Chapter 5, Angular Momentum Video Solutions, Quantum mechanics: Concepts and Applications | Numerade (2024)

If $\hat{L}_{ \pm}$and $\hat{R}_{ \pm}$are defined by $\hat{L}_{ \pm}=\hat{L}_x \pm i \hat{L}_y$ and $\hat{R}_{ \pm}=\hat{X} \pm i \hat{Y}$, prove the following commutators: $\left[\hat{L}_{ \pm}, \hat{R}_{ \pm}\right]=0,\left[\hat{L}_{ \pm}, \hat{R}_{\mp}\right]= \pm 2 \hbar \hat{Z}$.

If $\hat{L}_{ \pm}$and $\hat{R}_{ \pm}$are defined by $\hat{L}_{ \pm}=\hat{L}_x \pm i \hat{L}_y$ and $\hat{R}_{ \pm}=\hat{X} \pm i \hat{Y}$, prove the following commutators: $\left[\hat{L}_{ \pm}, \hat{Z}\right]=\mp \hbar \hat{X},\left[\hat{L}_z, \hat{R}_{\mp}\right]= \pm \hbar \hat{R}_{ \pm}$and $\left[\hat{L}_z, \hat{Z}\right]=0$.

Show that $\left[\hat{L}_z, \sin (2 \varphi)\right]=2 i \hbar\left(\sin ^2 \varphi-\cos ^2 \varphi\right)$, where $\varphi$ is the azimuthal angle. Hint: $[\hat{A}, \hat{B} \hat{C}]=\hat{B}[\hat{A}, \hat{C}]+[\hat{A}, \hat{B}] \hat{C}$

Using the properties of $\hat{J}_{+}$and $\hat{J}_{-}$, calculate $|j, \pm j\rangle$ and $|j, \pm m\rangle$ as functions of the action of $\hat{J}_{ \pm}$on the states $|j, \pm m\rangle$ and $|j, \pm j\rangle$, respectively.

Consider the operator $\hat{A}=\frac{1}{2}\left(\hat{J}_x \hat{J}_y+\hat{J}_y \hat{J}_x\right)$.
(a) Calculate the expectation value of $\hat{A}$ and $\hat{A}^2$ with respect of the state $|j, m\rangle$.
(b) Use the result of (a) to find an expression for $\hat{A}^2$ in terms of: $\hat{\vec{J}}^4, \hat{\vec{J}}^2, \hat{J}_z^2, \hat{J}_{+}^4, \hat{J}_{-}^4$.

Consider the expression $f(\theta, \varphi)=3 \sin \theta \cos \theta e^{i \varphi}-2\left(1-\cos ^2 \theta\right) e^{2 i \varphi}$.
(a) Write $f(\theta, \varphi)$ in terms of the spherical harmonics.
(b) Write the expression found in (a) in terms of the Cartesian coordinates.
(c) Is $f(\theta, \varphi)$ an eigenstate of $\hat{\vec{L}}^2$ or $\hat{L}_z$ ?
(d) Find the probability of measuring $2 \hbar$ for the $z$-component of the orbital angular momentum.

Find the expressions for the spherical harmonics $Y_{30}(\theta, \varphi)=\sqrt{7 / 16 \pi}\left(5 \cos ^3 \theta-3 \cos \theta\right)$ and $Y_{3, \pm 1}(\theta, \varphi)=\mp \sqrt{21 / 64 \pi} \sin \theta\left(5 \cos ^2 \theta-1\right) e^{ \pm i \varphi}$ in terms of the Cartesian coordinates $x, y, z$.

A particle of mass $m$ is fixed at one end of a rigid rod of negligible mass and length $R$. The other end of the rod rotates in the $x y$ plane about a bearing located at the origin, whose axis is in the $z$-direction.
(a) Write the the system's total energy in terms of its angular momentum $L$.
(b) Write down the time-independent Schrödinger equation of the system. Hint: In spherical coordinates, only $\varphi$ varies.
(c) Solve for the possible energy levels of the system, in terms of $m$ and the moment of inertia $I=m R^2$.
(d) Explain why there is no zero-point energy.

