Chapter 4, Transformations and observables Video Solutions, The Physics of Quantum Mechanics

Problem 1

Verify that $[\mathbf{J}, \mathbf{x} \cdot \mathbf{x}]=0$ and $[\mathbf{J}, \mathbf{x} \cdot \mathbf{p}]=0$ by using the commutation relations $\left[x_{i}, J_{j}\right]=\mathrm{i} \sum_{k} \epsilon_{i j k} x_{k}$ and $\left[p_{i}, J_{j}\right]=\mathrm{i} \sum_{k} \epsilon_{i j k} p_{k}$.

Alan Ghazarians

Numerade Educator

00:59

Problem 2

Show that the vector product $\mathbf{a} \times \mathbf{b}$ of two classical vectors transforms like a vector under rotations. Hint: A rotation matrix $\mathbf{R}$ satisfies the relations $\mathbf{R} \cdot \mathbf{R}^{T}=\mathbf{I}$ and $\operatorname{det}(\mathbf{R})=1$, which in tensor notation read $\sum_{p} R_{i p} R_{t p}=\delta_{i t}$ and $\sum_{i j k} \epsilon_{i j k} R_{i r} R_{j s} R_{k t}=\epsilon_{r s t}$

James Kiss

Numerade Educator

04:31

Problem 3

We have shown that $\left[v_{i}, J_{j}\right]=\mathrm{i} \sum_{k} \epsilon_{i j k} v_{k}$ for any operator whose components $v_{i}$ form a vector. The expectation value of this operator relation in any state $|\psi\rangle$ is then $\left\langle\psi\left|\left[v_{i}, J_{j}\right]\right| \psi\right\rangle=\mathrm{i} \sum_{k} \epsilon_{i j k}\left\langle\psi\left|v_{k}\right| \psi\right\rangle .$ Check that with $U(\boldsymbol{\alpha})=\mathrm{e}^{-\mathrm{i} \alpha \cdot \mathrm{J}}$ this relation is consistent under a further rotation $|\psi\rangle \rightarrow\left|\psi^{\prime}\right\rangle=U(\boldsymbol{\alpha})|\psi\rangle$ by evaluating both sides separately.

Melissa Munoz

Numerade Educator

03:44

Problem 4

The matrix for rotating an ordinary vector by $\phi$ around the $z$-axis is
$$
\mathbf{R}(\phi) \equiv\left(\begin{array}{ccc}
\cos \phi & -\sin \phi & 0 \\
\sin \phi & \cos \phi & 0 \\
0 & 0 & 1
\end{array}\right)
$$
By considering the form taken by $\mathbf{R}$ for infinitesimal $\phi$ calculate from $\mathbf{R}$ the matrix $\mathcal{J}_{z}$ that appears in $\mathbf{R}(\phi)=\exp \left(-\mathrm{i} \mathcal{J}_{z} \phi\right)$. Introduce new coordinates $u_{1} \equiv(-x+\mathrm{i} y) / \sqrt{2}, u_{2}=z$ and $u_{3} \equiv(x+\mathrm{i} y) / \sqrt{2}$. Write down the matrix $\mathbf{M}$ that appears in $\mathbf{u}=\mathbf{M} \cdot \mathbf{x}$ [where $\mathbf{x} \equiv(x, y, z)]$ and show that it is unitary. Then show that
$$
\mathcal{J}_{z}^{\prime} \equiv \mathbf{M} \cdot \mathcal{J}_{z} \cdot \mathbf{M}^{\dagger}
$$
is identical with $S_{z}$ in the set of spin-one Pauli analogues
$$
S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{array}\right), \quad S_{y}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}
0 & -\mathrm{i} & 0 \\
\mathrm{i} & 0 & -\mathrm{i} \\
0 & \mathrm{i} & 0
\end{array}\right), \quad S_{z}=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -1
\end{array}\right)
$$

Adriano Chikande

Numerade Educator

01:53

Problem 5

Determine the commutator $\left[\mathcal{J}_{x}^{\prime}, \mathcal{J}_{z}^{\prime}\right]$ of the generators used in Problem 4.4. Show that it is equal to $-\mathrm{i} \mathcal{J}_{y}^{\prime}$, where $\mathcal{J}_{y}^{\prime}$ is identical with $S_{y}$ in the set $(4.75)$.

