Chapter 6: Time-Independent Perturbation Theory

Question: In the text I asserted that the first-order corrections to an n-fold degenerate energy are the eigen values of the Wmatrix, and I justified this claim as the "natural" generalization of the case n = 2.

Prove it, by reproducing the steps in Section 6.2.1, starting with

${\psi }^0=\sum _{j=1}^n{\alpha }_j{\psi }_j^0$

(generalizing Equation 6.17), and ending by showing that the analog to Equation6.22 can be interpreted as the eigen value equation for the matrix W.

Question: Use the virial theorem (Problem 4.40) to prove Equation 6.55.

Question: In Problem 4.43you calculated the expectation value of${\mathit{r}}^{\mathbf{s}}$in the state${\mathbf{\psi }}_{\mathbf{321}}$. Check your answer for the special cases s = 0(trivial), s = -1(Equation 6.55), s = -2(Equation 6.56), and s = -3(Equation 6.64). Comment on the case s = -7.

Find the (lowest order) relativistic correction to the energy levels of the one-dimensional harmonic oscillator. Hint: Use the technique in Example 2.5 .

Show that${\mathit{P}}^{\mathbf{2}}$is Hermitian, but${\mathit{P}}^{\mathbf{4}}$is not, for hydrogen states with$\mathit{l}\mathbf{=}\mathbf{0}$. Hint: For such states$\mathit{\psi }$is independent of$\mathit{\theta }$and$\mathit{\varphi }$, so

localid="1656070791118" ${\mathit{p}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\frac{{\mathbf{\hslash }}^{\mathbf{2}}}{{\mathbf{r}}^{\mathbf{2}}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{\left(}{\mathbf{r}}^{\mathbf{2}}\frac{\mathbf{d}}{\mathbf{d}\mathbf{r}}\mathbf{\right)}$

(Equation 4.13). Using integration by parts, show that

localid="1656069411605" $\mathbf{<}\mathit{f}\mathbf{\mid }{\mathit{p}}^{\mathbf{2}}\mathit{g}\mathbf{>}\mathbf{=}\mathbf{-}\mathbf{├}\mathbf{4}\mathit{\pi }{\mathit{h}}^{\mathbf{2}}\mathbf{(}{\mathbf{r}}^{\mathbf{2}}\mathbf{f}\frac{\mathbf{d}\mathbf{g}}{\mathbf{d}\mathbf{r}}\mathbf{-}{\mathbf{r}}^{\mathbf{2}}\mathbf{g}\frac{\mathbf{d}\mathbf{f}}{\mathbf{d}\mathbf{r}}\mathbf{)}\mathbf{}\overline{)\underset{\mathbf{0}}{\overset{\mathbf{\infty }}{\mathbf{}}}\mathbf{}}\mathbf{+}\mathbf{<}{\mathbf{p}}^{\mathbf{2}}\mathbf{f}\mathbf{\mid }\mathbf{g}\mathbf{>}$

Check that the boundary term vanishes for${\mathit{\psi }}_{\mathbf{n}\mathbf{00}}$, which goes like

${\mathit{\psi }}_{\mathbf{n}\mathbf{00}}\mathbf{~}\frac{\mathbf{1}}{\sqrt{\mathbf{\pi }\mathbf{(}\mathbf{n}\mathbf{a}{\mathbf{)}}^{\mathbf{3}\mathbf{/}\mathbf{2}}}}\mathit{e}\mathit{x}\mathit{p}\mathbf{(}\mathbf{-}\mathit{r}\mathbf{/}\mathit{n}\mathit{a}\mathbf{)}$

near the origin. Now do the same for${\mathit{p}}^{\mathbf{4}}$, and show that the boundary terms do not vanish. In fact:

$<{\psi }_{n00}\mid p^4{\psi }_{m00}>\mathbf{=}\frac{\mathbf{8}{\mathbf{\hslash }}^{\mathbf{4}}}{{\mathbf{a}}^{\mathbf{4}}}\frac{\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{m}\mathbf{)}}{\mathbf{(}\mathbf{n}\mathbf{m}{\mathbf{)}}^{\mathbf{5}\mathbf{/}\mathbf{2}}}\mathbf{+}<p^4{\psi }_{n00}\mid {\psi }_{m00}>$

Question: Evaluate the following commutators :

a)$\left[L·S,L\right]$

b)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}\mathbf{S}\mathbf{]}$

c)role="math" localid="1658226147021" $\mathbf{}\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}\mathbf{J}\mathbf{]}$

d)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{L}}^{\mathbf{2}}\mathbf{]}$

e)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{S}}^{\mathbf{2}}\mathbf{]}$

f)$\mathbf{[}\mathbf{L}\mathbf{·}\mathbf{S}\mathbf{,}{\mathbf{J}}^{\mathbf{2}}\mathbf{]}$

