# Chapter 13 The Hydrogen Atom
## 13.1 The Eigenvalue Problem
**Exercise 13.1.1** Derive Eqs. (13.1.11) and (13.1.14) starting from Eqs. (13.1.8)-(13.1.10).



**Exercise 13.1.2** Derive the degeneracy formula, Eq. (13.1.18).

**Exercise 13.1.3** Starting from the recursion relation, obtain $\psi_{210}$ (normalized).

**Exercise 13.1.4** Recall from the last chapter [Eq. (12.6.19)] that as $r \rightarrow \infty$, $U_E \sim(r)^{m e^2 / \kappa \hbar^2} \mathrm{e}^{-\kappa r}$ in a Coulomb potential $V=-\mathrm{e}^2 / r$ $\left[\kappa=\left(2 m W / \hbar^2\right)^{1 / 2}\right]$. Show that this agrees with Eq. (13.1.26).

**Exercise 13.1.5** (Virial Theorem) Since $|n, l, m\rangle$ is a stationary state, $\langle\dot{\mathbf{\Omega}}\rangle=0$ for any $\mathbf{\Omega}$.\\ Consider $\mathbf{\Omega}=\mathbf{R} \cdot \mathbf{P}$ and use Ehrenfest's theorem to show that $\langle T\rangle=(-1 / 2)\langle V\rangle$ in the state $|n, l, m\rangle$.

## 13.2 The Degeneracy of the Hydrogen Spectrum
**Exercise 13.2.1** Let us see why the conservation of the Runge-Lenz vector $\mathbf{n}$ implies closed orbits.

(1) Express $\mathbf{n}$ in terms of $\mathbf{r}$ and $\mathbf{p}$ alone (get rid of $\mathbf{1}$).

(2) Since the particle is bound, it cannot escape to infinity. So, as we follow it from some arbitrary time onward, it must reach a point $\mathbf{r}_{\max}$ where its distance from the origin stops growing. Show that
$$
\mathbf{n}=\mathbf{r}_{\max}\left(2 E+\frac{\mathrm{e}^2}{r_{\max}}\right)
$$
## 13.3 Numerical Estimates and Comparison with Experiment
**Exercise 13.3.1** The pion has a range of $1~\mathrm{Fermi}=10^{-5} \text{\AA}$ as a mediator of nuclear force. Estimate its rest energy.

**Exercise 13.3.2** Estimate the de Broglie wavelength of an electron of kinetic energy $200~\mathrm{eV}$. (Recall $\lambda=2 \pi \hbar / p$.)
## 13.4 Multielectron Atoms and the Periodic Table
**Exercise 13.4.1** Show that if we ignore interelectron interactions, the energy levels of a multielectron atom go as $Z^2$. Since the Coulomb potential is $Z e / r$, why is the energy $\propto Z^2$?

**Exercise 13.4.2** Compare (roughly) the sizes of the uranium atom and the hydrogen atom. Assume levels fill in the order of increasing $n$, and that the nonrelativistic description holds. Ignore interelectron effects.

**Exercise 13.4.3** Visible light has a wavelength of approximately $5000 \text{\AA}$. Which of the series-Lyman, Balmer, Paschen-do you think was discovered first?

Chapter 13 The Hydrogen Atom

Chapter 12 Rotational Invariance and Angular Momentum

# Chapter 11 Symmetries and Their Consequences
## 11.1 Overview
## 11.2 Translational Invariance in Quantum Theory
**Exercise 11.2.1** Verify Eq. (11.2.11b).



**Exercise 11.2.2** Using $T^{\dagger}(\varepsilon)T(\varepsilon)=I$ to order $\varepsilon$, deduce that $G^{\dagger}=G$.

**Exercise 11.2.3** Recall that we found the finite rotation transformation from the infinitesimal one, by solving differential equations (Section 2.8). Verify that if, instead, you relate the transformed coordinates $\overline{x}$ and $\overline{y}$ to $x$ and $y$ by the infinite string of Poisson brackets, you get the same result, $\overline{x}=x \cos \theta-y \sin \theta$, etc. (Recall the series for $\sin \theta$, etc.)
## 11.3 Time Translational Invariance
## 11.4 Parity Invariance
**Exercise 11.4.1** Prove that if $[\Pi, H]=0$, a system that starts out in a state of even/odd parity maintains its parity. (Note that since parity is a discrete operation, it has no associated conservation law in classical mechanics.)

