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 |0&gt; and |1&gt; 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 /> &amp;lt;\hat{H}&amp;gt;=&amp;lt;\psi|\hat{H}|\psi&amp;gt;<br /> <br /> specifically I cannot figure out how to apply the derivatives in the momentum operator from N using this notation.

 

[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.