Chapter 18 Time-Dependent Perturbation Theory

18.1 The Problem

18.2 First-Order Perturbation Theory

Exercise 18.2.1 Show that if $H^1(t)=-e \mathscr{E} X /\left[1+(t / \tau)^2\right]$, then, to first order,

Exercise 18.2.2 A hydrogen atom is in the ground state at $t=-\infty$. An electric field $\mathbf{E}(t)=(\mathbf{k} \mathscr{E}) e^{-t^2 / \tau^2}$ is applied until $t=\infty$. Show that the probability that the atom ends up in any of the $n=2$ states is, to first order,

where $\omega=\left(E_{2 l m}-E_{100}\right) / \hbar$. Does the answer depend on whether or not we incorporate spin in the picture?

Exercise 18.2.3 Consider a particle in the ground state of a box of length $L$. Argue on semiclassical grounds that the natural time period associated with it is $T \simeq m L^2 / \hbar \pi$. If the box expands symmetrically to double its size in time $\tau \ll T$ what is the probability of catching the particle in the ground state of the new box? (See Exercise (5.2.1).)

Exercise 18.2.4 In the $\beta$ decay $\mathrm{H}^3$ (two neutrons + one proton in the nucleus) $\rightarrow\left(\mathrm{He}^3\right)^{+}$ (two protons + one neuron in the nucleus), the emitted electron has a kinetic energy of $16~\mathrm{keV}$. Argue that the sudden approximation may be used to describe the response of an electron that is initially in the $1 s$ state of $\mathrm{H}^3$. Show that the amplitude for it to be in the ground state of $\left(\mathrm{He}^3\right)^{+}$is $16(2)^{1 / 2} / 27$. What is the probability for it to be in the state

Exercise 18.2.5 An oscillator is in the ground state of $H=H^0+H^1$, where the time-independent perturbation $H^1$ is the linear potential ($-f x$). If at $t=0$, $H^1$ is abruptly turned off, show that the probability that the system is in the $n$th eigenstate of $H^0$ is given by the Poisson distribution

[Hint: Use the formula

where $[A, B]$ is a $c$ number.]

Exercise 18.2.6 Consider a system subject to a perturbation $H^1(t)=H^1 \delta(t)$. Show that if at $t=0^{-}$the system is in the state $\left|i^0\right\rangle$, the amplitude to be in a state $\left|f^0\right\rangle$ at $t=0^{+}$is, to first order,

Notice that (1) the state of the system does change instantaneously; (2) Even though the perturbation is "infinite" at $t=0$, we can still use first-order perturbation theory if the “area under it” is small enough.

18.3 Higher Orders in Perturbation Theory

Exercise 18.3.1 Derive Eq. (18.3.30).

Exercise 18.3.2 In the paramagnetic resonance problem Exercise 14.4.3 we moved to a frame rotating in real space. Show that this is also equivalent to a Hilbert space rotation, but that it takes us neither to the interaction nor the Heisenberg picture, except at resonance. What picture is it at resonance? (If $\mathbf{B}=B_0 \mathbf{k}+B \cos \omega t \mathbf{i}-B \sin \omega t \mathbf{j}$, associate $B_0$ with $H_S^0$ and $B$ with $H_S^1$.)

18.4 A General Discussion of Electromagnetric Interactions

Exercise 18.4.1 By taking the divergence of Eq. (18.4.5) show that the continuity equation must be obeyed if Maxwell's equations are to be mutually consistent.

Exercise 18.4.2 Calculate $\mathbf{E}$ and $\mathbf{B}$ corresponding to $(\mathbf{A}, \phi)$ and $(\mathbf{A}^{\prime}, \phi^{\prime})$ using Eqs. (18.4.7) and (18.4.9) and verify the above claim.

Exercise 18.4.3 Suppose we are given some A and $\phi$ that do not obey the Coulomb gauge conditions. Let us see how they can be transformed to the Coulomb gauge.

(1) Show that if we choose

and transform to ($\mathbf{A}^{\prime}, \phi^{\prime}$) then $\phi^{\prime}=0$. $\mathbf{A}^{\prime}$ is just $\mathbf{A}-\boldsymbol{\nabla} \Lambda$, with $\boldsymbol{\nabla} \cdot \mathbf{A}^{\prime}$ not necessarily zero.

(2) Show that if we gauge transform once more to ($\mathbf{A}^{\prime \prime}, \phi^{\prime \prime}$) via

then $\boldsymbol{\nabla} \cdot \mathbf{A}^{\prime \prime}=0$. [Hint: Recall $\nabla^2\left(1 /\left|\mathbf{r}-\mathbf{r}^{\prime}\right|\right)=-4 \pi \delta^3\left(\mathbf{r}-\mathbf{r}^{\prime}\right)$.]

(3) Verify that $\phi^{\prime \prime}$ is also zero by using $\nabla \cdot \mathbf{E}=0$.

(4) Show that if we want to make any further gauge transformations within the Coulomb gauge, $\Lambda$ must be time independent and obey $\nabla^2 \Lambda=0$. If we demand that $|\mathbf{A}| \rightarrow 0$ at spatial infinity, $\mathbf{A}$ becomes unique.

Exercise 18.4.4 (Proof of Gauge Invariance in the Schrödinger Approach)

(1) Write $H$ for a particle in the potentials $(A, \phi)$.

(2) Write down $H_{\Lambda}$, the Hamiltonian obtained by gauge transforming the potentials.

(3) Show that if $\psi(\mathbf{r}, t)$ is a solution to Schrödinger's equation with the Hamiltonian $H$, then $\psi_{\Lambda}(\mathbf{r}, t)$ given in Eq. (18.4.33) is the corresponding solution with $H \rightarrow H_{\Lambda}$.

18.5 Interaction of Atoms with Electromagnetic Radiation

Exercise 18.5.1 (1) By going through the derivation, argue that we can take the $\mathrm{e}^{\mathrm{i}\mathbf{k} \cdot \mathbf{r}}$ factor into account exactly, by replacing $\mathbf{p}_f$ by $\mathbf{p}_f-\hbar \mathbf{k}$ in Eq. (18.5.19).

(2) Verify the claim made above about the electron momentum distribution.

Exercise 18.5.2 (1) Estimate the photoelectric cross section when the ejected electron has a kinetic energy of $10~\mathrm{Ry}$. Compare it to the atom's geometric cross section $\simeq \pi a_0^2$.

(2) Show that if we consider photoemission from the $1 s$ state of a charge $Z$ atom, $\sigma \propto Z^5$, in the limit $p_f a_0 / Z \hbar \gg 1$.