**Exercise 11.4.2** A particle is in a potential
$$
V(x)=V_0 \sin (2 \pi x / a)
$$
which is invariant under the translations $x \rightarrow x+m a$, where $m$ is an integer. Is momentum conserved? Why not?

**Exercise 11.4.3** You are told that in a certain reaction, the electron comes out with its spin always parallel to its momentum. Argue that parity is violated.

**Exercise 11.4.4** We have treated parity as a mirror reflection. This is certainly true in one dimension, where $x \rightarrow-x$ may be viewed as the effect of reflecting through a (point) mirror at the origin. In higher dimensions when we use a plane mirror (say lying on the $x-y$ plane), only one ($z$) coordinate gets reversed, whereas the parity transformation reverses all three coordinates.

Verify that reflection on a mirror in the $x-y$ plane is the same as parity followed by $180^{\circ}$ rotation about the $z$ axis. Since rotational invariance holds for weak interactions, noninvariance under mirror reflection implies noninvariance under parity.
## 11.5 Time-Reversal Symmetry

Chapter 11 Symmetries and Their Consequences

# Chapter 10 Systems with $N$ Degrees of Freedom
## 10.1 $N$ Particles in One Dimension
**Exercise 10.1.1** Show the following:

(1)
$$
[\Omega_{1}^{(1)}\otimes I^{(2)}, I^{(1)}\otimes \Lambda_{2}^{(2)}]=0~\text{for any}~\Omega_{1}^{(1)}~\text{and}~\Lambda_{2}^{(2)}\tag{10.1.17a}
$$

(2)
$$
(\Omega_{1}^{(1)}\otimes\Gamma_{2}^{(2)})(\theta_{1}^{(1)}\otimes\Lambda_{2}^{(2)})=(\Omega\theta)_{1}^{(1)}\otimes(\Gamma\Lambda)_{2}^{(2)}\tag{10.1.17b}
$$

(3)If
$$
[\Omega_{1}^{(1)},\Lambda_{1}^{(1)}]=\Gamma_{1}^{(1)}
$$
then
$$
[\Omega_{1}^{(1)\otimes(2)},\Lambda_{1}^{(1)\otimes(2)}]=\Gamma_{1}^{(1)}\otimes I^{(2)}\tag{10.1.17c}
$$
and similarly with $1\to 2$.

(4)
$$
(\Omega_{1}^{(1)\otimes (2)}+\Omega_{2}^{(1)\otimes(2)})^{2}=(\Omega_{1}^{2})^{(1)}\otimes I^{(2)}+I^{(1)}\otimes (\Omega_{2}^{2})^{(2)}+2\Omega_{1}^{(1)}\otimes\Omega_{2}^{(2)}\tag{10.1.17d}
$$



**Exercise 10.1.2** Imagine a fictitious world in which the single-particle Hilbert space is two-dimensional. Let us denote the basis vectors by $|+\rangle$ and $|-\rangle$. Let
![image](https://i.upmath.me/svgb/U1HhiinOTM9NjDeMq9Yw1Ky15YpJyi9KSS3KTSwpyqyo5lJT0FZQU9BViEku4gKxEoE4CczTBbKSgTiFqzamsDQxJaYktaKkOjEvBcqFmGsENNcIaC4BY1OBOA1ubDoQZ3DVcqmoAAA)

be operators in $\mathbb{V}_1$ and $\mathbb{V}_2$, respectively (the $\pm$ signs label the basis vectors. Thus $b=\langle+| \sigma_1^{(1)}|-\rangle$ etc.) The space $\mathbb{V}_1 \otimes \mathbb{V}_2$ is spanned by four vectors $|+\rangle \otimes|+\rangle$, $|+\rangle \otimes|-\rangle$, $|-\rangle\otimes|+\rangle$, $|-\rangle\otimes|-\rangle$. Show (using the method of images or otherwise) that

(1) ![image](https://i.upmath.me/svgb/U4kpzkzPTYyvNqyNq9Yw1IzJL8nMTS3WMNKstUWTqoXKKXgCuWD5pPyilNSi3MSSosyKai41BW1tBSChCyR0QSxd3ZjkIi6wYCIQGwBxEpgGC-tChRBSIGGwxmSoUApctS5MNVyKq1YFAA)

(Recall that $\langle\alpha|\otimes\langle\beta|$ is the bra corresponding to $|\alpha\rangle\otimes|\beta\rangle$.)