Consider a system which is described by the state
$$
\psi(\theta, \varphi)=\sqrt{\frac{3}{8}} Y_{11}(\theta, \varphi)+\sqrt{\frac{1}{8}} Y_{10}(\theta, \varphi)+A Y_{1,-1}(\theta, \varphi),
$$
where $A$ is a real constant
(a) Calculate $A$ so that $|\psi\rangle$ is normalized.
(b) Find $\hat{L}_{+} \psi(\theta, \varphi)$.
(c) Calculate the expectation values of $\hat{L}_x$ and $\hat{\vec{L}}^2$ in the state $|\psi\rangle$.
(d) Find the probability associated with a measurement that gives zero for the $z$-component of the angular momentum.
(e) Calculate $\left\langle\Phi\left|\hat{L}_z\right| \psi\right\rangle$ and $\left\langle\Phi\left|\hat{L}_{-}\right| \psi\right\rangle$ where
$$
\Phi(\theta, \varphi)=\sqrt{\frac{8}{15}} Y_{21}(\theta, \varphi)+\sqrt{\frac{4}{15}} Y_{10}(\theta, \varphi)+\sqrt{\frac{3}{15}} Y_{2,-1}(\theta, \varphi) .
$$

(a) Using the commutation relations of angular momentum, verify the validity of the (Jacobi) identity: $\left[\hat{J}_x,\left[\hat{J}_y, \hat{J}_z\right]\right]+\left[\hat{J}_y,\left[\hat{J}_z, \hat{J}_x\right]\right]+\left[\hat{J}_z,\left[\hat{J}_x, \hat{J}_y\right]\right]=0$.
(b)Prove the following identity: $\left[\hat{J}_x^2, \hat{J}_y^2\right]=\left[\hat{J}_y^2, \hat{J}_z^2\right]=\left[\hat{J}_z^2, \hat{J}_x^2\right]$.
(c) Calculate the expressions of $\hat{L}_{-} \hat{L}_{+} Y_{l m}(\theta, \varphi)$ and $\hat{L}_{+} \hat{L}_{-} Y_{l m}(\theta, \varphi)$, and then infer the commutator $\left[\hat{L}_{+} \hat{L}_{-}, \hat{L}_{-} \hat{L}_{+}\right] Y_{l m}(\theta, \varphi)$.

PK

(a) Show the following commutation relations:
$$
\begin{array}{lll}
{\left[\hat{P}_y, \hat{L}_y\right]=0,} & {\left[\hat{P}_y, \hat{L}_z\right]=i \hbar \hat{P}_x,} & {\left[\hat{P}_y, \hat{L}_x\right]=-i \hbar \hat{P}_z,} \\
{\left[\hat{P}_z, \hat{L}_z\right]=0,} & {\left[\hat{P}_z, \hat{L}_x\right]=i \hbar \hat{P}_y,} & {\left[\hat{P}_z, \hat{L}_y\right]=-i \hbar \hat{P}_x .}
\end{array}
$$
(b) Use the results of (a) to infer by means of a cyclic permutation the expressions for $\left[\hat{P}_x, \hat{L}_x\right],\left[\hat{P}_x, \hat{L}_y\right]$ and $\left[\hat{P}_x, \hat{L}_z\right]$.
(c) Use the results of (a) and (b) to calculate $\left[\hat{P}^2, \hat{L}_x\right],\left[\hat{R}^2, \hat{L}_y\right]$ and $\left[\hat{R}^2, \hat{L}_x\right]$, where $\hat{P}^2=\hat{P}_x^2+\hat{P}_y^2+\hat{P}_z^2$

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Consider a particle whose wave function is given by $\psi(x, y, z)=A\left[(x+z) y+z^2\right] / r^2-A / 3$, where $A$ is a constant.
(a) Is $\psi$ an eigenstate of $\hat{\vec{L}}^2$ ? If yes, what is the corresponding eigenvalue? Is it also an eigenstate of $\hat{L}_z$ ?
(b) Find the constant $A$ so that $\psi$ is normalized.
(c) Find the relative probabilities for measuring the various values of $\hat{L}_z$ and $\hat{L}^2$, and then calculate the expectation values of $\hat{L}_z$ and $\hat{\vec{L}}^2$.
(d) Calculate $\hat{L}_{ \pm}|\psi\rangle$ and then infer $\left\langle\psi\left|\hat{L}_{ \pm}\right| \psi\right\rangle$.