Hint: L and S satisfy the fundamental commutation relations for angular momentum (Equations 4.99 and 4.134 ), but they commute with each other.

$$\begin{gathered} \left[L_X,L_Y\right]\mathbf{=}{\mathbf{ihL}}_{\mathbf{z}}\mathbf{;}\mathbf{[}{\mathbf{L}}_{\mathbf{y}}\mathbf{,}{\mathbf{L}}_{\mathbf{z}}\mathbf{]}\mathbf{=}{\mathbf{ihL}}_{\mathbf{x}}\mathbf{;}\mathbf{[}{\mathbf{L}}_{\mathbf{z}}\mathbf{,}{\mathbf{L}}_{\mathbf{x}}\mathbf{]}\mathbf{=}{\mathbf{ihL}}_{\mathbf{y}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{4}\mathbf{.}\mathbf{99} \\ \mathbf{[}{\mathbf{S}}_{\mathbf{X}}\mathbf{,}{\mathbf{S}}_{\mathbf{Y}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{z}}\mathbf{;}\mathbf{[}{\mathbf{S}}_{\mathbf{y}}\mathbf{,}{\mathbf{S}}_{\mathbf{z}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{x}}\mathbf{;}\mathbf{[}{\mathbf{S}}_{\mathbf{z}}\mathbf{,}{\mathbf{S}}_{\mathbf{x}}\mathbf{]}\mathbf{=}{\mathbf{ihS}}_{\mathbf{y}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{4}\mathbf{.}\mathbf{134} \end{gathered}$$

Question: Derive the fine structure formula (Equation 6.66) from the relativistic correction (Equation 6.57) and the spin-orbit coupling (Equation 6.65). Hint: Note tha $\mathrm{j}=\mathrm{l}\pm \frac{1}{2}$t; treat the plus sign and the minus sign separately, and you'll find that you get the same final answer either way.

Question: The most prominent feature of the hydrogen spectrum in the visible region is the red Balmer line, coming from the transition n = 3to n = 2. First of all, determine the wavelength and frequency of this line according to the Bohr Theory. Fine structure splits this line into several closely spaced lines; the question is: How many, and what is their spacing? Hint: First determine how many sublevels the n = 2level splits into, and find ${\mathbf{E}}_{\mathbf{fs}}^{\mathbf{1}}$for each of these, in eV. Then do the same for n = 3. Draw an energy level diagram showing all possible transitions from n = 3to n = 2. The energy released (in the form of a photon) is role="math" localid="1658311193797" $\mathbf{(}{\mathbf{E}}_{\mathbf{3}}\mathbf{-}{\mathbf{E}}_{\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{∆}\mathbf{E}$, the first part being common to all of them, and the $\mathbf{∆}\mathbf{E}$(due to fine structure) varying from one transition to the next. Find $\mathbf{∆}\mathbf{E}$(in eV) for each transition. Finally, convert to photon frequency, and determine the spacing between adjacent spectral lines (in Hz- -not the frequency interval between each line and the unperturbed line (which is, of course, unobservable), but the frequency interval between each line and the next one. Your final answer should take the form: "The red Balmer line splits into (???)lines. In order of increasing frequency, they come from the transitionsto (1) j =(???),toj =(???) ,(2) j =(???) to j =(???)……. The frequency spacing between line (1)and line (2)is (???) Hz, the spacing between line (2)and (3) line (???) Hzis……..”

Question: The exact fine-structure formula for hydrogen (obtained from the Dirac equation without recourse to perturbation theory) is 16

${\mathbf{E}}_{\mathbf{nj}}\mathbf{=}{\mathbf{mc}}^{\mathbf{2}}\left\{{\left[1+{\left(\frac{a}{n-\left(j+{\displaystyle \frac{1}{2}}\right)+\sqrt{{\left(j+\frac{1}{2}\right)}^2-a^2}}\right)}^2\right]}^{-\frac{1}{2}}-1\right\}$

Expand to order ${\mathbf{\alpha }}^{\mathbf{4}}$(noting that $\mathbf{\alpha }\mathbf{\ll }\mathbf{1}$), and show that you recoverEquation .

Suppose we put a delta-function bump in the center of the infinite square well:

${\mathbf{H}}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{\alpha \delta }\left(x-a/2\right)$

where$\mathit{a}$is a constant.

(a) Find the first-order correction to the allowed energies. Explain why the energies are not perturbed for even$\mathbf{n}$.

(b) Find the first three nonzero terms in the expansion (Equation 6.13) of the correction to the ground state,${\mathit{\Psi }}_{\mathbf{1}}^{\mathbf{1}}$.