(2) $\sigma_{2}^{(1)\otimes(2)}=\begin{pmatrix}
e & f & 0 & 0\\
g & h & 0 & 0\\
0 & 0 & e & f\\
0 & 0 & g & h
\end{pmatrix}$

(3) $(\sigma_{1}\sigma_{2})^{(1)\otimes(2)}=\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}=\begin{pmatrix}
ae & af & be & bf\\
ag & ah & bg & bh\\
ce & cf & de & df\\
cg & ch & dg & dh
\end{pmatrix}$

Do part (3) in two ways, by taking the matrix product of $\sigma_{1}^{(1)\otimes (2)}$ and $\sigma_{2}^{(1)\otimes (2)}$ and by directly computing the matrix elements of $\sigma_{1}^{(1)}\otimes\sigma_{2}^{(2)}$.

**Exercise 10.1.3** Consider the Hamiltonian of the coupled mass system:
$$
\mathscr{H}=\dfrac{p_{1}^{2}}{2m}+\dfrac{p_{2}^{2}}{2m}+\dfrac{1}{2}m\omega^{2}[x_{1}^{2}+x_{2}^{2}+(x_{1}-x_{2})^{2}]
$$
We know from Example 1.8.6 that $\mathscr{H}$ can be decoupled if we use normal coordinates
$$
x_{\text{I},\text{II}}=\dfrac{x_{1}\pm x_{2}}{2^{1/2}}
$$
and the corresponding momenta
$$
p_{\text{I},\text{II}}=\dfrac{p_{1}\pm p_{2}}{2^{1/2}}
$$

(1) Rewrite $\mathscr{H}$ in terms of normal coordinates. Verify that the normal coordinates are also canonical, i.e., that
$$
\{x_{i},p_{j}\}=\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
Now quantize the system, promoting these variables to operators obeying
$$
[X_{i},P_{j}]=\mathrm{i}\hbar\delta_{ij}~\text{etc.;}\qquad i,j=\text{I},\text{II}
$$
Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{\text{I}}$ and $X_{\text{II}}$.

(2) Quantize the system directly, by promoting $x_{1}$, $x_{2}$, $p_{1}$, and $p_{2}$ to quantum operators. Write the eigenvalue equation for $H$ in the simultaneous eigenbasis of $X_{1}$ and $X_{2}$. Now change from $x_{1}$, $x_{2}$ (and of course $\partial/\partial x_{1}$, $\partial/\partial x_{2}$) to $x_{\text{I}}$, $x_{\text{II}}$ (and $\partial/\partial x_{\text{I}}$, $\partial/\partial x_{\text{II}}$) in the differential equation. You should end up with the result form part (1).
## 10.2 More Particles in More Dimensions
**Exercise 10.2.1** (Particle in a Three-Dimensional Box)
Recall that a particle in a one-dimensional box extending from $x=0$ to $L$ is confined to the region $0 \leqslant x \leqslant L$; its wave function vanishes at the edges $x=0$ and $L$ and beyond (Exercise 5.2.5). Consider now a particle confined in a three-dimensional cubic box of volume $L^3$. Choosing as the origin one of its corners, and the $x, y$, and $z$ axes along the three edges meeting there, show that the normalized energy eigenfunctions are
$$
\psi_E(x, y, z)=\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_x \pi x}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_y \pi y}{L}\right)\left(\frac{2}{L}\right)^{1 / 2} \sin \left(\frac{n_z \pi z}{L}\right)
$$
where
$$
E=\frac{\hbar^2 \pi^2}{2 M L^2}\left(n_x^2+n_y^2+n_z^2\right)
$$
and $n_i$ are positive integers.

