## Nouredine Zettili

## Chapter 6

## Three-Dimensional Problems - all with Video Answers

## Educators

Chapter Questions

Consider a spinless particle of mass $m$ confined to move in the $x y$ plane under the influence of a harmonic oscillator potential $\hat{V}(x, y)=\frac{1}{2} m \omega^2\left(\hat{x}^2+\hat{y}^2\right)$ for all values of $x$ and $y$.

(a) Show that the Hamiltonian $\hat{H}$ of this particle can be written as a sum of two familiar onedimensional Hamiltonians, $\hat{H}_1$ and $\hat{H}_2$. Then show that $\hat{H}$ commutes with $\hat{L}_z=\hat{X} \hat{P}_y-\hat{Y} \hat{P}_x$.

(b) Find the expression for the energy levels $E_{n_1 n_2}$.

(c) Find the energies of the four lowest states and their corresponding degeneracies.

(d) Find the degeneracy $g_n$ of the $n$th excited state as a function of the quantum number $n$ $\left(n=n_1+n_2\right)$.

(e) If the state vector of the $n$th excited state is $|n\rangle=\left|n_1\right\rangle\left|n_2\right\rangle$ or

$$

\langle x y \mid n\rangle=\left\langle x \mid n_1\right\rangle\left\langle y \mid n_2\right\rangle=\psi_{n_1}(x) \psi_{n_2}(y),

$$

calculate the expectation value of the operator $\hat{A}=\hat{x}^4+\hat{y}^2$ in the state $\ln \langle$ as a function of the quantum numbers $n_1$ and $n_2$.

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Consider a muonic atom which consists of a nucleus of charge $Z e$ and a negative muon moving around it; the muon's charge is $-e$ and its mass is 207 times the mass of the electron, $m_{\mu^{-}}=$ $207 m_e$. For a muonic atom with $Z=6$ calculate

(a) the radius of the first Bohr orbit

(b) the energy of the ground, first and second excited states

(c) the frequency associated with the transitions $n_i=2 \rightarrow n_f=1, n_i=3 \rightarrow n_f=1$ and $n_i=3 \rightarrow n_f=2$.

Raj Bala

Numerade Educator

A hydrogen atom has the wave function $\Psi_{n l m}(\vec{r})$, where $n=4, l=3, m=3$.

(a) What is the magnitude of the angular momentum of the electron around the proton?

(b) What is the angle between the angular momentum vector and the $z$-axis? Can this angle be reduced by changing $n$ or $m$ if $l$ is held constant? What is the physical significance of this result?

(c) Sketch the probability of finding the electron a distance $r$ from the proton.

Narayan Hari

Numerade Educator

An electron in a hydrogen atom is in the energy eigenstate

$$

\psi_{2,1,-1}(r, \theta, \varphi)=N r e^{-r / 2 a_0} Y_{1,-1}(\theta, \varphi) .

$$

(a) Find the normalization constant, $N$.

(b) What is the probability per unit volume of finding the electron at $r=a_0, \theta=45^{\circ}$, $\varphi=60^{\circ}$ ?

(c) What is the probability per unit radial interval $(d r)$ of finding the electron at $r=2 a_0$ ? (One must take an average over $\theta$ and $\varphi$ at $r=2 a_0$.)

(d) If $\hat{L}^2$ and $\hat{L}_z$ are made, what will be the results?

Narayan Hari

Numerade Educator

Consider a hydrogen atom which is in its ground state; the ground state wave function is given by

$$

\Psi(r, \theta, \varphi)=\frac{1}{\sqrt{\pi a_0^3}} e^{-r / a_0}

$$

where $a_0$ is the Bohr radius.

(a) Find the most probable distance between the electron and the proton for a hydrogen atom in its ground state.

(b) Find the average distance between the electron and the proton.

Victor Salazar

Numerade Educator

A hydrogen atom is in the state

$$

\Psi(\vec{r}, 0)=\frac{1}{\sqrt{2}} \phi_{300}(\vec{r})+\frac{1}{\sqrt{3}} \phi_{311}(\vec{r})+\frac{1}{\sqrt{6}} \phi_{322}(\vec{r})

$$

at time $t=0$.

(a) What is the time-dependent wave function?

(b) If a measurement of the energy is made, what values could be found and with what probabilities?

(c) Repeat (b) for $\hat{L}^2$ and $\hat{L}_z$.

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A particle of mass $m$ moves in the $x y$ plane in the potential

$$

V(x, y)=\left\{\begin{array}{ll}

\frac{1}{2} m \omega^2 y^2 & \text { for all } y \text { and } 0<x<a \\

+\infty & \text { elsewhere }

\end{array} .\right.

$$

(a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations.

(b) Find the normalized eigenfunctions and the eigenenergies.

