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Calculate Expectation Value of Hamiltonian using Dirac Notation?

  1. Nov 14, 2011 #1
    1. The problem statement, all variables and given/known data
    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.



    2. Relevant 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

    3. 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.
     
  2. jcsd
  3. Nov 14, 2011 #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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