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What does the asterisk mean?

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What does the asterisk mean?

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gabbagabbahey

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The asterisk stand for complex conjugation.

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And the

Calculate: [tex]\left\langle\Psi\left|A(hat)\right|\right\Psi\rangle[/tex]

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gabbagabbahey

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Are you giving me a pop quiz?Calculate: [tex]\left\langle\Psi\left|A(hat)\right|\right\Psi\rangle[/tex]

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.

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gabbagabbahey

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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)?In the case of [tex]\Psi=c[/tex]_{1}[tex]\Psi[/tex]_{1}[tex] + c[/tex]_{2}[tex]\Psi[/tex]_{2}[tex] + ... + c[/tex]_{n}[tex]\Psi[/tex]_{n}

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?

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What effect does a normalized function vs a 'orthonormal' function have on the 'expectation value'? Sorry I'm being thrown into this terminology very rudely. Any help is greatly appreciated.

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gabbagabbahey

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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?

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