**Exercise 10.2.2** Quantize the two-dimensional oscillator for which
$$
\mathscr{H}=\frac{p_x^2+p_y^2}{2 m}+\frac{1}{2} m \omega_x^2 x^2+\frac{1}{2} m \omega_y^2 y^2
$$
(1) Show that the allowed energies are
$$
E=\left(n_x+1 / 2\right) \hbar \omega_x+\left(n_y+1 / 2\right) \hbar \omega_y, \quad n_x, n_y=0,1,2, \ldots
$$
(2) Write down the corresponding wave functions in terms of single oscillator wave functions. Verify that they have definite parity (even/odd) number $x \rightarrow-x, y \rightarrow-y$ and that the parity depends only on $n=n_x+n_y$.

(3) Consider next the isotropic oscillator $\left(\omega_x=\omega_y\right)$. Write explicit, normalized eigenfunctions of the first three states (that is, for the cases $n=0$ and 1). Reexpress your results in terms of polar coordinates $\rho$ and $\phi$ (for later use). Show that the degeneracy of a level with $E=(n+1) \hbar \omega$ is $(n+1)$.

**Exercise 10.2.3** Quantize the three-dimensional isotropic oscillator for which
$$
\mathscr{H}=\frac{p_x^2+p_y^2+p_z^2}{2 m}+\frac{1}{2} m \omega^2\left(x^2+y^2+z^2\right)
$$
(1) Show that $E=(n+3 / 2) \hbar \omega$; $n=n_x+n_y+n_z$; $n_x, n_y, n_z=0,1,2, \ldots$.

(2) Write the corresponding eigenfunctions in terms of single-oscillator wave functions and verify that the parity of the level with a given $n$ is $(-1)^n$. Reexpress the first four states in terms of spherical coordinates. Show that the degeneracy of a level with energy $E=$ $(n+3 / 2) \hbar \omega$ is $(n+1)(n+2) / 2$.
## 10.3 Identical Particles
**Exercise 10.3.1** Two identical bosons are found to be in states $|\phi\rangle$ and $|\psi\rangle$. Write down the normalized state vector describing the system when $\langle\phi \mid \psi\rangle \neq 0$.

**Exercise 10.3.2** When an energy measurement is made on a system of three bosons in a box, the $n$ values obtained were $3$, $3$, and $4$. Write down a symmetrized, normalized state vector.

**Exercise 10.3.3** Imagine a situation in which there are three particles and only three states $a$, $b$, and $c$ available to them. Show that the total number of allowed, distinct configurations for this system is

(1) 27 if they are labeled

(2) 10 if they are bosons

(3) 1 if they are fermions

**Exercise 10.3.4** Two identical particles of mass $m$ are in a one-dimensional box of length $L$. Energy measurement of the system yields the value $E_{\text{sys}}=\hbar^2 \pi^2/mL^2$. Write down the state vector of the system. Repeat for $E_{\text{sys}}=5 \hbar^2 \pi^2 / 2 m L^2$. (There are two possible vectors in this case.) You are not told if they are bosons or fermions. You may assume that the only degrees of freedom are orbital.

**Exercise 10.3.5** Consider the exchange operator $P_{12}$ whose action on the $X$ basis is
$$
P_{12}\left|x_1, x_2\right\rangle=\left|x_2, x_1\right\rangle
$$
(1) Show that $P_{12}$ has eigenvalues $\pm 1$. (It is Hermitian and unitary.)

(2) Show that its action on the basis ket $\left|\omega_1, \omega_2\right\rangle$ is also to exchange the labels $1$ and $2$, and hence that $\mathbb{V}_{S/A}$ are its eigenspaces with eigenvalues $\pm 1$.

(3) Show that $P_{12} X_1 P_{12}=X_2$, $P_{12} X_2 P_{12}=X_1$ and similarly for $P_1$ and $P_2$. Then show that $P_{12} \Omega\left(X_1, P_1 ; X_2, P_2\right) P_{12}=\Omega\left(X_2, P_2 ; X_1, P_1\right)$. [Consider the action on $\left|x_1, x_2\right\rangle$ or $\left|p_1, p_2\right\rangle$. As for the functions of $X$ and $P$, assume they are given by power series and consider any term in the series. If you need help, peek into the discussion leading to Eq. (11.2.22).]

