Perturbation Theory

The basic idea of perturbation theory is very simple: we split the Hamiltonian into a piece we know how to solve (the ``reference'' or ``unperturbed'' Hamiltonian) and a piece we don't know how to solve (the ``perturbation''). As long as the perburbation is small compared to the unperturbed Hamiltonian, perturbation theory tells us how to correct the solutions to the unperturbed problem to approximately account for the influence of the perturbation. For example, perturbation theory can be used to approximately solve an anharmonic oscillator problem with the Hamiltonian

$${\hat H} = - \frac{\hbar^2}{2\mu} \frac{d^2}{dx^2} + \frac{1}{2} k x^2 + \frac{1}{6} \gamma x^3.$$ (132)

Here, since we know how to solve the harmonic oscillator problem (see 5.2), we make that part the unperturbed Hamiltonian (denoted ${\hat H}^{(0)}$), and the new, anharmonic term is the perturbation (denoted ${\hat H}^{(1)}$):

$${\hat H}^{(0)} = -\frac{\hbar^2}{2\mu} \frac{d^2}{dx^2}+\frac{1}{2} k x^2,$$ (133)

$${\hat H}^{(1)} = + \frac{1}{6} \gamma x^3.$$ (134)

Perturbation theory solves such a problem in two steps. First, obtain the eigenfunctions and eigenvalues of the unperturbed Hamiltonian, ${\hat H}^{(0)}$:

$${\hat H}^{(0)} \Psi^{(0)}_n = E_n^{(0)} \Psi^{(0)}_n.$$ (135)

Second, correct these eigenvalues and/or eigenfunctions to account for the perturbation's influence. Perturbation theory gives these corrections as an infinite series of terms, which become smaller and smaller for well-behaved systems:

$$E_n = E_n^{(0)} + E_n^{(1)} + E_n^{(2)} + \cdots$$ (136)

$$\Psi_n = \Psi_n^{(0)} + \Psi_n^{(1)} + \Psi_n^{(2)} + \cdots$$ (137)

Quite frequently, the corrections are only taken through first or second order (i.e., superscripts (1) or (2)). According to perturbation theory, the first-order correction to the energy is

$$E_n^{(1)} = \int \Psi_n^{(0)*} {\hat H}^{(1)} \Psi_n^{(0)},$$ (138)

and the second-order correction is

$$E_n^{(2)} = \int \Psi_n^{(0)*} {\hat H}^{(1)} \Psi_n^{(1)}.$$ (139)

One can see that the first-order correction to the wavefunction, $\Psi_n^{(1)}$, seems to be needed to compute the second-order energy correction. However, it turns out that the correction $\Psi_n^{(1)}$ can be written in terms of the zeroth-order wavefunction as

$$\Psi_n^{(1)} = \sum_{i \neq n} \Psi_i^{(0)} \frac{\int \Psi_i^{(0)*} {\hat H}^{(1)} \Psi_n^{(0)}}{ E_n^{(0)} - E_i^{(0)}}.$$ (140)

Substituting this in the expression for $E_n^{(2)}$, we obtain

$$E_n^{(2)} = \sum_{i \neq n} \frac{\vert\int \Psi_n^{(0)*} {\hat H}^{(1)} \Psi_i^{(0)}\vert^2}{ E_n^{(0)} - E_i^{(0)}}.$$ (141)

Going back to the anharmonic oscillator example, the ground state wavefunction for the unperturbed problem is just (from section 5.2)

$$E_0^{(0)} = \frac{1}{2} \hbar \omega,$$ (142)

$$\Psi_0^{(0)}(x) = N_0 H_0(\alpha^{1/2} x) e^{- \alpha x^2 / 2}$$ (143)

$$= \left( \frac{\alpha}{\pi} \right)^{1/4} e^{- \alpha x^2 / 2}.$$ (144)

The first-order correction to the ground state energy would be

$$E_0^{(1)} = \left( \frac{\alpha}{\pi} \right)^{1/2} \int_{-\infty}^{\infty} \frac{1}{6} \gamma x^3 e^{- \alpha x^2} dx.$$ (145)

It turns out in this case that $E_0^{(1)} = 0$, since the integrand is odd. Does this mean that the anharmonic energy levels are the same as for the harmonic oscillator? No, because there are higher-order corrections such as $E_0^{(2)}$ which are not necessarily zero.