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.