Problem 1.
- (20 points total) In one dimension, a particle of mass $m$ is in a harmonic potential with potential energy $V_{0}(x)=\frac{1}{2} m w_{0}^{2} x^{2}$. For times $t<0$ the system is in its ground state. Then at $t=0$, the potential is suddenly changed to $V_{1}(x)=\frac{1}{2} m \omega_{1}^{2} x^{2}$. Here $\omega_{0}$ and $\omega_{1}$ are constants that represent the classical frequency of oscillation. (a) (5 points) What is $\langle E\rangle$ right before this change? (b) (5 points) What is $\langle E\rangle$ right after this change? Hint: (i) The state of the system does not discontinuously change and so right after the change it has the same wave function it had right before the change. (ii) For all energy eigenstates of a harmonic oscillator, $\langle V(x)\rangle=\langle K\rangle$, where $V(x)$ is the potential energy and $K$ is the kinetic energy (iii) For a harmonic oscillator, relate $\langle V(x)\rangle$ and $\langle K\rangle$ to $\langle E\rangle$. (c) (5 points) What is $\langle E\rangle$ at a function of time $t$ where $t>0$. Hint: use the equation for $\frac{d}{d t}\langle Q\rangle$ from the formula page below to determine how $\langle E\rangle$ depends on time.
(d) (5 points) For any time $t>0$, what is the probability of measuring the energy to be the new ground state energy $E=h \omega_{1} / 2 ?$
Problem 2.
- (20 points total) A free particle in one dimension of mass $m$ has a wavefunction at $t=0$
$$
\psi(x)=\frac{1}{\left(2 \pi \sigma^{2}\right)^{1 / 4}} \exp \left(\frac{-x^{2}}{4 \sigma^{2}}\right) \exp (i k x)
$$
Here “free particle” means that the potential energy is zero. $k$ and $\sigma$ are real constants.
(a) $(10$ points) Calculate $\langle p\rangle$ at $t=0$.
(b) $\left(10\right.$ points) Calculate $\langle p\rangle$ for any time $t>0$. Hint: use the equation for $\frac{d}{d t}\langle\hat{Q}\rangle$ from the formula page below to determine how $\langle p\rangle$ depends on time.
Problem 3.
- (20 points total) A particle has two states that it can be in. One is where it is at a site $A$, and the other at an adjacent site $B$. These can be represented by the two basis states
$$
\psi_{A}=\left(\begin{array}{l}
1 \
0
\end{array}\right) \text { and } \psi_{B}=\left(\begin{array}{l}
0 \
1
\end{array}\right)
$$
meaning that in state $\psi_{A}$, the particle is only at site $A$ and in state $\psi_{B}$, that the particle is only at site $B$. The general state of the system is a linear combination of these two states. In this basis, the Hamiltonian is
$$
\hat{H}=\hbar \omega\left(\begin{array}{ll}
0 & 1 \
1 & 0
\end{array}\right)
$$
where $\omega$ is a constant with units of frequency.
(a) (10 points) Find the energy eigenstates and eigenvalues for this Hamiltonian.
(b) (10 points) Assume that at $t=0$, the particle is observed to be on site $A$. Calculate the probability that if observed again, at a time $t$, that the particle will be found to be on site $B$.
Problem 4.
- ( 20 points total) A particle, with mass $m$ is confined to a two dimensional rectangular box of length $2 a$ along the $x$ axis, and $a$ along the $y$ axis. That is, the potential energy is 0 if $0<x<2 a$ and $0<y<a$, and is infinite otherwise.
(a) (10 points) Calculate the ground state and first excited state wave functions and their corresponding energies. For this part of the problem, you do not need to consider the particle’s spin.
(b) (10 points) Consider the same situation but with the addition of a second identical particle. Both particles are spin $1 / 2$ fermions. Calculate the total ground state energy and two particle wave function. This wave function should include the spin.
Problem 5.
- (20 points)
Two spin $1 / 2$ particles are prepared in the state $|\rightarrow \rightarrow\rangle$ that is, each spin is in an eigenstate of its $S_{x}$ operator with eigenvalue $h / 2$. This is analogous to the notation $|\uparrow \uparrow\rangle$ representing each spin being in an eigenstate of its $S_{z}$ operator with eigenvalue $\hbar / 2$.
What is the probability that the system will be observed in the state $|s m\rangle$ where $s$ is the total angular momentum and $m$ is the total angular momentum in the $z$ direction. Find the probabilities for all four possible states.
