Finding the Expectation value for a wave function

Question

I have a question about finding the expectation value. Here's the question:

So I have this wave function $$|⟩=\frac{\sqrt{6}}{}\sum_{n=1}^∞ C_|n⟩ $$

where the eigenvectors |n⟩ form an orthonormal basis and:

$$C_n=\frac{(-1)^n}{n} $$

So what I need to do consider an operator |5⟩⟨2| - I don't know what that means first of all.

And I need to find the expectation value corresponding to this operator for a particle in the state |⟩.

I know that in order to calculate the expectation value:

$$ <Q> =⟨|\hat Q|⟩ $$

So my question is what does |5⟩⟨2| actually mean? And going on, what shall I do next to calculate the expectation value.

Thanks for reading.

Answer 1

The operator $|5\rangle\langle2|$ is exactly what it looks like. If you apply it to any state $|\phi\rangle$ you get:

\begin{equation} (|5\rangle\langle2|)|\phi\rangle=\langle2|\phi\rangle|5\rangle \end{equation} That is, it gives you the ket $|5\rangle$ multiplied by the projection of the state $|\phi\rangle$ on $|2\rangle$.

In general, remember that an operator is defined by how it acts on the states of your vector space.

Can you now evaluate the expectation value?

Answer 2

The sequence is the following $$ \langle\phi|Q|\phi\rangle=\langle\phi|5\rangle\langle 2|\phi\rangle. $$ The number 5 and 2 refer to $n=5$ and $n=2$ occupation values and so, to the states $|5\rangle$ and $|2\rangle$ respectively. Please, note that $$ \langle m| n\rangle=\delta_{mn} $$ with $m,n=1,2,\ldots$ and $\delta_{mn}=1$ for $m=n$ and = otherwise. Then, using all this, one has $$ \langle\phi|Q|\phi\rangle=\frac{6}{\pi^2}\sum_{m=1}^\infty\sum_{n=1}^\infty C_mC_n\langle m|5\rangle\langle 2|n\rangle. $$ Could you go on from here?

Answer 3

So first of all, your state seems to be well-defined and normalised, as, $$ \left\langle \phi\left|\phi\right\rangle \right.=1 $$ on account of, $$ \zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots=\frac{\pi^{2}}{6} $$ https://en.wikipedia.org/wiki/Basel_problem

But you have a problem: $\left|5\right\rangle \left\langle 2\right|$ is not a self-adjoing operator, so it's not a valid observable. A self-adjoint extension of it would be: $$ \frac{1}{2}\left(\left|5\right\rangle \left\langle 2\right|+\left|2\right\rangle \left\langle 5\right|\right) $$ Once you do that, it's a simple matter of "sandwiching" the operator with the state in the <bra|ket> way and using orthogonality. Other way to interpret $\left|5\right\rangle \left\langle 2\right|$ is as a transition amplitude of going from $\left|5\right\rangle $ to $\left\langle 2\right|$. It's the words "expected value" that do not sit well with the particular operator you've proposed. I hope that was helpful, and I didn't make any idiotic mistake, to which I seem to be prone lately.