Homework Help: Operator Terminology

The asterisk stand for complex conjugation.

 

Are you giving me a pop quiz?

You need to show us an attempt in order to receive help with your homework problem. Simply stating the problem does not qualify as an attempt.

 

Okay, well when you say:

Do you mean [itex]|\Psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle+\ldots+c_n|\psi_n\rangle[/itex] (abstract form), or do you mean [itex]\Psi(\textbf{r})=c_1\psi_1(\textbf{r})+c_2\psi_2(\textbf{r})+\ldots+c_n\psi_n(\textbf{r})[/itex] (all the wavefunctions are expanded in the position basis)?

More importantly, what do you know about [itex]\{|\psi_1\rangle,|\psi_2\rangle,\ldots,|\psi_n\rangle\}[/itex]? For example, are they orthogonal? Normalized?

 

Orthonormal means that the eigenfunctions are both orthogonal and normalized to unity.

In abstract form this means that [itex]\langle\psi_i|\psi_j\rangle=\delta_{ij}[/itex], where [tex]\delta_{ij}=\left\{\begin{array}{lr} 0, & i\neq j \\ 1, & i=j \end{array}\right.[/itex] is the Kronecker delta.

When you expand the eigenfunctions in the position basis (i.e. [itex]\psi(\textbf{r})=\langle\hat{\mathbf{r}}|\psi\rangle[/itex]), you get

[tex]\langle\psi_i|\psi_j\rangle=\int_{-\infty}^{\infty}\psi_i^{*}(\textbf{r})\psi_j(\textbf{r})d\tau=\delta_{ij}[/tex]


Start by calculating [itex]\hat{A}|\Psi\rangle[/itex]...what do you get for that?