16.4: Exercises - Time Dependence

An asterisk denotes a harder problem, which you are nevertheless encouraged to try!

1. An easy one to start with! A particle moving in the infinite 1-d square well potential

$$V (x) = 0 \quad \text{ for } |x| < a, \quad V (x) = \infty \quad \text{ for } |x| > a \nonumber$$

is set up in the initial state $(t = 0)$ described by the wavefunction

$$\Psi (x, 0) \equiv \psi (x) = [u_1(x) + u_2(x)] / \sqrt{2} \nonumber$$

where $u_1(x)$, $u_2(x)$ are the energy eigenfunctions corresponding to the energy eigenvalues $E_1$ and $E_2$ respectively. Sketch the probability density at $t = 0$.

What is the wavefunction at time $t$?

Calculate the probabilities $P_1$ and $P_2$ that at $t = 0$ a measurement of the total energy yields the results $E_1$ and $E_2$ respectively. Do $P_1$ and $P_2$ change with time?

Calculate the probabilities $P_+(t)$ and $P_−(t)$ that at time $t$ the particle is in the intervals $0 < x < a$ and $−a < x < 0$ respectively and try to interpret your results.

2. A system has just two independent states, $|1\rangle$ and $|2\rangle$, represented by the column matrices

$$|1\rangle \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix} \quad \text{ and } \quad |2\rangle \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix} \nonumber$$

With respect to these two states, the Hamiltonian has a time-independent matrix representation

$$\begin{pmatrix} E & V \\ V & E \end{pmatrix} \nonumber$$

where $E$ and $V$ are both real.

Show that the probability of a transition from the state $|1\rangle$ to the state $|2\rangle$ in the time interval $t$ is given without approximation by

$$p(t) = \sin^2 \left( \frac{V t}{\hbar} \right) \nonumber$$

[Hint: expand the general state $|\Psi, t\rangle$ in terms of $|1 \rangle$ and $|2 \rangle$ and substitute in the TDSE. Note that $|1 \rangle$ and $|2 \rangle$ are not energy eigenstates!]

Compute the transition probability using first-order time-dependent perturbation theory, taking the unperturbed Hamiltonian matrix to be that for which $|1 \rangle$ and $|2 \rangle$ are energy eigenstates. By comparing with the exact result, deduce the conditions under which you expect the approximation to be good.

3.*A 1-d harmonic oscillator of charge $q$ is acted upon by a uniform electric field which may be considered to be a perturbation and which has time dependence of the form

$$\mathcal{E}(t) = \frac{A}{\sqrt{\pi} \tau} \text{ exp } \left\{ −(t/\tau )^2 \right\} \nonumber$$

Assuming that when $t = −\infty$, the oscillator is in its ground state, evaluate the probability that it is in its first excited state at $t = +\infty$ using time-dependent perturbation theory. You may assume that

$$\int^{\infty}_{−\infty} \text{ exp}(−y^2 ) dy = \sqrt{\pi} \nonumber$$

$$\langle n + 1|\hat{x}|n \rangle = \sqrt{\frac{ (n + 1)\hbar }{2m\omega}} \nonumber$$

$$\langle n + i|\hat{x}|n \rangle = 0 \quad − 1 > i > 1 \nonumber$$

Discuss the behavior of the transition probability and the applicability of the perturbation theory result when (a) $\tau \ll \frac{1}{\omega}$, and (b) $\tau \gg \frac{1}{\omega}$.

4. The Hamiltonian which describes the interaction of a static spin-$\frac{1}{2}$ particle with an external magnetic field, $\underline{B}$, is

$$\hat{H} = −\underline{\hat{\mu}} \cdot \underline{B} \nonumber$$

When $\underline{B}$ is a static uniform magnetic field in the $z$-direction, $\underline{B}_0 = (0, 0, B_0)$, the matrix representation of $\hat{H}_0$ is simply

$$− \frac{1}{2} \gamma B_0\hbar \begin{pmatrix} 1 & 0 \\ 0 & −1 \end{pmatrix} \nonumber$$

with eigenvalues $\mp \frac{1}{2} \gamma B_0\hbar$ and for this time-independent Hamiltonian, the energy eigenstates are represented by the 2-component column matrices

$$| \uparrow \rangle \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix} \text{ and } | \downarrow \rangle \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix} \nonumber$$

Now consider superimposing on the static field $\underline{B}_0$ a time-dependent magnetic field of constant magnitude $B_1$, rotating in the $x − y$ plane with constant angular frequency $\omega$:

$$\underline{B}_1 (t) = (B_1 \cos \omega t, B_1 \sin \omega t, 0) \nonumber$$

If the Hamiltonian is now written as $\hat{H} (t) = \hat{H} 0 + \hat{V} (t)$, write down a matrix representation of $\hat{V} (t)$.

Any spin state can be written

$$|\Psi , t \rangle = c_1(t) \text{ exp}(−iE_{\uparrow}t/\hbar )| \uparrow \rangle + c_2(t) \text{ exp} (−iE_{\downarrow}t/\hbar )| \downarrow \rangle \nonumber$$

Obtain, without approximation, the coupled equations for the amplitudes $c_1(t), c_2(t)$.

*If initially at $t = 0$ the system is in the spin-down state, show that the probability that at time $t$, the system is in the spin-up state is given without approximation by

$$p_1(t) = |c_1(t)|^2 = A \sin^2 \left\{ \frac{1}{2} \left[ (\gamma B_1)^2 + (\omega + \gamma B_0)^2 \right]^{1/2} t \right\} \nonumber$$

where

$$A = \frac{(\gamma B_1)^2}{\{[(\gamma B_1)^2 + (\omega + \gamma B_0)^2 ]\}} \nonumber$$

What is the corresponding probability, $p_2(t)$, that the system is in the spin-down state? Sketch $p_1(t)$ and $p_2(t)$ as functions of time.