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Hamiltonian is defined by $H_I = \hbar \omega (\hat{a}^+ \hat{a} + 1/2)$

What is the expectation value of the energy on the number state

$$\vert \psi \rangle = \frac{1}{\sqrt{2}} ( \vert 1 \rangle + \vert 2 \rangle )$$

So I think that its

$$\langle E \rangle = \langle \psi \vert H_I \vert \psi \rangle$$

$$ = \hbar \omega (\langle \psi \vert \hat{a}^+ \hat{a} \vert \psi \rangle + 1/2) $$

$$ = \hbar \omega ((\langle 2 \vert + \langle 1 \vert) \hat{a}^+ \hat{a} (\vert 1 \rangle + \vert 2 \rangle) + 1/2) $$

Now we use that $\hat{n} = \hat{a}^+ \hat{a}$, and then I get confused...does the last expresion become

$$ = \hbar \omega (\langle 1 \vert \hat n \vert 1 \rangle + \langle 2 \vert \hat n \vert 2 \rangle + 1/2) $$

or

$$ = \hbar \omega (\langle 1 \vert \hat n \vert 1 \rangle +\langle 2 \vert \hat n \vert 2 \rangle +\langle 1 \vert \hat n \vert 2 \rangle + \langle 2 \vert \hat n \vert 1 \rangle + 1/2) $$

any advice?


EDIT I think I have the answer

E = $$ = \hbar \omega (\langle 1 \vert \hat n \vert 1 \rangle +\langle 2 \vert \hat n \vert 2 \rangle +\langle 1 \vert \hat n \vert 2 \rangle + \langle 2 \vert \hat n \vert 1 \rangle + 1/2) $$

Where

$$\langle i \vert \hat n \vert j \rangle = j \langle i \vert j \rangle = j \delta_{ij}$$

So $\langle E \rangle = \hbar \omega (3 + 1/2) = \frac{7 \hbar \omega}{2}$

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Yep, perfect, that it's exactly! –  Flint72 Apr 12 at 16:23

1 Answer 1

up vote 1 down vote accepted

Well, I guess the answer is it becomes both! You're almost there!

But then the third and fourth terms in your equation will vanish.

So, since $\hat{n}$ is the number operator, a number state, such as $\vert m \rangle$ say will be an 'eigenfunction' or the $\hat{n}$ operator with 'eigenvalue' m.

It seems that you probably already know this judging by how far you've already gotten.

So then we will get

$$ \langle E \rangle = \hbar \omega (\langle 1 \vert \hat n \vert 1 \rangle +\langle 2 \vert \hat n \vert 2 \rangle +\langle 1 \vert \hat n \vert 2 \rangle + \langle 2 \vert \hat n \vert 1 \rangle + 1/2) $$

$$ = \hbar \omega (\langle 1 \vert (1) \vert 1 \rangle +\langle 2 \vert \hat (2) \vert 2 \rangle +\langle 1 \vert (2) \vert 2 \rangle + \langle 2 \vert (1) \vert 1 \rangle + 1/2) $$

$$ = \hbar \omega ( 1 \langle 1 \vert 1 \rangle + 2 \langle 2 \vert 2 \rangle + 2 \langle 1 \vert 2 \rangle + 1 \langle 2 \vert 1 \rangle + 1/2) $$

$$ = \hbar \omega ( 1 (1) + 2 (1) + 2 (0) + 1 (0) + 1/2) $$

$$ = \hbar \omega ( 3 + 1/2) $$ $$ = \frac{7}{2}\hbar \omega $$

This is because the states $ \vert n \rangle $ and $ \vert m \rangle $ are orthogonal, so their amplitude vanishes.

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