Calculate Expectation Value of Hamiltonian using Dirac Notation?

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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.
 

Answers and Replies

  • #2
fzero
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You can no doubt find the relationship between [itex]\hat{a}, \hat{a}^\dagger[/itex] and [itex]\hat{x},\hat{p}[/itex] in your textbook. However, there is a more direct way to solve the problem when you note that the states [itex]|n\rangle[/itex] are actually labeled by the corresponding eigenvalue of the number operator [itex]\hat{N}[/itex]. Use this fact to compute the value of [itex]\langle n|\hat{H}|m\rangle[/itex] and use that to compute [itex]\langle \psi|\hat{H}|\psi\rangle[/itex].
 

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