Chapter 6, Three-Dimensional Problems Video Solutions, Quantum mechanics: Concepts and Applications | Numerade (2024)

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 $n_2$.

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.

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?

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.

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 $\hat{L}_z$.

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.

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 $\hat{\vec{L}}^2$ ?

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.

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.

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.

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

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 $\psi(\vec{r}, 0)$.

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

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 .

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.