Quantum mechanics

Eugen Merzbacher

Chapter 12 Spherically Symmetric Potentials - all with Video Answers

Educators

Chapter Questions

Problem 1

Compute (or obtain from mathematical tables) and plot the 10 lowest energy eigenvalues of a particle in an infinitely deep, spherically symmetric square well, and label the states by their appropriate quantum numbers.

Suzanne W.

Numerade Educator

02:22

Problem 2

If the ground state of a particle in a spherical square well is just barely bound, show that the well depth and radius parameters $V_0$ and $a$ are related to the binding energy by the expansion
$$
\frac{2 m V_0 a^2}{\hbar^2}=\frac{\pi^2}{4}+2 \kappa a+\left(1-\frac{4}{\pi^2}\right)(\kappa a)^2+\cdot \cdot
$$
where
$$
\hbar \kappa=\sqrt{-2 m E}
$$

The deuteron is bound with an energy of $2.226 \mathrm{MeV}$ and has no discrete excited states. If the deuteron is represented by a nucleon, with reduced mass, moving in a square well with $a=1.5$ fermi, estimate the depth of the potential.

Kratika Bhadauria

Numerade Educator

02:22

Problem 3

Given an attractive central potential of the form
$$
V(r)=-V_0 e^{-r / a}
$$
solve the Schrödinger equation for the $S$ states by making the substitution
$$
\xi=e^{-r^2 2 a}
$$

Obtain an equation for the eigenvalues. Estimate the value of $V_0$, if the state of the deuteron is to be described with an exponential potential (see Problem 2 for data).

Kratika Bhadauria

Numerade Educator

Problem 4

Show that, if a square well just binds an energy level of angular momentum $\ell(\neq 0)$, its parameters satisfy the condition
$$
j_{\ell-1}\left(\sqrt{\frac{2 m V_0 a^2}{\hbar^2}}\right)=0
$$
(Use recurrence formulas for Bessel functions from standard texts.)

Check back soon!

View

Problem 5

Assuming the eigenfunctions for the hydrogen atom to be of the form $r^\beta e^{-\alpha r} Y_{\ell}^m$, with undetermined parameters $\alpha$ and $\beta$, solve the Schrödinger equation. Are all eigenfunctions and eigenvalues obtained this way?

Victor Salazar

Numerade Educator

Problem 6

Apply the WKB method to an attractive Coulomb potential, and derive an approximate formula for the $S$-state energy levels of hydrogen.

Check back soon!

06:25

Problem 7

Compute the probability that the electron in a hydrogen atom will be found at a distance from the nucleus greater than its energy would permit on the classical theory. Make the calculation for the $n=1$ and 2 levels.

Vysakh M

Numerade Educator

01:29

Problem 8

Calculate the probability distribution for the momentum of the electron in the ground state of a hydrogen atom. Obtain the expectation value of $p_x^2$ from this or from the virial theorem. Also calculate $\left\langle x^2\right\rangle$ from the ground state wave function, and verify the uncertainty relation for this state.

Dominador Tan

Numerade Educator

15:48

Problem 9

Solve the Schrodinger equation for the three-dimensional isotropic harmonic oscillator, $V=(1 / 2) m \omega^2 r^2$, by separation of variables in Cartesian and in spherical polar coordinates. In the latter case, assume the eigenfunctions to be of the form
$$
\psi(r, \theta, \varphi)=r^{\ell} \exp \left(-\frac{m \omega}{2 \hbar} r^2\right) f(r) Y_{\ell}^m(\theta, \varphi)
$$
and show that $f(r)$ can be expressed as an associated Laguerre polynomial (or a confluent hypergeometric function) of the variable $m \omega r^2 / \hbar$ with half-integral indices. Obtain the eigenvalues and establish the correspondence between the two sets of quantum numbers. For the lowest two energy eigenvalues, show the relation between the eigenfunctions obtained by the two methods.

Mahnoor Amin

Numerade Educator

01:44

Problem 10

For the isotropic harmonic oscillator of Problem 9, obtain a formula for the degree of degeneracy in terms of the energy. For large energies (large quantum numbers), compare the density of energy eigenstates in the oscillator and in a cubic box.

