Calculate Expectation Value of Hamiltonian using Dirac Notation?

[kaplac]

Homework Statement

I have the state:
$|\psi>=cos(\theta)|0>+sin(\theta)|1>$
where $\theta$ is an arbitrary real number and $|\psi>$ is normalized.
And [itex]|0> and |1> refer to the ground state and first excited state of the harmonic oscillator.<br /> <br /> Calculate the expectation value of the Hamiltonian for the harmonic oscillator.<br /> <br /> <br /> <br /> <h2>Homework Equations</h2><br /> $\hat{H}=\hbar*\omega(\hat{N}+1/2)$<br /> where<br /> $\hat{N}=\hat{adagger}*\hat{a}$<br /> The product of the raising and lowering operators<br /> <br /> I also know that <br /> [a,adagger]=1<br /> <br /> <h2>The Attempt at a Solution</h2><br /> 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. <br /> But I am curious as to how I can solve this using Dirac notation<br /> $<\hat{H}>=<\psi|\hat{H}|\psi>$<br /> <br /> specifically I cannot figure out how to apply the derivatives in the momentum operator from N using this notation.[/itex]

 

[fzero]

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