Mehdi Hatefipour

Numerade Educator

A particle of mass $m$ moves in the $x y$ plane in a two-dimensional rectangular well

By reducing the time independent Schrödinger equation to a set of more familiar one-dimensional equations, find the normalized wave functions and the energy levels of this particle.

Ren Jie Tuieng

Numerade Educator

Consider an anisotropic three-dimensional harmonic oscillator potential

$$

V(x, y, z)=\frac{1}{2} m\left(\omega_x^2 x^2+\omega_y^2 y^2+\omega_z^2 z^2\right)

$$

(a) Evaluate the energy levels in terms of $\omega_x, \omega_y$, and $\omega_z$.

(b) Find the three lowest levels for the case $\omega_x=\omega_y=2 \omega_z / 3$, and determine the degeneracy of each level.

(c) Do you expect the wave functions to be eigenfunctions of $\hat{\vec{L}}^2$ ?

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The wave function of an electron in a hydrogen atom is given by

$$

\psi_{21 m_l m_z}(r, \theta, \varphi)=R_{21}(r)\left[\frac{1}{\sqrt{3}} Y_{10}(\theta, \varphi)\left|\frac{1}{2}, \frac{1}{2}\right\rangle+\sqrt{\frac{2}{3}} Y_{11}(\theta, \varphi)\left|\frac{1}{2},-\frac{1}{2}\right\rangle\right],

$$

where $\left|\frac{1}{2}, \pm \frac{1}{2}\right\rangle$ are the spin state vectors.

(a) Calculate the $z$-component of the electron's total angular momentum; that is, calculate $\hat{J}_z \psi_{21 m}$,

(b) If you measure the $z$-component of the electron's spin angular momentum, what values will you obtain? What are the corresponding probabilities.

(c) If you measure $\hat{J}^2$, what values will you obtain? What are the corresponding probabilities.

Narayan Hari

Numerade Educator

Two atoms, each of mass $m$, are bound together to form a molecule; the effective potential is given by

$$

V(r)=V_0\left[1-e^{-\left(r-r_0\right) / a}\right]^2-V_0

$$

where $r$ is the separation of the two atoms (i.e., separation between the nuclei) and $r_0$ is their equilibrium separation.

(a) Write down the time-independent Schrödinger equation for the nuclei and reduce it to an equivalent single particle problem; show how to separate the angular part of each wave function.

(b) Find, to lowest order, the energy levels corresponding to the s states.

Narayan Hari

Numerade Educator

Consider a spinless particle of mass $m$ which is confined to move under the influence of a three-dimensional potential

$$

\hat{V}(x, y, z)= \begin{cases}0 & \text { for } 0<x<a, \quad 0<y<a, 0<z<b \\ +\infty & \text { elsewhere }\end{cases}

$$

(a) Find the expression for the energy levels $E_{n_z n_y n_z}$ and their corresponding wave functions.

(b) If $a=2 b$ find the energies of the five lowest states and their degeneracies.

(c) From (b) find the degeneracy infer the general expression for the degeneracy $g_n$ of the $n$th excited state.

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Calculate the width of the probability density distribution for $r$ for the hydrogen atom in its ground state: $\Delta r=\sqrt{\left\langle r^2\right\rangle_{10}-\langle r\rangle_{10}^2}$.

Zachary Warner

Numerade Educator

Consider a hydrogen atom whose wave function is given at time $t=0$ by

$$

\psi(\vec{r}, 0)=\frac{A}{\sqrt{\pi}}\left(\frac{1}{a_0}\right)^{3 / 2} e^{-r / a_0}+\frac{1}{\sqrt{2 \pi}}\left(\frac{z-\sqrt{2} x}{r}\right) R_{21}(r),

$$

Where $A$ is a real constant, $a_0$ is the Bohr radius, and $R_{21}(r)$ is the radial wave function: $R_{21}(r)=1 / \sqrt{6}\left(1 / a_0\right)^{3 / 2}\left(r / 2 a_0\right) e^{-r / 2 a_0}$.

(a) Write down $\psi(\vec{r}, 0)$ in terms of $\sum_{n l m} \phi_{n l m}(\vec{r})$ where $\phi_{n l m}(\vec{r})$ is the hydrogen wave function $\phi_{n l m}(\vec{r})=R_{n l}(r) Y_{l m}(\theta, \varphi)$.

(b) Find $A$ so that $\psi(\vec{r}, 0)$ is normalized, $\int \phi_{n^{\prime} l^{\prime} m^{\prime}}^*(\vec{r}) \phi_{n l m}(\vec{r}) d^3 r=\delta_{n^{\prime} n} \delta_{l^{\prime}} \delta_{m^{\prime} m}$.

(c) Write down the wave function $\psi(\vec{r}, t)$ at any later time $t$.

(d) Is $\psi(\vec{r}, 0)$ an eigenfunction of $\hat{\vec{L}}^2$ and $\vec{L}_2$ ? If yes, what are the eigenvalues?

