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  {\bf University of Kentucky, Physics 521 \\
    Homework \#16, Rev. A, due Wednesday, 2018-03-21 }
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{\bf 0.} Griffiths [2ed]
Ch.~6 \#1,  %p254 delta in inf sq well
      \#2,  %p255 sho k->k'
      \#5,  %p256 sho 2nd order
      \#9,  %p266 deg 3x3 matrix
      \#16, %p276 spin commutators
      \#36% %p289 Stark effect
.

{\bf 1.} Show that in {\bf third-order} perturbation theory,
\[ E_{n}^{3}=\sum_{l,m\neq
    n}\frac{V_{nl}V_{lm}V_{mn}}{\Delta_{nl}\Delta_{nm}}-V_{nn}\sum_{m\neq
    n}\frac{|V_{nm}|^{2}}{\Delta^{2}_{nm}}, \] where
$V_{mn}\equiv\langle\psi_{m}^{0}|H'|\psi_{n}^{0}\rangle$ and
$\Delta_{mn}=E_{m}-E_{n}$ [Griffiths, p.  256].

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