Chapter 14, The Principles of Quantum Dynamics Video Solutions, Quantum mechanics

Problem 1

A particle of charge $q$ moves in a uniform magnetic field $B$ which is directed along the $z$ axis. Using a gauge in which $A_z=0$, show that $q=\left(c p_x-q A_x\right) / q B$ and $p=\left(c p_y-q A_y\right) / c$ may be used as suitable canonically conjugate coordinate and momentum together with the pair $z, p_z$. Derive the energy spectrum and the eigenfunctions in the $q$-representation. Discuss the remaining degeneracy. Propose alternative methods for solving this eigenvalue problem.

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Problem 2

A linear harmonic oscillator is subjected to a spatially uniform external force $F(t)=C \eta(t) e^{-\lambda t}$ where $\lambda$ is a positive constant and $\eta(t)$ the Heaviside step function (A.23). If the oscillator is in the ground state at $t<0$, calculate the probability of finding it at time $t$ in an oscillator eigenstate with quantum number $n$. Assuming $C=\left(\hbar m \lambda^3\right)^{1 / 2}$, examine the variation of the transition probabilities with $n$ and with the ratio $\lambda / \omega, \omega$ being the natural frequency of the harmonic oscillator.

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Problem 3

If the term $V(t)$ in the Hamiltonian changes suddenly ("impulsively"') between time $t$ and $t+\Delta t$, in a time $\Delta t$ short compared with all relevant periods, and assuming only that $\left[V\left(t^{\prime}\right), V\left(t^{\prime \prime}\right)\right]=0$ during the impulse, show that the time development operator is given by
$$
T(t+\Delta t, t)=\exp \left[-\frac{i}{\hbar} \int_t^{t+\Delta t} V\left(t^{\prime}\right) d t^{\prime}\right]
$$

Note especially that the state vector remains unchanged during a sudden change of $V$ by a finite amount.

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Problem 4

. A linear harmonic oscillator in its ground state is exposed to a spatially constant force which at $t=0$ is suddenly removed. Compute the transition probabilities to the excited states of the oscillator. Use the generating function for Hermite polynomials to obtain a general formula. How much energy is transferred?

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Problem 5

In the nuclear beta decay of a tritium atom $\left({ }^3 \mathrm{H}\right)$ in its ground state, an electron is emitted and the nucleus changes into an ${ }^3 \mathrm{He}$ nucleus. Assume that the change is sudden, and compute the probability that the atom is found in the ground state of the helium ion after the emission. Compute the probability of atomic excitation to the $2 S$ and $2 P$ states of the helium ion. How probable is excitation to higher levels, including the continuum?

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Problem 6

A linear harmonic oscillator, with energy eigenstates $|n\rangle$, is subjected to a timedependent interaction between the ground state $|0\rangle$ and the first excited state:
$$
V(t)=F(t)|1\rangle\left\langle 0\left|+F^*(t)\right| 0\right\rangle\langle 1|
$$
(a) Derive the coupled equations of motion for the probability amplitudes $\langle n \mid \Psi(t)\rangle$.
(b) If $F(t)=\sqrt{2} \hbar \omega \eta(t)$, obtain the energy eigenvalues and the stationary states for $t>0$.
(c) If the system is in the ground state of the oscillator before $t=0$, calculate $\langle n \mid \Psi(t)\rangle$ for $t>0$.

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Problem 7

At $t<0$ a system is in a coherent state $|\alpha\rangle$ (eigenstate of $a$ ) of an oscillator and subjected to an impulsive interaction
$$
V(t)=\frac{i \hbar}{2} \zeta\left(a^2-a^{\dagger 2}\right) \delta(t)
$$
where $\zeta$ is a real-valued parameter. Show that the sudden change generates a squeezed state. If the oscillator frequency is $\omega$, derive the time dependence of the variances
$$
\left[\Delta\left(\frac{a+a^{\dagger}}{\sqrt{2}}\right)\right]^2 \text { and }\left[\Delta\left(\frac{a-a^{\dagger}}{\sqrt{2} i}\right)\right]^2
$$

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