Chapter 17, Rotations and Other Symmetry Operations Video Solutions, Quantum mechanics

Problem 1

In the notation of $(17.64)$ the state of a spin one-half particle with sharp total angular momentum $j, m$, is:
$$
a^{\hdashline y_{j-1 / 2}^{i m}}+b^{Q \mathrm{Y}_{j+1 / 2}^m}
$$

Assume this state to be an eigenstate of the Hamiltonian with no degeneracy other than that demanded by rotation invariance.
(a) If $H$ conserves parity, how are the coefficients $a$ and $b$ restricted?
(b) If $H$ is invariant under time reversal, show that $a / b$ must be imaginary.
(c) Verify explicitly that the expectation value of the electric dipole moment $-e \mathbf{r}$ vanishes if either parity is conserved or time reversal invariance holds (or both).

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13:51

Problem 2

A particle with spin one-half (lambda hyperon) decays at rest $(\ell=0)$ into two particles with spin one-half (nucleon) and spin zero (pion). (See Figure 17.2.)
(a) Show that, in the representation in which the relative momentum of the decay products is diagonal, the final state wave functions corresponding to $m= \pm 1 / 2$ may, because of conservation of total angular momentum, be written in the form
$$
\begin{aligned}
\left\langle\mathbf{p} \left\lvert\, \frac{1}{2} \frac{1}{2}\right.\right\rangle & =A_S \alpha+A_P\left(\cos \theta \alpha+e^{i \varphi} \sin \theta \beta\right) \\
\left\langle\mathbf{p} \left\lvert\, \frac{1}{2}\right.,-\frac{1}{2}\right\rangle & =A_S \beta-A_P\left(\cos \theta \beta-e^{-i \varphi} \sin \theta \alpha\right)
\end{aligned}
$$
where $\theta$ is the angle between the momentum vector $\mathbf{p}$ of the spin zero particle and the $z$-axis of quantization. (Neglect any interactions between the decay products in the final state.) "
(b) If the initial spin one-half particle is in a state with polarization $\mathbf{P}(|\mathbf{P}| \leq 1)$, show that the (unnormalized) angular distribution can be written as $W=1+\lambda \mathbf{P} \cdot \hat{\mathbf{p}}$. Evaluate $\lambda$ in terms of $A_s$ and $A_p$.

Robert Zaballa

Numerade Educator

Problem 3

From the results of Problem 2, construct the density matrix $\rho$ for the spin state of the final spin one-half particle (nucleon).
(a) Prove that the polarization of the spin one-half decay products can be written in the form
$$
\langle\boldsymbol{\sigma}\rangle=\operatorname{trace}(\rho \boldsymbol{\sigma})=\frac{1}{1-\lambda \hat{\mathbf{p}}_n \cdot \mathbf{P}}\left[-\left(\lambda-\hat{\mathbf{p}}_n \cdot \mathbf{P}\right) \hat{\mathbf{p}}_n+\mu \hat{\mathbf{p}}_n \times \mathbf{P}+\nu\left(\hat{\mathbf{p}}_n \times \mathbf{P}\right) \times \hat{\mathbf{p}}_n\right]
$$
where $\hat{\boldsymbol{p}}_n$ is the unit vector in the direction of emission of the spin one-half particle, and $\mathbf{P}$ denotes the initial polarization of the decaying particles. Determine the realvalued quantitięs $\mu$ and $\nu$ in terms of $A_s$ and $A_P$.
(b) Show that $\lambda^2+\mu^2+\nu^2=1$.
(c) Discuss the simplifications that occur in the expressions for the angular distribution and final state polarization if conservation of parity or invariance under time reversal is assumed for the decay-inducing interaction.

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Problem 4

The Hamiltonian of the positronium atom in the $1 S$ state in a magnetic field $B$ along the $z$ axis is to good approximation
$$
H=A \mathbf{S}_1 \cdot \mathbf{S}_2+\frac{e B}{m c}\left(S_{1_2}-S_{2_2}\right)
$$
if all higher energy states are neglected. The electron is labeled as particle 1 and the positron as particle 2 . Using the coupled representation in which $\mathbf{S}^2=\left(\mathbf{S}_1+\mathbf{S}_2\right)^2$ and $S_z=S_{1_z}+S_{2_z}$ are diagonal, obtain the energy eigenvalues and eigenvectors and classify them according to the quantum numbers associated with constants of the motion.

Empirically, it is known that for $B=0$ the frequency of the $1^3 S \rightarrow 1^1 S$ transition is $2.0338 \times 10^5 \mathrm{MHz}$ and that the mean lifetimes for annihilation are $10^{-10} \mathrm{~s}$ for the singlet state (two-photon decay) and $10^{-7} \mathrm{~s}$ for the triplet state (three-photon decay). Estimate the magnetic field strength $B$ which will cause the lifetime of the longer lived $m=0$ state to be reduced ("quenched") to $10^{-8} \mathrm{~s}$.

