Midterm practice

Short Questions

1. Write the time dependent Schrödinger equation in three dimensions and the position representation for a free particle.

2. Write the time independent Schrödinger equation in three dimensions and the position representation for a free particle. Give the eigenfunctions and eigenvalues.

3. Given an operator $\hat{O}$, give its expectation value for state $|\psi ⟩$ in bra-ket notation, and in the position and momentum representations. You can write $⟨\overrightarrow{r}|\psi ⟩=\psi (\overrightarrow{r})$ and $⟨\overrightarrow{p}|\psi ⟩=\psi (\overrightarrow{p})$

4. Can a Hermitian operator have eigenvalues of the form $e^{i\varphi }$? If yes, what are the possible values of $\varphi$?

5. From the following operators that we have encountered in class, circle those that are Hermitian.

   • $\hat{y},{\hat{p}}_y,{\hat{p}}_z-\hat{z}$
   • ${\hat{a}}^{†}\hat{a}$
   • $\hat{R}(\theta )=\left(\begin{array}{ccc}\cos (\theta )& -\sin (\theta )\sin (\theta )& \cos (\theta )\end{array}\right)$
   • ${\hat{L}}_x-{\hat{L}}_y$
   • ${\hat{\sigma }}_y=\left(\begin{array}{ccc}0& -ii& 0\end{array}\right)$
   • ${\hat{a}}^{†}$

6. Give the commutator between the $\hat{x}$ and $\widehat{p_x}$ operators. What are the physical consequences of this relation.

7. Give the commutator between the $\hat{x}$ and $\widehat{p_y}$ operators. What are the physical consequences of this relation.

8. Consider a system that at time $t=0$ is described by the state $|\psi (t=0)⟩=\sqrt{\frac{2}{9}|{\psi }_1⟩-\sqrt{\frac{7}{9}}|{\psi }_2⟩}$ where $|{\psi }_1⟩$ and $|{\psi }_2⟩$ are eigenstates of the system’s Hamiltonian with eigenvalues $E_1$ and $E_2$. Give the energy expectation value of $|\psi (t=0)⟩$ and an expression for the time dependent state $|\psi (t)⟩.$

9. True of False? For a free particle with energy $E$ incident on an infinitely thick, square barrier of potential energy $V$, there is zero probability of reflection from the barrier when $E>V$.

10. How do the eigenvalues of the one-dimensional particle in a box of length $L$ depend of the quantum number $n$ and the length of the box?

11. Give the eigenvalues of a one-dimensional harmonic oscillator of frequency $\omega$. What is the energy spacing? What is the degeneracy of each level.

12. Give a set of wave functions that are eigenstates to both ${\hat{L}}^2$ and ${\hat{L}}_z.$ Use Dirac notation. Give the eigenvalue equations for both operators. What are the possible values of the eigenvalues?

A Quantum Mechanical Swing Voter

Consider a quantum mechanical swing voter that can exist in any linear combination of the orthonormal states $|D⟩$ and $|R⟩$. Asking who they will vote for is equivalent to making a measurement with the operator $\hat{V}$, for which $|D⟩$ and $|R⟩$ are the two eigenstates. Having investigated the Hamiltonian $\hat{H}$ of the swing voter, you have discovered the following two states: $$|A⟩=\frac{1}{\sqrt{10}}|D⟩+\frac{3}{\sqrt{10}}|R⟩\text{ with }\hat{H}|A⟩=4|A⟩$$

$$|B⟩=\frac{3}{\sqrt{10}}|D⟩-\frac{1}{\sqrt{10}}|R⟩\text{ with }\hat{H}|B⟩=2|B⟩$$

Neuscamman MT1 2016

i) Show that the states $|A⟩$ and $|B⟩$ are orthonormal.

ii) Let $\hslash$ = 1, show that if at time t = 0 you ask this voter who they will vote for, then you will be able to predict with complete certainty when you ask again at time $t=\pi$.

