Confusion about Dirac notation

[Philip Land]

Using that $\hat{a} =a = \sqrt{\frac{mw}{2 \hbar}} \hat{x} +\frac{i}{\sqrt{2mw \hbar}} \hat{p}$ and $a \dagger = \sqrt{\frac{mw}{2 \hbar}} \hat{x} -\frac{i}{\sqrt{2mw \hbar}} \hat{p}$

We can solve for x in term of the lowering and raising operator.

Now, recently I read a derivation of $<n| \hat{x} |m> (1)$.

Question 1: n and m were never specified, so what does the above expression actually mean?

By substitution, we can rewrite (1) to $\sqrt{\frac{ \hbar}{2mw}} <n| (a + a \dagger )|m>$(2)

Question 2: I'm a little confused about how I can simplify the above expression. I'm not super familiar with Dirac notation. I know very well the definition of the raising and lowering operators. But can someone fill in the blanks of how they get from (2) to $\sqrt{\frac{ \hbar}{2mw}} \cdot ( \sqrt{m} \delta_{n, m-1} + \sqrt{m+1} \delta_{n, m+1})$? No relation between n and m is defined.

That is not clear to me.

 

[DrClaude]

More context would be necessary for 100% certainty, but these appear to be eigenkets of the harmonic oscillator, such that
$$
\hat{H} | n \rangle = \hbar \omega \left(n + \frac{1}{2} \right) | n \rangle
$$

As for the second question, you need to know the action of the raising and lowering operators of those eigenkets: $\hat{a} | n \rangle = ?$. I'll let you look it up in your textbook.

 

[Philip Land]

More context would be necessary for 100% certainty, but these appear to be eigenkets of the harmonic oscillator, such that
$$
\hat{H} | n \rangle = \hbar \omega \left(n + \frac{1}{2} \right) | n \rangle
$$

As for the second question, you need to know the action of the raising and lowering operators of those eigenkets: $\hat{a} | n \rangle = ?$. I'll let you look it up in your textbook.

Yes you are right that we are looking at harmonic oscillators.

And yes I know the definitions of the ladder operators (figure attached).

But I still don't follow, because I don't know how to apply the sum of the ladder operators on the two states with the Dirac-notation in (2). Simply an algebra problem.

Can I rewrite (2) so I can see perhaps more clearly about what's going on?

 

[stevendaryl]

So given those facts about $\hat{a}$ and $\hat{a}^\dagger$, you immediately get:

$(\hat{a} + \hat{a}^\dagger) |m\rangle = \sqrt{m} |m -1\rangle + \sqrt{m+1} |m+1\rangle$

The final fact that you need is: What is the value of $\langle n|m\rangle$? The whole point of an orthonormal basis is that different basis elements are orthogonal (they give 0 inner product).