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Problem 5

An alternative to the usual representation for states of a particle with spin one-half, in which the simultaneous eigenstates of $\mathbf{r}$ and $\sigma_z$ are used as a basis, is to employ a basis spanned by the simultaneous eigenstates of $\mathbf{r}$ and $\kappa=\boldsymbol{\sigma} \cdot \hat{\mathbf{r}} / 2$. Show that the operators $\mathbf{S}^2, \mathbf{J}^2, J_z, \kappa$ commute and that their eigenfunctions may be represented as
$$
\left\langle\hat{\mathbf{r}} \kappa^{\prime} \mid j m \kappa^{\prime \prime}\right\rangle \propto D_{m \kappa^{\prime}}^{G *}(\varphi, \theta) \delta_{\kappa^{\prime} \kappa^*}
$$
where $\varphi, \theta$ denotes the rotation that turns the $z$ axis into the direction of $\mathbf{r}$. Can this representation be generalized to particles of higher spin? Can an analogous basis be constructed in the momentum representation ${ }^{10}$

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01:08

Problem 6

The magnetic moment operator for a nucleon of mass $m_n$ is $\boldsymbol{\mu}=e\left(g_{\ell} \mathbf{L}+g_s \mathbf{S}\right) / 2 m_n c$, where $g_{\ell}=1$ and $g_s=5.587$ for a proton, $g_{\ell}=0$ and $g_S=-3.826$ for a neutron. In a central field with an additional spin-orbit interaction, the nucleons move in shells characterized by the quantum numbers $\ell$ and $j=\ell \pm 1 / 2$. Calculate the magnetic moment of a single nucleon as a function of $j$ for the two kinds of nucleons, distinguishing the two cases $j=\ell+1 / 2$ and $j=\ell-1 / 2$. Plot $j$ times the effective gyromagnetic ratio versus $j$, connecting in each case the points by straight-line seg. ments (Schmidt lines).

Chai Santi

Numerade Educator

Problem 7

The angular momentum operator or generator of infinitesimal rotations may be rep resented in terms of Euler angles, which specify the orientation of a rectangula coordinate system $O x^{\prime} y^{\prime} z^{\prime}$ anchored in a rigid body (Figure 17.1). In this representation, the components of angular momentum along the $z, y^{\prime \prime}$, and $z^{\prime}$ axes are
$$
J_z=\frac{\hbar}{i} \frac{\partial}{\partial \alpha}, \quad J_{y^{\prime \prime}}=\frac{\hbar}{i} \frac{\partial}{\partial \beta}, \quad J_{z^{\prime}}=\frac{\hbar}{i} \frac{\partial}{\partial \gamma}
$$
(a) Prove that the components of angular momentum along the $x$ and $y$ axes can be expressed in terms of the Euler angles as
$$
\begin{aligned}
J_x & =\frac{\hbar}{i}\left(-\sin \alpha \frac{\partial}{\partial \beta}-\cos \alpha \cot \beta \frac{\partial}{\partial \alpha}+\frac{\cos \alpha}{\sin \beta} \frac{\partial}{\partial \gamma}\right) \\
J_y & =\frac{\hbar}{i}\left(\cos \alpha \frac{\partial}{\partial \beta}-\sin \alpha \cot \beta \frac{\partial}{\partial \alpha}+\frac{\sin \alpha}{\sin \beta} \frac{\partial}{\partial \gamma}\right)
\end{aligned}
$$
(b) Check the commutation relations for the operators $J_x, J_y, J_z$.
(c) Work out the operator $\mathbf{J}^2$ in terms of the Euler angles.
(d) Using relation (17.37), show that the matrix elements $D_{m^{\prime} m}^{(e)}(\alpha, \beta, \gamma)$ are eigenfunctions of the differential operators $J_z, J_{z^{\prime}}$ and $\mathrm{J}^2$ with eigenvalues $m^{\prime} \hbar, m \hbar$, and $\ell(\ell+1) \hbar^2$, respectively.

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05:42

Problem 8

In the quantum mechanical treatment of a rigid body with principal moments of inertia, $I_1, I_2$, and $I_3$, it is convenient to use Euler angles as the coordinates to specify the orientation of the system. If the $z^{\prime}$ axis is the axis of a symmetric top, its Hamiltonian is
$$
H=\frac{1}{2 I_1}\left(\mathbf{J}^2-\mathbf{J}_{z^{\prime}}^2\right)+\frac{1}{2 I_3} J_{z^{\prime}}^2
$$

Using the results of Problem 7, obtain the energy eigenvalues and the corresponding eigenfunctions."

Eduard Sanchez

Numerade Educator

Problem 9

A system that is invariant under rotation is perturbed by a quadrupole interaction
$$
V=\sum_{q=-2}^2 C_q T_2^q
$$
where the $C_q$ are constant coefficients and $T_2^q$ are the components of an irreducible spherical tensor operator, defined by one of its components:
$$
T_2^2=\left(J_x+i J_y\right)^2
$$
(b) Deduce the conditions for the coefficients $C_q$ if $V$ is to be Hermitian.
(b) Consider the effect of the quadrupole perturbation on the manifold of a degenerate energy eigenstate of the unperturbed system with angular momentum quantum number $j$, neglecting all other unperturbed energy eigenstates. What is the effect of the perturbation on the manifold of an unperturbed $j=1 / 2$ state?
(c) If $C_{ \pm 2}=C_0$ and $C_{ \pm 1}=0$, calculate the perturbed energies for a $j=1$ state, and plot the energy splittings as a function of the interaction strength $C_0$. Derive the corresponding unperturbed energy eigenstates.

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