Chapter 8, Time dependence Video Solutions, Quantum Mechanics

Problem 1

The wavefunction of a particle in a one-dimensional infinite-sided potential well of width $2 a$ is $\psi=2^{-1 / 2}\left(u_{1}+u_{2}\right)$ at time $t=0$ where $u_{1}$ and $u_{2}$ are the two lowest energy eigenfunctions. Find an expression for the position probability distribution as a function of time and show that it is periodic with angular frequency $\omega=3 \pi^{2} h / 8 m a^{2}$. Sketch this probability distribution at times $0, \pi / 2 \omega, \pi / \omega$ and $3 \pi / 2 \omega .$

Laszlo Zalavari

Numerade Educator

05:23

Problem 2

Show that the expectation value of the position of a particle in a harmonic oscillator potential oscillates sinusoidally with the classical frequency if the system is not in an energy eigenstate.

Hafiz Shahzaib

Numerade Educator

00:59

Problem 3

Show that the expectation value of the angular momentum of an electron in a magnetic field B precesses about the direction of $\mathbf{B}$ with an angular frequency $e B / m_{e}$, unless it is in an energy eigenstate.

Mayukh Banik

Numerade Educator

01:29

Problem 4

What is the probability of finding the resulting ${ }^{3} \mathrm{He}^{+}$ion in (i) its $2 s$ and (ii) one of its $2 p$ states, following the $\beta$ decay of a ${ }^{3} \mathrm{H}$ atom initially in its ground state?

Nick Johnson

Numerade Educator

02:19

Problem 5

The spring constant of a harmonic oscillator in its ground state is suddenly doubled. Calculate the probability that a subsequent energy measurement will find the new oscillator in (i) its ground state, (ii) its first excited state, and (iii) its second excited state.

Penny Riley

Numerade Educator

01:26

Problem 6

In the experiment on spin interference described in section 8.1, the magnetic fields used had a magnitude of $0.5 T$ and the path lengths were each $7 \times 10^{-5} \mathrm{~m}$. Show that a minimum in the centre of the diffraction pattern is to be expected for neutrons with a wavelength of $3.89 \times 10^{-10} \mathrm{~m}$, given that the neutron has a magnetic moment of magnitude $1.91\left(e \hbar / 2 m_{n}\right), m_{n}$ being the neutron mass.

Narayan Hari

Numerade Educator

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Problem 7

A particle, initially in an energy eigenstate of an infinite-sided potential well, is subject to a perturbation of the form $V_{0} x \cos \omega t$. Show that transitions are possible between the states $u_{n}$ and $u_{m}$ only if $n+m$ is odd.

Victor Salazar

Numerade Educator

04:20

Problem 8

The amplitude $H^{\prime \prime}$ associated with magnetic dipole transitions turns out to be proportional to the operator representing the angular-momentum vector. Show that in a one-electron atom the selection rules for such transitions are $\Delta l=0, \Delta m=\pm 1$ or $0 .$ Which of these apply when the angularmomentum vector is (i) parallel and (ii) perpendicular to the $z$ axis?
Hint: Express $L_{x}$ and $L_{y}$ in terms of ladder operators.

Hafiz Shahzaib

Numerade Educator

07:07

Problem 9

Given that $y H_{n}=\frac{1}{2} H_{n+1}+n H_{n-1}$ where the $H_{n}$ 's are Hermite polynomials, show that the selection rule for electric dipole transitions in a one-dimensional harmonic oscillator is $\Delta n=\pm 1$.

Hafiz Shahzaib

Numerade Educator

07:06

Problem 10

A certain physical system has a Hamiltonian operator of the form $\hat{H}_{0}+\hat{H}^{\prime \prime} \cos \omega t$ where $\hat{H}_{0}$ and $\hat{H}^{\prime \prime}$ are time independent, but $H^{\prime \prime}$ need not be a small perturbation. The operator $\hat{H}_{0}$ has only two eigenstates whose eigenfunctions are $u_{1}$ and $u_{2}$ respectively. Show that the expression $a u_{1} \exp \left(-i E_{1} t / \hbar\right)+b u_{2} \exp \left(-i E_{2} t / \hbar\right)$ is a solution to the time-dependent Schrödinger equation in this case if
$$
\frac{d a}{d t}=\frac{b}{i \hbar} H_{21}^{\prime \prime} \cos \omega t e^{i \omega_{21} t}
$$
and
$$
\frac{d b}{d t}=\frac{a}{i \hbar} H_{21}^{\prime \prime} \cos \omega t e^{i \omega_{12} t}
$$
in the usual notation, provided $H_{11}^{\prime \prime}=H_{22}^{\prime \prime}=0$. Show that if $\omega=\omega_{12}$ and if high-frequency terms can be ignored, then
$$
a=\cos (\Omega t-\phi) \quad b=\sin (\Omega t-\phi)
$$
are solutions to these equations where $\Omega=\left|H_{12}^{\prime \prime}\right| / 2 \hbar$ and $\phi$ is a constant. Compare these results with the discussion of quantum oscillations in the text.

Andrew Eddins

Emory University

02:35

Problem 11

A particle moves in a one-dimensional potential well given by
$$
V=V_{0} \quad(-a \leqslant x \leqslant a) \quad V=0 \quad(a<|x|<b) \quad V=\infty \quad(|x|>b)
$$
Assuming that the magnitudes of $a, b$, and $V_{0}$ are such that the ground-state energy is less than $V_{0}$ and that there is a small, but finite, probability of tunnelling through the central barrier, draw sketch graphs of the two lowest energy eigenfunctions of this system. Discuss the evolution in time of such a system if the particle is known to be initially on one side of the barrier. Consider the response of this system to a perturbation whose frequency corresponds to the difference between the energies of the two lowest states and compare the properties of this system with those of the ammonia molecule.

Mayukh Banik

Numerade Educator