Phy5646/Non-degenerate Perturbation Theory - Problem 3

(Submitted by Team 1)

This example taken from "Quantum Physics" 3rd ed., Stephen Gasiorowicz, p. 177.

Problem: A charged particle in a simple harmonic oscillator, for which ${H}_{0}={\frac {p^{2}}{2m}}+{\frac {mw^{2}}{2}}$, subject to a constant electric field so that ${H}^{'}=q{\mathcal {E}}x$. Calculate the energy shift for the $n^{th}$ level to first and second order in $(q{\mathcal {E}})$. (Hint: Use the operators $a$ and $a^{\dagger }$ for the evaluation of the matrix elements).

Solution: (a) To first order we need to calculate ${H}^{'}=q{\mathcal {E}}\langle n|x|n\rangle$. It is easy to show that $\langle n|x|n\rangle =0$. One way is to use the relation

$x={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })$

and since $A|n\rangle ={\sqrt {n}}|n-1\rangle$ and $A^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle$ we see that $\langle n|x|n\rangle =0$.

(b) The second-order term involves

$q^{2}{\mathcal {E}}^{2}\sum _{k\neq n}{\frac {|\langle k|x|n\rangle |^{2}}{\hbar \omega (n-k)}}={\frac {q^{2}{\mathcal {E}}^{2}}{\hbar \omega }}{\frac {\hbar }{2m\omega }}\sum _{k\neq n}{\frac {|\langle k|a+a^{\dagger }|n\rangle |^{2}}{n-k}}$

The only contributions come from $k=n-1$ and $k=n+1$, so that

$\sum _{k\neq n}{\frac {|\langle k|a+a^{\dagger }|n\rangle |^{2}}{n-k}}=|{\sqrt {n}}|^{2}-|{\sqrt {n+1}}|^{2}=-1$

and thus

$E_{n}^{(2)}={\frac {-q^{2}{\mathcal {E}}^{2}}{2m\omega }}$

The result is independent of $n$. We can check for its correctness by noting that the total potential energy is

${\frac {1}{2}}m\omega ^{2}x^{2}+q{\mathcal {E}}x={\frac {1}{2}}m\omega ^{2}(x^{2}+{\frac {2q{\mathcal {E}}}{m\omega ^{2}}}x)={\frac {1}{2}}m\omega ^{2}(x+{\frac {q{\mathcal {E}}}{m\omega ^{2}}})^{2}-{\frac {q^{2}{\mathcal {E}}^{2}}{2m\omega ^{2}}}$

Thus the perturbation shifts the center of the potential by $-{\frac {q{\mathcal {E}}}{m\omega ^{2}}}$ and lowers the energy by $-{\frac {q^{2}{\mathcal {E}}^{2}}{2m\omega ^{2}}}$, which agrees with our second-order result.