Comprehensive Study Notes

This chapter discusses the second major quantum-mechanical approximation method, perturbation theory.

Suppose we have a system with a time-independent Hamiltonian operator $\hat{H}$ and we are unable to solve the Schrödinger equation

\(
\begin{equation}
\hat{H} \psi{n}=E{n} \psi_{n} \tag{9.1}
\end{equation}
\)

for the eigenfunctions and eigenvalues of the bound stationary states. Suppose also that the Hamiltonian $\hat{H}$ is only slightly different from the Hamiltonian $\hat{H}^{0}$ of a system whose Schrödinger equation

\(
\begin{equation}
\hat{H}^{0} \psi{n}^{(0)}=E{n}^{(0)} \psi_{n}^{(0)} \tag{9.2}
\end{equation}
\)

we can solve. An example is the one-dimensional anharmonic oscillator with

\(
\begin{equation}
\hat{H}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{1}{2} k x^{2}+c x^{3}+d x^{4} \tag{9.3}
\end{equation}
\)

The Hamiltonian (9.3) is closely related to the Hamiltonian

\(
\begin{equation}
\hat{H}^{0}=-\frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}}+\frac{1}{2} k x^{2} \tag{9.4}
\end{equation}
\)

of the harmonic oscillator. If the constants $c$ and $d$ in (9.3) are small, we expect the eigenfunctions and eigenvalues of the anharmonic oscillator to be closely related to those of the harmonic oscillator.

We shall call the system with Hamiltonian $\hat{H}^{0}$ the unperturbed system. The system with Hamiltonian $\hat{H}$ is the perturbed system. The difference between the two Hamiltonians is the perturbation $\hat{H}^{\prime}$ :

\(
\begin{equation}
\hat{H}^{\prime} \equiv \hat{H}-\hat{H}^{0} \tag{9.5}
\end{equation}
\)

\(
\begin{equation}
\hat{H}=\hat{H}^{0}+\hat{H}^{\prime} \tag{9.6}
\end{equation}
\)

(The prime does not denote differentiation.) For the anharmonic oscillator with Hamiltonian (9.3), the perturbation on the related harmonic oscillator is $\hat{H}^{\prime}=c x^{3}+d x^{4}$.

In $\hat{H}^{0} \psi{n}^{(0)}=E{n}^{(0)} \psi{n}^{(0)}$ [Eq. (9.2)], $E{n}^{(0)}$ and $\psi_{n}^{(0)}$ are called the unperturbed energy and unperturbed wave function of state $n$. For $\hat{H}^{0}$ equal to the harmonic-oscillator

Hamiltonian (9.4), $E_{n}^{(0)}$ is $\left(n+\frac{1}{2}\right) h \nu$ [Eq. (4.45)], where $n$ is a nonnegative integer. ( $n$ is used instead of $v$ for consistency with the perturbation-theory notation.) Note that the superscript ${ }^{(0)}$ does not mean the ground state. Perturbation theory can be applied to any state. The subscript $n$ labels the state we are dealing with. The superscript ${ }^{(0)}$ denotes the unperturbed system.

Our task is to relate the unknown eigenvalues and eigenfunctions of the perturbed system to the known eigenvalues and eigenfunctions of the unperturbed system. To aid in doing so, we shall imagine that the perturbation is applied gradually, giving a continuous change from the unperturbed to the perturbed system. Mathematically, this corresponds to inserting a parameter $\lambda$ into the Hamiltonian, so that

\(
\begin{equation}
\hat{H}=\hat{H}^{0}+\lambda \hat{H}^{\prime} \tag{9.7}
\end{equation}
\)

When $\lambda$ is zero, we have the unperturbed system. As $\lambda$ increases, the perturbation grows larger, and at $\lambda=1$ the perturbation is fully "turned on." We inserted $\lambda$ to help relate the perturbed and unperturbed eigenfunctions, and ultimately we shall set $\lambda=1$, thereby eliminating it.

Sections 9.1 to 9.7 deal with time-independent Hamiltonians and stationary states. Section 9.8 deals with time-dependent perturbations.