Quantum Mechanics

Leonard I Schiff

Chapter 7 Approximation Methods For Stationary Piroislems - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that Eq. (26.6) is valid, making use of the form $(26.4)$ for $U_{r}(e)$ and the commutation relations for the components of $\mathbf{r}$ and $\mathbf{p}$.

Willis James

Numerade Educator

00:59

Make use of the invariance of the scalar product of any two vectors under rotations, in order to show that the rows and columns of the rotation matrix $R$ are respectively orthonormal to each other. Show also that the transpose of $R$ is equal to the inverse of $R$ and that the determinant of $R$ is equal to $\pm 1 .$

Raj Bala

Numerade Educator

02:10

Problem 3

Show that the three commutation relations (27.14) are valid, making use of the form (27.7) for $\mathrm{L}$ and the commutation relations for the components of $\mathrm{r}$ and $\mathrm{p}$.

Willis James

Numerade Educator

01:52

Problem 4

Show that the three matrices S defined in Eqs. (27.11) satisfy the relations $\mathrm{S} \times \mathrm{S}=i \hbar \mathrm{S}$. Show also that $\mathrm{S}^{2}=2 \hbar^{2}$

Hoan Nguyen

Numerade Educator

03:34

Problem 5

Show that the matrix elements of $\mathrm{r}$ for states that are rotated through the infinitesimal vector $\phi$ are equal to the corresponding matrix elements of $\mathbf{r}_{R}=\mathbf{r}+\phi \times \mathbf{r}$ for the original states.

Manisha Sarker

Numerade Educator

View

Problem 6

Show that the matrix elements of $J$ for states that are rotated through the infinitesimal vector $\phi$ are equal to the corresponding matrix elements of $\mathrm{J}_{R}=\mathrm{J}+\boldsymbol{\phi} \times \mathrm{J}$ for the original states.

Nick Johnson

Numerade Educator

01:11

Problem 7

Show that the eigenvalues of $S_{z}$ given in (27.11) are the same as those of $J_{z}$ given in $(27.26)$ for $j=1 .$ Then find the most general unitary matrix that transforms $S_{z}$ into $J_{z}: U S_{z} U^{\dagger}=J_{z} .$ Choose the arbitrary parameters in $U$ so that it also transforms $S_{x}$ into $J_{x}$ and $S_{y}$ into $J_{y} .$ Into what does this $U$ transform the vector wave function $\psi_{\alpha}$ of Eq. (27.8)?

Adriano Chikande

Numerade Educator

01:07

Problem 8

Establish Eq. (27.28) by using the definitions (27.27) and the properties of the spherical harmonics given in Sec. 14 .

Chai Santi

Numerade Educator

04:50

Problem 9

Obtain an explicit expression for $U_{R}(\phi)=\exp (-i \phi \cdot \mathrm{J} / \hbar)$ in the form of a $2 \times 2$ matrix when J is given by Eq. (27.26) with $j=\frac{1}{2}$. Let the vector $\phi$ have the magnitude $\phi$ and the polar angles $\theta$ and $\varphi$. Show explicitly that your matrix for $U_{R}(\phi)$ is unitary and that it is equal to $-1$ when $\phi=2 \pi$

Victor Salazar

Numerade Educator

03:14

Problem 10

Show that the matrices $\lambda_{i}(j=1, \ldots, 8)$ defined by Eqs. (27.37) and $(27.40)$ satisfy the commutation relations $(27.41)$ and $(27.42) .$ Then use these commutation relations (not the original matrix representation of the $\lambda_{i}$ ) to show that each $\lambda_{j}$ commutes with the Casimir operator $C$ defined by Eq. (27.43).

Nick Johnson

Numerade Educator

01:16

Problem 11

Show that the 28 commutators of the eight operators $(27.44)$, computed from the commutation relations between the components of $\mathbf{r}$ and $\mathbf{p}$, agree with the commutators of the $\lambda_{j}$ when the identifications $(27.45)$ and $(27.46)$ are adopted.

Lottie Adams

Numerade Educator

02:22

Problem 12

Use the methods and formulas of Sec. 28 to calculate the matrix of ClebschGordan coefficients in the case $j_{1}=\frac{3}{2}, j_{2}=\frac{1}{2} . \quad$ Compare your results with the wallet card.

Sam Limsuwannarot

Numerade Educator

02:39

Problem 13

A deuteron has spin 1. Use the Wigner-Eckart theorem to find the ratios of the expectation values of the electric quadrupole moment operator $Q(2,0)$ for the three orientations of the deuteron: $m=1,0,-1$

Keshav Singh

Numerade Educator

04:20

Problem 14

Show that the momentum inversion operator $U$ defined after Eq. (29.17), that has the property $U \psi_{\alpha}(\mathrm{p})=\psi_{\alpha}(-\mathrm{p})$, is unitary.

Hafiz Shahzaib

Numerade Educator

13:31

Problem 15

Show that, if $\psi_{\alpha^{\prime}}=T \psi_{\alpha}$ and $\psi_{\beta^{\prime}}=T \psi_{\beta}$, then $\left(\psi_{\alpha^{\prime}}, \psi_{\beta^{\prime}}\right)=\left(\psi_{\alpha, \psi} \psi\right)^{*}=\left(\psi_{\beta}, \psi_{\alpha}\right)$.
From this, show that the norm of a state vector is unchanged by time reversal.

Chris Trentman

Numerade Educator

02:40

Problem 16

Show explicitly that $U=e^{-i \pi s_{y} / \hbar}$ satisfies the two equations (29.19). Make use of the commutation properties of $S_{v}$ and the operators $S_{\pm} \equiv S_{z} \pm i S_{x}$

Lottie Adams

Numerade Educator

01:02

Problem 17

Show by direct expansion that, for $s=\frac{1}{2}, e^{-i \pi s_{y} / \hbar}=-i \sigma_{y}$

Raj Bala

Numerade Educator

12:01

Problem 18

Show by a general argument that $T^{2}=\pm 1 .$ Make use of the form $T=U K$ and of the fact that two successive time reversals take a state into itself so that $T^{2}$ is a multiple of the unit matrix. Do not make use of the particular form for $T$ given in Eq. (29.20).

Anas Venkitta

Numerade Educator

03:43

Problem 19

A charged particle with spin operator $S$ is assumed to possess an electric dipole moment operator $\mu \mathrm{S}$, where $\mu$ is a numerical constant, so that the hamiltonian for this particle in any electric field E contains the interaction term $-\mu \mathrm{S} \cdot \mathrm{E} .$ Show that neither space inversion nor time reversal is a symmetry operation for this particle moving in a spherically symmetric electrostatic potential $\phi(r)$, even when no external electric field is present.

Mahnoor Amin

Numerade Educator

02:58

Problem 20

Find the lowest energy eigenfunction of the hydrogen atom in the coordinate representation, starting from the work of Sec. 30. Proceed by finding the analog of Eq. (25.8) for the lowest state of the linear oscillator; then solve it in analogy with Eq. (25.14).

Yaqub Khan

Numerade Educator