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Normalised Energy Eigenfunction (Probability with Dirac Notation)

  1. Oct 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Normalised energy eigenfunction for ground state of a harmonic oscillator in one dimension is:

    〈x|n〉=α^(1/2)/π^(1/4) exp(-□(1/2) α^2 x^2)

    n = 0

    α^2=mω/h

    suppose now that the oscillator is prepared in the state:

    〈x|ψ〉=σ^(1/2)/π^(1/4) exp(-(1/2) σ^2 x^2)


    What is the probability that a measurement of the energy gives the result E_0 = 1/2 hω?


    2. Relevant equations



    3. The attempt at a solution

    I squared both states as that is the probability of the states, but I cannot see where the energy eigenvalue would fit into it.

    It probably something glaringly obvious, but I haven't done this in a long time and certainly not with dirac notation. Please help.
     
  2. jcsd
  3. Oct 17, 2011 #2
    I squared the states to get probabilities, but now I'm left with 2 probability amplitudes that I'm stumped on how to use.

    What can I use the energy for?

    I know n = 0, but do I have to find the expectation value of getting the energy?
     
  4. Oct 18, 2011 #3

    dextercioby

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    Apply the definition. The probability should be equal to the square modulul of some complex number, namely the scalar product of the 2 states involved. Which are the 2 states involved ?
     
  5. Oct 18, 2011 #4
    Since both states are real, their conjugates equal their normal counterparts. So I end up using an Identity relation to superimpose the states anyway (underhanded trick from my QM lecturer :p ).

    Ok so I'm multiplying and integrating over unity, hopefully a useful co-efficient should pop out which I should then modulus square for the probability? Will show my working a hot second.
     
  6. Oct 18, 2011 #5
    How do I integrate e^[(-1/2)x^2] dx from -∞ to ∞ ?
     
  7. Oct 18, 2011 #6

    dextercioby

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    It's the typical gaussian integration. You should find its value in your QM book, or statistics book. Look it up.
     
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