Comprehensive Study Notes

The Schrödinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic-repulsion terms in the Hamiltonian, the Schrödinger equation for many-electron atoms and molecules cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9 . To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter.

Before starting, we introduce some notation. The definite integral over all space of an operator sandwiched between two functions occurs often, and various abbreviations are used:

\(
\begin{equation}
\int f_{m}^{} \hat{A} f{n} d \tau \equiv\left\langle f{m}\right| \hat{A}\left|f_{n}\right\rangle \equiv\langle m| \hat{A}|n\rangle \tag{7.1}
\end{}
\)

where $f{m}$ and $f{n}$ are two functions. If it is clear what functions are meant, we can use just the indexes, as indicated in (7.1). The above notation, introduced by Dirac, is called bracket notation. Another notation is

\(
\begin{equation}
\int f_{m}^{} \hat{A} f{n} d \tau \equiv A{m n} \tag{7.2}
\end{}
\)

The notations $A_{m n}$ and $\langle m| \hat{A}|n\rangle$ imply that we use the complex conjugate of the function whose letter appears first. The definite integral $\langle m| \hat{A}|n\rangle$ is called a matrix element of the operator $\hat{A}$. Matrices are rectangular arrays of numbers and obey certain rules of combination (see Section 7.10).

For the definite integral over all space between two functions, we write

\(
\begin{equation}
\int f_{m}^{} f{n} d \tau \equiv\left\langle f{m} \mid f{n}\right\rangle \equiv\left(f{m}, f_{n}\right) \equiv\langle m \mid n\rangle \tag{7.3}
\end{}
\)

Note that

\(
\langle f| \hat{B}|g\rangle=\langle f \mid \hat{B} g\rangle
\)

where $f$ and $g$ are functions. Since $\left(\int f{m}^{*} f{n} d \tau\right)^{}=\int f_{n}^{} f_{m} d \tau$, we have the identity

\(
\begin{equation}
\langle m \mid n\rangle^{}=\langle n \mid m\rangle \tag{7.4}
\end{}
\)

Since the complex conjugate of $f_{m}$ in (7.1) is taken, it follows that

\(
\begin{equation}
\langle c f| \hat{B}|g\rangle=c^{}\langle f| \hat{B}|g\rangle \quad \text { and } \quad\langle f| \hat{B}|c g\rangle=c\langle f| \hat{B}|g\rangle \tag{7.5}
\end{}
\)

where $\hat{B}$ is a linear operator and $c$ is a constant.