The following text is an example of how to use the ideas set out in the perturbation theory writeup. We consider a one-dimensional quantum harmonic oscillator under the influence of a weak electric field of the form -eλx.

We use the Dirac formalism of Quantum Mechanics and write a,a^{†} for the annihilation and creation operators of the harmonic oscillator. In the notation of the perturbation theory writeup we therefore have

H(λ) = H_{0} + λ H'
= {hω (a a^{†} + 1/2)} + λ {-ex}
= {hω (a a^{†} + 1/2)} + λ {-e sqrt(h/2mω) (a+a^{†})}
= {hω (a a^{†} + 1/2)} + λ {-ec (a+a^{†})}

where we have set c=sqrt(h/2mω) to make the notation easier. Here h represents the angular form of Planck's constant. We let |n> be the eigenfunctions of H_{0} with corresponding energies ε_{n}. We assume a Taylor Series expansion of the form

En(λ) = ε_{n} + λF_{n} + λ²G_{n} + ...
|ψ_{n}(λ)> = N (|n> + λ|φ_{n}> + λ²|χ_{n}> + ...)

Using the perturbation theory results, the first order shift in energy is given by

F_{n} = <n|H'|n>
= <n|-ec(a + a^{†})|n>
= - ec(<n|a|n> + <n|a^{†}|n>)
= - ec(<n+1|n>sqrt(n+1) + <n|n+1>sqrt(n+1))
= 0.

The second order shift in energy is given by

G_{n} = -sum(|<r|H'|n>|²/(ε_{r} - ε_{n}), r ≠ n)
= -e²c² sum(|<r|(a+a^{†})|n>|²/(hω(r-n)), r ≠ n)
= e²c²/hω sum(|<r+1|n>sqrt(r+1)+<r|n+1>sqrt(n+1)|²/(n-r), r ≠ n)
= e²/(2mω²) sum(|δ_{(r+1)n}sqrt(r+1)+δ_{r(n+1)}sqrt(n+1)|²/(n-r), r ≠ n)
= e²/(2mω²) (n-(n+1))
= -e²/(2mω²)

where δ is the Kronecker delta.

#### Verification

For this simple case, it turns out that we can solve the Hamiltonian explicitly, which allows us to verify the above results.

H(λ) = H_{0} + λ H'
= {p²/(2m) + mω²x²} + {-eλx}
= p²/(2m) + mω²(x² - 2eλx/(mω²))
= p²/(2m) + mω²(x² - eλx/(mω²))² - mω²e²λ²/(2m²w^{4})
= p²/(2m) + mω²(x² - eλx/(mω²))² - e²λ²/(2mω²).

The effect of the electric field is to shift the harmonic oscillator and alter it by a fixed potential. The energy levels are therefore

E_{n}(λ) = hω(n + 1/2) - e²λ²/(2mω²).

For this example perturbation theory has given us the exact result: this is a very special case.