(4) Show that the Hamiltonian and propagator for two identical particles are left unaffected under $H \rightarrow P_{12} H P_{12}$ and $U \rightarrow P_{12} U P_{12}$. Given this, show that any eigenstate of $P_{12}$ continues to remain an eigenstate with the same eigenvalue as time passes, i.e., elements of $\mathbb{V}_{S/A}$ never leave the symmetric or antisymmetric subspaces they start in.

**Exercise 10.3.6** Consider a composite object such as the hydrogen atom. Will it behave as a boson or fermion? Argue in general that objects containing an even/odd number of fermions will behave as bosons/fermions.

Chapter 10 Systems with N Degrees of Freedom

# Chapter 9 The Heisenberg Uncertainty Relations
## 9.1 Introduction
## 9.2 Derivation of the Uncertainty Relations
## 9.3 The Minimum Uncertainty Packet
## 9.4 Applications of the Uncertainty Principle
**Exercise 9.4.1** Consider the oscillator in the state $|n=1\rangle$ and verify that
$$
\left\langle\frac{1}{X^2}\right\rangle \simeq \frac{1}{\left\langle X^2\right\rangle} \simeq \frac{m \omega}{\hbar}
$$



**Exercise 9.4.2** (1) By referring to the table of integrals in Appendix A.2, verify that
$$
\psi=\frac{1}{\left(\pi a_0^3\right)^{1 / 2}} e^{-r / a_0}, \quad r=\left(x^2+y^2+z^2\right)^{1 / 2}
$$
is a normalized wave function (of the ground state of hydrogen). Note that in three dimensions the normalization condition is
$$
\begin{aligned}
\langle\psi \mid \psi\rangle & =\int \psi^*(r, \theta, \phi) \psi(r, \theta, \phi) r^2 \mathrm{d} r \mathrm{d}(\cos \theta) \mathrm{d} \phi \\
& =4 \pi \int \psi^*(r) \psi(r) r^2 \mathrm{d} r=1
\end{aligned}
$$
for a function of just $r$.

(2) Calculate $(\Delta X)^2$ in this state [argue that $(\Delta X)^2=\dfrac{1}{3}\left\langle r^2\right\rangle$] and 
regain the result quoted in Eq. (9.4.9).

(3) Show that $\langle 1 / r\rangle \simeq 1 /\langle r\rangle \simeq m e^2 / \hbar^2$ in this state.

**Exercise 9.4.3** Ignore the fact that the hydrogen atom is a three-dimensional system and pretend that
$$
H=\frac{P^2}{2 m}-\frac{e^2}{\left(R^2\right)^{1 / 2}} \quad\left(P^2=P_x^2+P_y^2+P_z^2, R^2=X^2+Y^2+Z^2\right)
$$
corresponds to a one-dimensional problem. Assuming
$$
\Delta P \cdot \Delta R \geqslant \hbar / 2
$$
estimate the ground-state energy.

**Exercise 9.4.4** Compute $\Delta T \cdot \Delta X$, where $T=P^2/2m$. Why is this relation not so famous?

## 9.5 The Energy-Time Unvertainty Relation

Chapter 13 The Hydrogen Atom

13.1 The Eigenvalue Problem

Chapter 12 Rotational Invariance and Angular Momentum

12.1 Translations in Two Dimensions

Chapter 11 Symmetries and Their Consequences

11.1 Overview

11.2 Translational Invariance in Quantum Theory

Chapter 10 Systems with NNN Degrees of Freedom

10.1 NNN Particles in One Dimension

Chapter 9 The Heisenberg Uncertainty Relations

9.1 Introduction

9.2 Derivation of the Uncertainty Relations

9.3 The Minimum Uncertainty Packet

9.4 Applications of the Uncertainty Principle

Chapter 10 Systems with $N$ Degrees of Freedom

10.1 $N$ Particles in One Dimension