assuming $\psi_E$ is an eigenstate in a potential that vanishes as $r \rightarrow \infty$, find $E$. (Match leading terms in Schrödinger's equation.)

(3) Having found $E$, consider finite $r$ and find $V(r)$.

Exercise 12.6.2 Provide the steps connecting Eq. (12.6.3) and Eq. (12.6.5).

Exercise 12.6.3 Show that Eq. (12.6.7b) follows from Eq. (12.6.7a).

Exercise 12.6.4 (1) Show that

(consider a test function).

(2) Show that

[Hint: First show that $\nabla^2(1 / r)=0$ if $r \neq 0$. To see what happens at $r=0$, consider a small sphere centered at the origin and use Gauss's law and the identity $\nabla^2 \phi=\nabla \cdot \nabla \phi$]. (Or compare this equation to Poisson's equation in electrostatics $\nabla^2 \phi=-4 \pi \rho$. Here $\rho=\delta^3(\mathbf{r})$, which represents a unit point charge at the origin. In this case we know from Coulomb's law that $\phi=1 / r$.)

Exercise 12.6.5 Show that $D_l$ is nondegenerate in the space of functions $U$ that vanish as $r \rightarrow 0$. (Recall the proof of Theorem 15, Section 5.6.) Note that $U_{El}$ is nondegenerate even for $E>0$. This means that $E$, $l$, and $m$, label a state fully in three dimensions.

Exercise 12.6.6 (1) Verify that Eqs. (12.6.21) and (12.6.22) are equivalent to Eq. (12.6.20).

(2) Verify Eq. (12.6.24).

Exercise 12.6.7 Verify that $j_{0}$ and $j_{1}$ have the limits given by Eq. (12.6.33).

Exercise 12.6.8 Find the energy levels of a particle in a spherical box of radius $r_{0}$ in the $l=0$ sector.

Exercise 12.6.9 Show that the quantization condition for $l=0$ bound states in a spherical well of depth $-V_0$ and radius $r_0$ is

where $k^{\prime}$ is the wave number inside the well and $\mathrm{i}\kappa$ is the complex wave number for the exponential tail outside. Show that there are no bound states for $V_0<\pi^2 \hbar^2 / 8 \mu r_0^2$. (Recall Exercise 5.2.6.)

Exercise 12.6.10 Verify Eq. (12.6.41) given that

(1) $\displaystyle\int_{-1}^1 P_l(\cos \theta) P_{l^{\prime}}(\cos \theta) \mathrm{d}(\cos \theta)=[2 /(2 l+1)] \delta_{l l^{\prime}}$.

(2) $P_l(x)=\dfrac{1}{2^{l} l!} \dfrac{\mathrm{d}^l\left(x^2-1\right)^{\prime}}{\mathrm{d} x^l}$

(3) $\displaystyle\int_0^1\left(1-x^2\right)^m \mathrm{d} x=\frac{(2 m)!!}{(2 m+1)!!}$

[Hint: Consider the limit $k r \rightarrow 0$ after projecting out $C_l$.]

Exercise 12.6.11 (1) By combining Eqs. (12.6.48) and (12.6.49) derive the two-term recursion relation. Argue that $C_0 \neq 0$ if $U$ is to have the right properties near $y=0$. Derive the quantizations condition, Eq. (12.6.50).

(2) Calculate the degeneracy and parity at each $n$ and compare with Exercise 10.2.3, where the problem was solved in Cartesian coordinates.

(3) Construct the normalized eigenfunction $\psi_{nlm}$ for $n=0$ and $1$. Write them as linear combinations of the $n=0$ and $n=1$ eigenfunctions obtained in Cartesian coordinates.