Problem Sheet: Week 15 - Quantum Mechanics

Exercise 7.3

Consider a particle trapped inside an infinite square well potential, with initial wavefunction given by

\psi_\init(x) = \frac{1}{\sqrt{2}}\left(u_1(x) + u_2(x)\right).

If the energy of the particle is measured, what energies does it have, and with what probabilities?
Using the superposition principle, or otherwise, write down the state of the particle at time $t$ , assuming that at $t = 0$ the wavefunction is $\psi(x,0) = \psi_\init(x)$ .
Confirm that the wavefunction from part 2 satisfies the SE (5.2).
Calculate $\pd(x,t)$ . It will be helpful to denote $\omega = (E_2 - E_1)/\hbar$ .
The motion of the particle is periodic. What is the period $T$ ?
Using your favourite sketching software, plot the probability density of the particle at $t = 0$ , $t = T/4$ , $T = T/2$ and $T = 3T/4$ .

Answers

$E_1$ and $E_2$ only; $\prob(E_1) = \frac{1}{2}$ and $\prob(E_2) = \frac{1}{2}$ .
$\psi(x,t) = \frac{1}{\sqrt{2}}\left(e^{-iE_1t/\hbar}u_1(x) + u_2(x)e^{-iE_2t/\hbar}\right)$
n/a (show that).
$\pd(x,t) = \frac{1}{2}\left(|u_1(x)|^2 + |u_2(x)|^2 + 2 u_1(x)u_2(x)\cos \omega t\right)$ .
$T = 2\pi /\omega$ .
n/a (sketch).

Exercise 7.5

Consider a particle confined inside an infinite square well with wavefunction

\psi_\init(x) = \begin{cases} \sqrt{\frac{3}{L^3}}x& \text{if } 0 \leq x < L, \\ 0 &\text{otherwise. } \end{cases}

In this exercise we will write this wavefunction as a superposition of the energy eigenfunctions of the infinite square well, as in (7.15).

If the energy of the particle is measured, what is the probability it has energies $E_1$ , $E_2$ and $E_3$ ?
What is the total probability that the particle has energy at least $E_4$ ?
Using the superposition principle, write down the wavefunction of the particle at time $t$ , assuming that at $t = 0$ the wavefunction is $\psi(x,0) = \psi_\init(x)$ .

Answers

n/a (sketch and show that).
n/a (show that).
$\prob(E_1) = \frac{6}{\pi^2}$ , $\prob(E_2) = \frac{6}{4 \pi^2}$ , $\prob(E_3) = \frac{6}{9\pi^2}$ .
$\prob(E_4 \text{ or higher}) = 1-\frac{49}{6\pi^2}$
$\psi(x,t) = -\frac{\sqrt{6}}{\pi}\sum_{n=1}^\infty \frac{(-1)^n}{n} e^{-iE_n t/\hbar} u_n(x).$

Exercise 8.1

The three dimensionful quantities for the quantum harmonic are the mass of the particle $M$ , the angular frequency ω (which specifies the spring constant $k = M\omega^2$ ), and $\hbar$ . Write down the dimensions of each of these quantities.
Find a combination of these quantities that has the dimension of length. That is, let
$\vph \ell = M^\alpha \omega^\beta \hbar^\gamma,$
(8.34)
and find the exponents α, β and γ that give $\ell$ the dimension of length.
Find a combination of $M$ , ω and $\hbar$ that have the dimension of momentum. That is, let
$\vph q = M^\mu \omega^\nu \hbar^\lambda,$
(8.35)
and find the exponents μ, ν and λ that give $q$ the dimension of momentum.

Answers

$[M] = \mathsf{M}$ , $[\omega] = \mathsf{T}^{-1}$ , $[\hbar] = \mathsf{M}\mathsf{L}^{2} \mathsf{T}^{-1}$ .
$\alpha = -\frac{1}{2}$ , $\beta = -\frac{1}{2}$ , $\gamma = \frac{1}{2}$ .
$\mu = \frac{1}{2}$ , $\nu = \frac{1}{2}$ , $\lambda = \frac{1}{2}$ .

Exercise 8.2

Show that for arbitrary operators $\hat{A}$ , $\hat{B}$ , $\hat{C}$ and $\hat{D}$ that
$[\hat{A} + \hat{B}, \hat{C} + \hat{D}] = [\hat{A},\hat{C}] + [\hat{A},\hat{D}] + [\hat{B},\hat{C}] + [\hat{B}\,\hat{D}].$
(8.36)
Show that for arbitrary operators $\hat{A}$ and $\hat{B}$ , and arbitrary complex constants α and β that
$[\alpha \hat{A}, \beta \hat{B}] = \alpha\beta [\hat{A},\hat{B}].$
(8.37)
Using parts 1 and 2, and the explicit form of $\hat{H}$ and $\hat{a}^\dagger$ , show that
$\vph[\hat{H}, \hat{a}^\dagger] = \frac{1}{2\sqrt{2}M\ell}[\hat{P}^2,\hat{X}] - \frac{i\ell}{2\sqrt{2}M\hbar}[\hat{P}^2,\hat{P}] +\frac{M\omega^2}{2\sqrt{2}\ell}[\hat{X}^2,\hat{X}] - \frac{iM\omega^2 \ell}{2\sqrt{2}\hbar}[\hat{X}^2,\hat{P}].$
(8.38)
Use (8.9), show that the other two commutators vanish, and use the definition of $\ell$ in (8.5) to show that
$[\hat{H},\hat{a}^\dagger] = \hbar \omega \hat{a}^\dagger.$
(8.39)

Answers