3 Particles in a Box

Consider three particles, each of mass $m$, in a $1\mathrm{D}$ box with the confinement potential $(U)$ for each particle given by:

$$\begin{array}{c}U(x_{\alpha })=\\ \{\begin{array}{l}0,\le x_{\alpha }\le \ell \\ \mathrm{\infty },\text{ otherwise }\end{array}\end{array}$$

where $x_{\alpha }$ is the position of particle $\alpha =1,2,3.$ This problem is equivalent to the particle in a $3\mathrm{D}$ box.

a) Assuming that the particles do not interact with each other, write down the Hamiltonian $\hat{\mathcal{H}}$ of the system. Use the notation $x_{\alpha }$ for the position operator and ${\hat{p}}_{\alpha }=-i\hslash \frac{\partial }{\partial x_{\alpha }}$ for the momentum operator of each particle ($\hslash =h/2\pi$ and $h$ is Planck’s constant).

b) Find the eigenvalues $E_{n_1,n_2,n_3}$ and eigenstates ${\psi }_{n_1,n_2,n_3}(x_1,x_2,x_3)$ by solving the stationary Schrodinger equation, where $n_{\alpha }$ is the quantum number for particle $\alpha$. Use the eigenvalues and eigenstates for a particle of mass $m$ in a $1\mathrm{D}$ box of size $\ell ,$ which are given by $E_n=\frac{h^2{n}^2}{8m{\ell }^2}$ and ${\psi }_n(x)=\sqrt{\frac{2}{\ell }}\sin \left(\frac{n\pi x}{\ell }\right),$ respectively, where $$n=1,2,3,\cdots \mathrm{\infty }$$

c) The system is prepared in the ground state. What is the probability of finding all three particles in $0\le x_{\alpha }\le \frac{\ell }{2}?$ You can use symmetry arguments without performing any integral (but express the integral explicitly).

d) Qualitatively, explain how your answer to part c) will change if the particles repel each other?

2D Harmonic Oscillator

Consider a particle of mass $m=1$ in a $2D$ isotropic harmonic potential with a force constant $k=1.$ The Hamiltonian is given by: $$\hat{\mathcal{H}}=-\frac{1}{2}(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2})+\frac{1}{2}(x^2+y^2)$$ where we assume that $\hslash =1$ (this is an arbitrary choice of units).

a) Express the eigenenergies, $E_{n_x,n_y},$ of the particle in terms of the quantum numbers $n_x$ and $n_y$. What are the possible values of $n_x$ and $n_y?$ Note: The eigenenergies of a particle in a $1\mathrm{D}$ harmonic potential are given by: $E_n=\hslash \omega (n+\frac{1}{2}),$ where $n=0,1,\cdots \mathrm{\infty }$ is the quantum number and $\omega =\sqrt{k/m}$ is the frequency of the oscillator.

b) The particle is initially prepared in a superposition of the first two lowest excited states, given by: $\psi (x,y,t=0)={\psi }_{1,0}(x,y)+$ $\frac{1}{2}{\psi }_{0,1}(x,y),$ where ${\psi }_{n_x,n_y}(x,y)$ are the eigenstates with eigenenergies $E_{n_x,n_y}$ found in part (a). Normalize $\psi (x,y,t=0)$ and obtain the solution for $\psi (x,y,t).$ Reminder: $$|\psi (t)⟩=\sum _{n_x}\sum _{n_y}{c}_{n_x,n_y}(0)e^{-i{E}_{n_x,n_y}t/h}|{\psi }_{n_x,n_y}⟩$$ where $\hat{\mathcal{H}}|{\psi }_{n_x,n_y}⟩=E_{n_x,n_y}|{\psi }_{n_x,n_y}⟩$ are the stationary solutions of $\hat{\mathcal{H}}$ and $c_{n_x,n_y}(0)=⟨{\psi }_{n_x,n_y}\mid \psi (0)⟩$ are the expansion coefficients. Hint: There is no need to perform any integrals.

c) Does the probability density $P(x,y)$ of finding the particle at position $x,y,$ depend on time? Explain your answer.

d) The expectation value of the energy, $⟨\hat{\mathcal{H}}⟩,$ is measured as a function of time for the normalized state found in part (b). What is the probability of finding the values $E=1$ and $E=2$ at time $t?$