Lottie Adams

Numerade Educator

Problem 11

Starting with the radial equation
$$
-\frac{\hbar^2}{2 m} \frac{d^2 u}{d r^2}+\frac{\hbar^2 \ell(\ell+1)}{2 m r^2} u-\frac{Z e^2}{r} u=E u
$$
for the hydrogenic atom, show that the transformation
$$
r=\alpha \bar{r}^2, \quad u=\sqrt{\bar{r}} \bar{u},
$$
produces an equation for $\bar{u}(\vec{r})$ that, with appropriate choices of the constants, is equivalent to the radial equation for the isotropic oscillator. Exhibit the relation between the energy eigenvalues and the radial quantum numbers for the two systems.

Check back soon!

24:25

Problem 12

The initial $(t=0)$ state of an isotropic harmonic oscillator is known to be an eigenstate of $L_z$ with eigenvalue zero and a superposition of the ground and first excited states. Assuming that the expectation value of the coordinate $z$ has at this time its largest possible value, determine the wave function for all times as completely as possible.

Dr. Rajveer Singh

Numerade Educator

Problem 13

Solve the energy eigenvalue problem for the two-dimensional isotropic harmonic oscillator. Assume that the eigenfunctions are of the form
$$
\psi=\rho^{\ell} \exp \left(-\frac{m \omega}{2 \hbar} \rho^2\right) f(\rho) \exp ( \pm i \ell \varphi)
$$
where $\rho$ and $\varphi$ are plane polar coordinates and $\ell$ is a nonnegative integer. Show that $f(\rho)$ can be expressed as an associated Laguerre polynomial of the variable $m \omega \rho^2 / \hbar$ and determine the eigenvalues. Solve the same problem in Cartesian coordinates, and establish the correspondence between the two methods. Discuss a few simple eigenfunctions.

Check back soon!

Problem 14

Apply the variational method to the ground state $(\ell=0)$ of a particle moving in an attractive central potential $V(r)=A r^n$ (integer $n \geq-1$ ), using
$$
R(r)=e^{-\beta r}
$$
as a trial wave function with variational parameter $\beta$. For $n=-1$ and +2 , compare the results with the exact ground state energies.

Check back soon!

Problem 15

Apply the variational method to the ground state $(\ell=0)$ of a particle moving in an attractive (Yukawa or screened Coulomb or Debye) potential
$$
V(r)=-V_0 \frac{e^{-r / a}}{r / a} \quad\left(V_0>0\right)
$$

Use as a trial function
$$
R(r)=e^{-\gamma r i a}
$$
with an adjustable parameter $\gamma$. Obtain the "best" trial wave function of this form and deduce a relation between $\gamma$ and the strength parameter $2 m V_0 a^2 / \hbar^2$. Evaluate $\gamma$ and calculate an upper bound to the energy for $2 m V_0 a^2 / \hbar^2=2.7$.
Are there any excited bound states?
Show that in the limit of the Coulomb potential ( $V_0 \rightarrow 0, a \rightarrow \infty, V_0 a$ finite) the correct energy and wave function for the hydrogenic atom are obtained.

Check back soon!

06:09

Problem 16

Using first-order perturbation theory, estimate the correction to the ground state energy of a hydrogenic atom due to the finite size of the nucleus. Under the assumption that the nucleus is much smaller than the atomic radius, show that the energy change is approximately proportional to the nuclear mean square radius. Evaluate the correction for a uniformly charged spherical nucleus of radius $R$. Is the level shift due to the finite nuclear size observable? Consider both electronic and muonic atoms.

Hafiz Shahzaib

Numerade Educator

06:09

Problem 17

An electron is moving in the Coulomb field of a point charge $Z e$, modified by the presence of a uniformly charged spherical shell of charge $-Z^{\prime} e$ and radius $R$, centered at the point charge. Perform a first-order perturbation calculation of the hydrogenic $1 S, 2 S$, and $2 P$ energy levels. For some representative values of $Z=Z^{\prime}$, estimate the limit that must be placed on $R$ so that none of the lowest three energy levels shift by more than 5 percent of the distance between the unperturbed first excited and ground state energy levels.

Hafiz Shahzaib

Numerade Educator