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## Homework Statement

I have the state:

[itex]|\psi>=cos(\theta)|0>+sin(\theta)|1>[/itex]

where [itex]\theta[/itex] is an arbitrary real number and [itex]|\psi>[/itex] is normalized.

And [itex]|0> and |1> refer to the ground state and first excited state of the harmonic oscillator.

Calculate the expectation value of the Hamiltonian for the harmonic oscillator.

## Homework Equations

[itex]\hat{H}=\hbar*\omega(\hat{N}+1/2)[/itex]

where

[itex]\hat{N}=\hat{adagger}*\hat{a}[/itex]

The product of the raising and lowering operators

I also know that

[a,adagger]=1

## The Attempt at a Solution

So far I know that I can solve this by converting the two states, 0 and 1, to the wave functions and solving the integral.

But I am curious as to how I can solve this using Dirac notation

[itex]<\hat{H}>=<\psi|\hat{H}|\psi>[/itex]

specifically I cannot figure out how to apply the derivatives in the momentum operator from N using this notation.