# 16.4: Exercises - Time Dependence

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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.

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