Lottie Adams

Numerade Educator

09:44

Problem 6

Show that if $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ are non-parallel vectors, $\boldsymbol{\alpha}$ is not invariant under the combined rotation $\mathbf{R}(\boldsymbol{\alpha}) \mathbf{R}(\boldsymbol{\beta})$. Hence show that
$$
\mathbf{R}^{\mathrm{T}}(\boldsymbol{\beta}) \mathbf{R}^{\mathrm{T}}(\boldsymbol{\alpha}) \mathbf{R}(\boldsymbol{\beta}) \mathbf{R}(\boldsymbol{\alpha})
$$
is not the identity operation. Explain the physical significance of this result.

Cullen Miller

Numerade Educator

01:14

Problem 7

In this problem you derive the wavefunction
$$
\langle\mathbf{x} \mid \mathbf{p}\rangle=\mathrm{e}^{\mathrm{ip} \cdot \mathbf{x} / \hbar}
$$
of a state of well-defined momentum from the properties of the translation operator $U(\mathbf{a}) .$ The state $|\mathbf{k}\rangle$ is one of well-defined momentum $\hbar \mathbf{k}$. How would you characterise the state $\left|\mathbf{k}^{\prime}\right\rangle \equiv U(\mathbf{a})|\mathbf{k}\rangle ?$ Show that the wavefunctions of these states are related by $u_{\mathbf{k}^{\prime}}(\mathbf{x})=\mathrm{e}^{-\mathrm{ia} \cdot \mathbf{k}} u_{\mathbf{k}}(\mathbf{x})$ and $u_{\mathbf{k}^{\prime}}(\mathbf{x})=u_{\mathbf{k}}(\mathbf{x}-\mathbf{a}) .$ Hence obtain equation $(4.76)$

Ren Jie Tuieng

Numerade Educator

03:12

Problem 8

By expanding the anticommutator on the left and then applying the third rule of the set $(2.22)$, show that any three operators satisfy the identity
$$
[\{A, B\}, C]=\{A,[B, C]\}+\{[A, C], B\}
$$

Supratim Pal

Numerade Educator

11:41

Problem 9

Define $G$ in terms of the parity operator $P$ by
$$
G \equiv \frac{1}{2}(1-P)
$$
Show that $G$ is Hermitian and that $G^{n}=G$ for positive integer $n$. Explain this result in terms of the eigenkets and eigenvalues of $G$. Show further that $P=U(\pi)$ where $U(s) \equiv \mathrm{e}^{\mathrm{i} s G}$.

Dr. Rajveer Singh

Numerade Educator

01:42

Problem 10

Let $P$ be the parity operator and $S$ an arbitrary scalar operator. Explain why $P$ and $S$ must commute.

Dominador Tan

Numerade Educator

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Problem 11

In this problem we consider discrete transformations other than that associated with parity. Let $\mathcal{S}$ be a linear transformation on ordinary three-dimensional space that effects a reflection in a plane. Let $S$ be the associated operator on kets. Explain the physical relationship between the kets $|\psi\rangle$ and $\left|\psi^{\prime}\right\rangle \equiv S|\psi\rangle .$ Explain why we can write
$$
\mathcal{S}\langle\psi|\mathbf{x}| \psi\rangle=\left\langle\psi\left|S^{\dagger} \mathbf{x} S\right| \psi\right\rangle
$$
What are the possible eigenvalues of $S ?$
Given that $\mathcal{S}$ reflects in the plane through the origin with unit normal $\hat{\mathbf{n}}$, show, by means of a diagram or otherwise, that its matrix is given by
$$
\mathcal{S}_{i j}=\delta_{i j}-2 n_{i} n_{j}
$$
Determine the form of this matrix in the case that $\mathbf{n}=(1,-1,0) / \sqrt{2}$. Show that in this case $S x=y S$ and give an alternative expression for $S y$. Show that a potential of the form
$$
V(\mathbf{x})=f(R)+\lambda x y, \quad \text { where } \quad R \equiv \sqrt{x^{2}+y^{2}}
$$
satisfies $V(\mathcal{S} \mathbf{x})=V(\mathbf{x})$ and explain the geometrical significance of this equation. Show that $[S, V]=0$. Given that $E$ is an eigenvalue of $H=$ $p^{2} / 2 m+V$ that has a unique eigenket $|E\rangle$, what equation does $|E\rangle$ satisfy in addition to $H|E\rangle=E|E\rangle ?$

Victor Salazar

Numerade Educator

04:17

Problem 12

Show that the operator defined by $\langle x, y|S| \psi\rangle=\langle y, x \mid \psi\rangle$ is Hermitian.

Victor Salazar

Numerade Educator