Consider a system which is in the state
$$
\psi(\theta, \varphi)=\sqrt{\frac{2}{13}} Y_{3,-3}+\sqrt{\frac{3}{13}} Y_{3,-2}+\sqrt{\frac{3}{13}} Y_{30}+\sqrt{\frac{3}{13}} Y_{3,-2}+\sqrt{\frac{2}{13}} Y_{33} .
$$
(a) If $\hat{L}_z$ were measured, what values will one obtain and with what probabilities?
(b) If after a measurement of $\hat{L}_z$ we find $l_z=2 \hbar$, calculate the uncertainties $\Delta L_x$ and $\Delta L_y$ and their product $\Delta L_x \Delta L_y$.
(a) Find $\left\langle\psi\left|\hat{L}_x\right| \psi\right\rangle$ and $\left\langle\psi\left|\hat{L}_y\right| \psi\right\rangle$.

Consider a system of total angular momentum $j=1$. We are interested here in the measurement of $\hat{J}_y$; its matrix is given by
$$
\hat{J}_y=\frac{\hbar}{\sqrt{2}}\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & -i \\
0 & i & 0
\end{array}\right) .
$$
(a) What are the possible values will we obtain when measuring $\hat{J}_y$ ?
(b) Calculate $\left\langle\hat{J}_z\right\rangle,\left\langle\hat{J}_z^2\right\rangle$ and $\Delta J_z$ if the system is in the state $j_y=\hbar$.
(c) Repeat (b) for $\left\langle\hat{J}_x\right\rangle,\left\langle\hat{J}_x^2\right\rangle$ and $\Delta J_x$.

Consider a system of total angular momentum $j=1$. We want to carry out measurements on
$$
\hat{J}_z=\hbar\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1
\end{array}\right) .
$$
(a) What are the possible values will we obtain when measuring $\hat{J}_z$ ?
(b) Calculate $\left\langle\hat{J}_x\right\rangle,\left\langle\hat{J}_x^2\right\rangle$ and $\Delta J_x$ if the system is in the state $j_z=-\hbar$.
(c) Repeat (b) for $\left\langle\hat{J}_y\right\rangle,\left\langle\hat{J}_y^2\right\rangle$ and $\Delta J_y$.

(a) Find the eigenvalues and eigenstates of the spin operator $\vec{S}$ of an electron in the direction of a unit vector $\vec{n}$; assume that $\vec{n}$ lies in the $y z$ plane.
(b) Find the probability of measuring $\hat{S}_z=-\hbar / 2$.
(c) Assuming that the eigenvectors of the spin calculated in (a) correspond to $t=0$, find these eigenvectors at time $t$.

Consider a system whose spin direction is in the $x y$ plane.
(a) Find the eigenvalues and eigenstates of the spin operator $\vec{S}$ of an electron in the direction of a unit vector $\vec{n}$; assume that $\vec{n}$ lies in the $x y$ plane.
(b) Find the probability of measuring $\hat{S}_z=\hbar / 2$.
(c) Assuming that the eigenvectors of the spin calculated in (a) correspond to $t=0$, find these eigenvectors at time $t$.

Consider a system whose wave function is $\psi(x, y, z)=\frac{1}{2} Y_{00}+\frac{1}{\sqrt{3}} Y_{11}+\frac{1}{2} Y_{1,-1}+\frac{1}{\sqrt{6}} Y_{22}$.
(a) Is $\psi(x, y, z)$ normalized?
(b) Is $\psi(x, y, z)$ an eigenstate of $\hat{\hat{L}^2}$ or $\hat{L}_z$ ?
(c) Calculate $\hat{L}_{ \pm} \psi(x, y, z)$ and $\left\langle\psi\left|\hat{L}_{ \pm}\right| \psi\right\rangle$.
(d) If a measurement of the $z$-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results $0, \hbar,-\hbar$ and $2 \hbar$.

