Physics Asked by user289904 on June 23, 2021
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.
The operator $|5ranglelangle2|$ is exactly what it looks like. If you apply it to any state $|phirangle$ you get:
begin{equation} (|5ranglelangle2|)|phirangle=langle2|phirangle|5rangle end{equation} That is, it gives you the ket $|5rangle$ multiplied by the projection of the state $|phirangle$ on $|2rangle$.
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?
Answered by Karim Chahine on June 23, 2021
The sequence is the following $$ langlephi|Q|phirangle=langlephi|5ranglelangle 2|phirangle. $$ The number 5 and 2 refer to $n=5$ and $n=2$ occupation values and so, to the states $|5rangle$ and $|2rangle$ respectively. Please, note that $$ langle m| nrangle=delta_{mn} $$ with $m,n=1,2,ldots$ and $delta_{mn}=1$ for $m=n$ and = otherwise. Then, using all this, one has $$ langlephi|Q|phirangle=frac{6}{pi^2}sum_{m=1}^inftysum_{n=1}^infty C_mC_nlangle m|5ranglelangle 2|nrangle. $$ Could you go on from here?
Answered by Jon on June 23, 2021
So first of all, your state seems to be well-defined and normalised, as, $$ leftlangle phileft|phirightrangle right.=1 $$ 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: $left|5rightrangle leftlangle 2right|$ is not a self-adjoing operator, so it's not a valid observable. A self-adjoint extension of it would be: $$ frac{1}{2}left(left|5rightrangle leftlangle 2right|+left|2rightrangle leftlangle 5right|right) $$ 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 $left|5rightrangle leftlangle 2right|$ is as a transition amplitude of going from $left|5rightrangle $ to $leftlangle 2right|$. 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.
Answered by joigus on June 23, 2021
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