(idea) by Chris Fri Mar 30 2001 at 9:49:20

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) = H0 + 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 H0 with corresponding energies εn. We assume a Taylor Series expansion of the form

En(λ) = εn + λFn + λ²Gn + ...
|ψn(λ)> = N (|n> + λ|φn> + λ²|χn> + ...)

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

Fn = <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

Gn = -sum(|<r|H'|n>|²/(εr - εn), 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)nsqrt(r+1)+δr(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) = H0 + λ H'
     = {p²/(2m) + mω²x²} + {-eλx}
     = p²/(2m) + mω²(x² - 2eλx/(mω²))
     = p²/(2m) + mω²(x² - eλx/(mω²))² - mω²e²λ²/(2m²w4)
     = 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

En(λ) = hω(n + 1/2) - e²λ²/(2mω²).

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

I like it!