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Homework Help: 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:
    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
    The product of the raising and lowering operators

    I also know that

    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

    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


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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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