(e) If a measurement of the energy is made, what value could be found and with what probability?

(f) What is the probability that a measurement of $\hat{L}_z$ yields $1 \hbar$ ?

(g) Find the mean value of $r$ in the state $\psi(\vec{r}, 0)$.

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A particle of mass $m$ moves in the three-dimensional potential

$$

V(x, y, z)= \begin{cases}\frac{1}{2} m \omega^2 z^2 & \text { for } 0<x<a, 0<y<a \text { and } z>0 \\ +\infty & \text { elsewhere }\end{cases}

$$

(a) Write down the time-independent Schrödinger equation for this particle and reduce it to a set of familiar one-dimensional equations; then find the normalized wave function $\psi_{n_x n_y n_z}(x, y, z)$.

(b) Find the allowed eigenenergies of this particle and show that they can be written as: $E_{n_x n_y n_z}=E_{n_x n_y}+E_{n_z}$

(c) Find the four lowest energy levels in the $x y$ plane (i.e., $E_{n_x n_y}$ ) and their corresponding degeneracies.

CG

Coleman Green

Numerade Educator

Consider a pendulum undergoing small harmonic oscillations (with angular frequency $\omega=$ $\sqrt{g / l}$, where $g$ is the acceleration due to gravity and $l$ is the length of the pendulum). Show that the quantum energy levels and the corresponding degeneracies of the pendulum are given by $E_n=(n+1) \hbar \omega$ and $g_n=n+1$, respectively.

Prabhakar Kumar

Numerade Educator

Consider a proton that is trapped inside an infinite central potential well

$$

V(r)= \begin{cases}-V_0 & 0<r<a \\ +\infty & r \geq a\end{cases}

$$

where $V_0=5104.34 \mathrm{MeV}$ and $a=10 \mathrm{fm}$.

(a) Find the energy and the (normalized) radial wave function of this particle for the s states (i.e., $l=0$ ).

(b) Find the number of bound states that have energies lower than zero; you may use these values $m=938 \mathrm{MeV}$ and $\hbar c=197 \mathrm{MeV} \mathrm{fm}$.

(c) Calculate the energies of the levels that lie just bellow and just above the zero-energy level; express your answer in MeV .

Keshav Singh

Numerade Educator

Consider the function $\psi(\vec{r})=-A(x+i y) e^{-r / 2 a_0}$, where $a_0$ is the Bohr radius and $A$ is a real constant.

(a) Is $\psi(\vec{r})$ an eigenfunction to $\hat{\vec{L}}^2$ and $\hat{L}_z$ ? If yes, write $\psi(\vec{r})$ in terms of $R_{n l}(r) Y_{l m}(\theta, \varphi)$ and find the values of the quantum numbers $n, m, l ; R_{n l}(r)$ are the radial wave functions of the hydrogen atom.

(b) Find the constant $A$ so that $\psi(\vec{r})$ is normalized.

(c) Find the mean value of $r$ and the most probable value of $r$ in this state.

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The wave function of a hydrogen-like atom at time $t=0$ is

$$

\Psi(\vec{r}, 0)=\frac{1}{\sqrt{11}}\left[\sqrt{3} \psi_{21-1}(\vec{r})-\psi_{210}(\vec{r})+\sqrt{5} \psi_{211}(\vec{r})+\sqrt{2} \psi_{311}(\vec{r})\right],

$$

where $\psi_{n l m}(\vec{r})$ is a normalized eigenfunction (i.e., $\psi_{n l m}(\vec{r})=R_{n l}(r) Y_{l m}(\theta, \varphi)$ ).

(a) What is the time-dependent wave function?

(b) If a measurement of energy is made, what values could be found and with what probabilities?

(c) What is the probability for a measurement of $\hat{L}_z$ that yields $1 \hbar$ ?

(d) Calculate $\Delta r \Delta p_r$ with respect to the state

$$

\psi_{210}(\vec{r})=\frac{1}{\sqrt{6}}\left(\frac{1}{a_0}\right)^{3 / 2} \frac{r}{2 a_0} e^{-r / 2 a_0} Y_{10}(\theta, \varphi),

$$

and verify that $\Delta r \Delta p_r$ satisfies the Heisenberg uncertainty principle.

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A particle of mass $m$ moves in the potential $V(x, y, z)=V_1(x, y)+V_2(z)$ where

$$

V_1(x, y)=\frac{1}{2} m \omega^2\left(x^2+y^2\right), \quad V_2(z)= \begin{cases}0 & 0 \leq z \leq a \\ +\infty & \text { elsewhere }\end{cases}

$$

(a) Calculate the energy levels and the wave function of this particle.

(b) Let us now turn off $V_2(z)$ (i.e., $m$ is subject only to $V_1(x, y)$ ). Calculate the degeneracy $g_n$ of the $n$th energy level (note that $n=n_x+n_y$ ).

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