- #1
doggydan42
- 170
- 18
Homework Statement
Consider the state
$$\psi_\alpha = Ne^{\alpha \hat a^\dagger}\phi_0, $$
where ##\alpha## can be complex, and ##N = e^{-\frac{1}{2}|\alpha|^2}## normalizes ##\psi_\alpha##.
Find ##\hat N \psi_\alpha##.
Homework Equations
$$\hat N = \hat a^\dagger \hat a$$
$$\hat a\phi_n = \sqrt{n}\phi_{n-1}$$
$$\hat a^\dagger = \sqrt{n+1}\phi_{n+1}$$
$$\hat a \psi_\alpha = \alpha \psi_\alpha$$
$$\phi_n = \frac{1}{\sqrt{n!}}(\hat a^\dagger)^n \phi_0$$
The Attempt at a Solution
First applying ##\hat a## gives
$$\hat N \psi_\alpha = \hat a ^\dagger \alpha \psi_\alpha$$
Expanding the exponential gives
$$\alpha N \sum_{n=0}^\infty \frac{\alpha^n}{n!}(\hat a ^\dagger)^{n+1}\phi_0 = N \sum_{n=0}^\infty \frac{\alpha^{n+1}}{n!}(\hat a ^\dagger)^{n+1}\phi_0 = N \sum_{n=1}^\infty \frac{\alpha^{n}}{(n-1)!}(\hat a ^\dagger)^{n}\phi_0 = N \sum_{n=1}^\infty \frac{\alpha^{n}}{n!}n(\hat a ^\dagger)^{n}\phi_0$$
Because at n = 0 the sum becomes 0,
$$\hat N \psi_\alpha = N \sum_{n=0}^\infty \frac{\alpha^{n}}{n!}n(\hat a ^\dagger)^{n}\phi_0$$
From here, I am unsure of how to simplify the summation.