Chapter 17 Time-Independence Perturbation Theory

17.1 The Formalism

17.2 Some Examples

Exercise 17.2.1 Consider $H^1=\lambda x^4$ for the oscillator problem.

(1) Show that

(2) Argue that no matter how small $\lambda$ is, the perturbation expansion will break down for some large enough $n$. What is the physical reason?

Exercise 17.2.2 Consider a spin-1/2 particle with gyromagnetic ratio $\gamma$ in a magnetic field $\mathbf{B}=B \mathbf{i}+B_0 \mathbf{k}$. Treating $B$ as a perturbation, calculate the first- and second-order shifts in energy and first-order shift in wave function for the ground state. Then compare the exact answers expanded to the corresponding orders.

Exercise 17.2.3 In our study of the H atom, we assumed that the proton is a point charge $e$. This leads to the familiar Coulomb interaction $\left(-e^2/r\right)$ with the electron.

(1) Show that if the proton is a uniformly dense charge distribution of radius $R$, the interaction is

(2) Calculate the first-order shift in the ground-state energy of hydrogen due to this modification. You may assume $\mathrm{e}^{-R / a_0} \simeq 1$. You should find $E^1=2 e^2 R^2 / 5 a_0^3$.

Exercise 17.2.4 (1) Prove the Thomas-Reiche-Kuhn sum rule

where $|n\rangle$ and $\left|n^{\prime}\right\rangle$ are eigenstates of $H=P^2 / 2 m+V(X)$. [Hint: Eliminate the $E_{n^{\prime}}-E_n$ factor in favor of $H$.]

(2) Test the sum rule on the $n$th state of the oscillator.

Exercise 17.2.5 We have seen that if we neglect the repulsion $e^2 / r_{12}$ between the two electrons in the ground state of He , the energy is $-8 \mathrm{Ry}=-108.8~\mathrm{eV}$. Treating $e^2 / r_{12}$ as a perturbation, show that

so that $E_0^0+E_0^1=-5.5 . \mathrm{Ry}=-74.8~\mathrm{eV}$. Recall that the measured value is $-78.6~\mathrm{eV}$ and the variational estimate is $-77.5~\mathrm{eV}$. [Hint: $\left\langle H^1\right\rangle$ can be viewed as the interaction between two concentric, spherically symmetric exponentially falling charge distributions. Find the potential $\phi(r)$ due to one distribution and calculate the interaction energy between this potential and the other charge distribution.]

Exercise 17.2.6 Verify Eq. (17.2.34).

Exercise 17.2.7 For the oscillator, consider $H^1=-qfX$. Find an $\Omega$ that satisfies Eq. (17.2.38). Feed it into Eq. (17.2.39) for $E_n^2$ and compare with the earlier calculation.

Exercise 17.2.8 Fill in the steps connecting Eqs. (17.2.41) and (17.2.43). Try to use symmetry arguments to reduce the labor involved in evaluating the integrals.

17.3 Degenerate Pertubation Theory

Exercise 17.3.1 Use the dipole selection rules to show that $H^1$ has the above form and carry out the evaluation of $\Delta$.

Exercise 17.3.2 Consider a spin-1 particle (with no orbital degrees of freedom). Let $H=A S_z^2+B\left(S_x^2-S_y^2\right)$, where $S_i$ are $3 \times 3$ spin matrices, and $A \gg B$. Treating the $B$ term as a perturbation, find the eigenstates of $H^0=A S_z^2$ that are stable under the perturbation. Calculate the energy shifts to first order in $B$. How are these related to the exact answers?

Exercise 17.3.3 Consider the case where $H^0$ includes the Coulomb plus spin-orbit interaction and $H^1$ is the effect of a weak magnetic field $\mathbf{B}=B \mathbf{k}$. Using the appropriate basis, show that the first-order level shift is related to $j_z$ by

Sketch the levels for the $n=2$ level assuming that $E^1 \ll E_{\text{f.s.}}^1$.

Exercise 17.3.4 We discuss here some tricks for evaluating the expectation values of certain operators in the eigenstates of hydrogen.

(1) Suppose we want $\langle 1 / r\rangle_{nlm}$. Consider first $\langle\lambda / r\rangle$. We can interpret $\langle\lambda / r\rangle$ as the first order correction due to a perturbation $\lambda / r$. Now this problem can be solved exactly; we just replace $e^2$ by $e^2-\lambda$ everywhere. (Why?) So the exact energy, from Eq. (13.1.16) is $E(\lambda)=$ $-\left(e^2-\lambda\right)^2 m / 2 n^2 \hbar^2$. The first-order correction is the term linear in $\lambda$, that is, $E^1=m e^2 \lambda / n^2 \hbar^2=$ $\langle\lambda / r\rangle$, from which we get $\langle 1 / r\rangle=1 / n^2 a_0$, in agreement with Eq. (13.1.36). For later use, let us observe that as $E(\lambda)=E^0+E^1+\cdots=E(\lambda=0)+\lambda(\mathrm{d}E/\mathrm{d}\lambda)_{\lambda=0}+\cdots$, one way to extract $E^1$ from the exact answer is to calculate $\lambda(\mathrm{d}E/\mathrm{d}\lambda)_{\lambda=0}$.

(2) Consider now $\left\langle\lambda / r^2\right\rangle$. In this case, an exact solution is possible since the perturbation just modifies the centrifugal term as follows:

where $l^{\prime}$ is a function of $\lambda$. Now the dependence of $E$ on $l^{\prime}(\lambda)$ is, from Eq. (13.1.14),

Show that

Canceling $\lambda$ on both sides, we get Eq. (17.3.11).

(3) Consider finally $\left\langle l / r^3\right\rangle$. Since there is no such term in the Coulomb Hamiltonian, we resort to another trick. Consider the radial momentum operator, $p_r=-i \hbar(\partial / \partial r+1 / r)$, in terms of which we may write the radial part of the Hamiltonian

as $p_r^2 / 2 m$. (Verify this.) Using the fact that $\left\langle\left[H, p_r\right]\right\rangle=0$ in the energy eigenstates, and by explictly evaluating the commutator, show that

combining which with the result from part (2) we get Eq. (17.3.20).

(4) Find the mean kinetic energy using the trick from part (1), this time rescaling the mass. Regain the virial theorem.