Using the expression of $\hat{L}_{-}$in spherical coordinates, show the following two commutators: $\left[\hat{L}_{-}, e^{-i \varphi} \sin \theta\right]=0$ and $\left[\hat{L}_{-}, \cos \theta\right]=\hbar e^{-i \varphi} \sin \theta$

Consider a particle whose angular momentum is $l=1$.
(a) Find the eigenvalues and eigenvectors, $\left|1, m_x\right\rangle$, of $\hat{L}_x$.
(b) Express the state $\left|1, m_x=1\right\rangle$ as a linear superposition of the eigenstates of $\hat{L}_z$. Hint: you need first to find the eigenstates of $L_x$ and find which of them corresponds to the eigenvalue $m_x=1$; this eigenvector will be expanded in the $z$ basis.
(c) What is the probability of measuring $m_z=1$ when the particle in the eigenstate $\left|1, m_x=1\right\rangle$ ? What about the probability corresponding to measuring $m_z=0$ ?
(c) Suppose that a measurement of the $z$-component of angular momentum is performed, and that the result $m_z=1$ is obtained. Now we measure the $x$-component of angular momentum. What are the possible results and with what probabilities?

Make polar plots in the $x z$ plane with the square magnitude $\left|Y_{l m}\right|^2$ as the distance from the origin as a function of $\theta$ for the cases $l=1$ and $l=2$ for all values of $m$.

Consider a system which is given in the following angular momentum eigenstates $|l, m\rangle$ :
$$
|\psi\rangle=\frac{1}{\sqrt{7}}|1,-1\rangle+A|1,0\rangle+\sqrt{\frac{2}{7}}|1,1\rangle,
$$
where $A$ is a real constant
(a) Calculate $A$ so that $|\psi\rangle$ is normalized.
(b) Calculate the expectation values of $\hat{L}_x, \hat{L}_y, \hat{L}_z$, and $\hat{\vec{L}}^2$ in the state $|\psi\rangle$.
(c) Find the probability associated with a measurement that gives $1 \hbar$ for the $z$-component of the angular momentum.
(d) Calculate $\left\langle 1, m\left|\hat{L}_{+}^2\right| \psi\right\rangle$ and $\left\langle 1, m\left|\hat{L}_{-}^2\right| \psi\right\rangle$.

Consider the Pauli matrices
$$
\sigma_x=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right), \quad \sigma_y=\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right) \quad \sigma_z=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right) .
$$
(a) Verify that $\sigma_x^2=\sigma_y^2=\sigma_z^2=I$, where $I$ is the unit matrix
$$
I=\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)
$$
(b) Calculate the commutators $\left[\sigma_x, \sigma_y\right],\left[\sigma_x, \sigma_z\right]$, and $\left[\sigma_y, \sigma_z\right]$.
(c) Calculate the anticommutator $\sigma_x \sigma_y+\sigma_y \sigma_x$.
(d) Show that $e^{i \theta \sigma_y}=I \cos \theta+i \sigma_y \sin \theta$, where $I$ is the unit matrix.
(e) Derive an expression for $e^{i \theta \sigma_z}$ by analogy with the one for $\sigma_z$.

Consider a spin $\frac{3}{2}$ particle whose Hamiltonian is given by
$$
\hat{H}=\frac{\varepsilon_0}{\hbar^2}\left(\hat{S}_x^2-\hat{S}_y^2\right)-\frac{\varepsilon_0}{\hbar^2} \hat{S}_z^2
$$
where $\varepsilon_0$ is a constant having the dimensions of energy.
(a) Find the matrix of the Hamiltonian and diagonalize it to find the energy levels.
(b) Find the eigenvectors and verify that the energy levels are doubly degenerate.