Review of Operators

First, a little review. Recall that the square integrable functions form a vector space, much like the familiar 3D vector space.

$$\bgroup\color{black}\vec{r}=a\vec{v_1}+b\vec{v_2} \egroup$$

in 3D space becomes

$$\bgroup\color{black}\vert\psi\rangle=\lambda_1\vert\psi_1\rangle+\lambda_2\vert\psi_2\rangle .\egroup$$

The scalar product is defined as

$$\bgroup\color{black} \langle\phi\vert\psi\rangle=\int\limits_{-\infty}^\infty dx\;\phi^*(x)\psi(x) \egroup$$

and many of its properties can be easily deduced from the integral.

$$\bgroup\color{black} \langle\phi\vert\psi\rangle^*=\langle\psi\vert\phi\rangle \egroup$$

As in 3D space,

$$\bgroup\color{black}\vec{a}\cdot\vec{b}\leq\vert a\vert\;\vert b\vert \egroup$$

the magnitude of the dot product is limited by the magnitude of the vectors.

$$\bgroup\color{black}\langle\psi\vert\phi\rangle\leq\sqrt{\langle\psi\vert\psi\rangle\langle\phi\vert\phi\rangle} \egroup$$

This is called the Schwartz inequality.

Operators are associative but not commutative.

$$\bgroup\color{black} AB\vert\psi\rangle=A(B\vert\psi\rangle)=(AB)\vert\psi\rangle \egroup$$

An operator transforms one vector into another vector.

$$\bgroup\color{black} \vert\phi'\rangle=O\vert\phi\rangle \egroup$$

Eigenfunctions of Hermitian operators

$$\bgroup\color{black} H\vert i\rangle=E_i\vert i\rangle \egroup$$

form an orthonormal

$$\bgroup\color{black}\langle i\vert j\rangle=\delta_{ij} \egroup$$

complete set

$$\bgroup\color{black}\vert\psi\rangle=\sum\limits_i\langle i\v... ...\sum\limits_i \vert i\rangle \langle i\vert\psi\rangle .\egroup$$

Note that we can simply describe the $j^{th}$ eigenstate at $\vert j\rangle$.

Expanding the vectors $\vert\phi\rangle$ and $\vert\psi\rangle$,

$$\begin{eqnarray*} \vert\phi\rangle &=&\sum\limits_i b_i \vert i\rangle \\ \vert\psi\rangle &=&\sum\limits_i c_i \vert i\rangle \\ \end{eqnarray*}$$

we can take the dot product by multiplying the components, as in 3D space.

$$\bgroup\color{black}\langle\phi\vert\psi\rangle=\sum\limits_ib_i^*c_i \egroup$$

The expansion in energy eigenfunctions is a very nice way to do the time development of a wave function.

$$\bgroup\color{black}\vert\psi(t)\rangle=\sum\limits_i\langle i\vert\psi(0)\rangle \vert i\rangle e^{-iE_it/\hbar} \egroup$$

The basis of definite momentum states is not in the vector space, yet we can use this basis to form any state in the vector space.

$$\bgroup\color{black}\vert\psi\rangle={1\over\sqrt{2\pi\hbar}}\int\limits_{-\infty}^\infty dp \phi(p) \vert p\rangle \egroup$$

Any of these amplitudes can be used to define the state.

$$\begin{eqnarray*} c_i=\langle i\vert\psi\rangle \\ \psi(x)=\langle x\vert\psi\rangle \\ \phi(p)=\langle p\vert\psi\rangle \\ \end{eqnarray*}$$