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Eigenfunction energy levels in a harmonic well

  1. Dec 23, 2017 #1
    1. The problem statement, all variables and given/known data
    If the first two energy eigenfunctions are
    ## \psi _0(x) = (\frac {1}{\sqrt \pi a})^ \frac{1}{2} e^\frac{-x^2}{2a^2} ##,
    ## \psi _1(x) = (\frac {1}{2\sqrt \pi a})^ \frac{1}{2}\frac{2x}{a} e^\frac{-x^2}{2a^2} ##
    2. Relevant equations


    3. The attempt at a solution
    Would it then be correct to presume
    ## \psi _3(x) = (\frac {1}{4\sqrt \pi a})^ \frac{1}{2}\frac{4x}{a} e^\frac{-x^2}{2a^2} ##

    Thank you for your time in considering this.
     
  2. jcsd
  3. Dec 23, 2017 #2

    kuruman

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    It would not because it is not orthogonal to ##\psi_1(x)## but the same as ##\psi_1(x)##. Also, you do not state the question that the problem asks.
     
  4. Dec 23, 2017 #3
    No. In terms of ladder operators, the nth eigenfunction is given by

    [tex]

    |n \rangle \equiv \psi_{n}(x) = \frac{(a^\dagger)^n}{\sqrt{n!}} |0 \rangle

    [/tex]
     
  5. Dec 24, 2017 #4
    the specific question goes as so

    For this equation

    ## \Psi (x,0) = \frac {1}{\sqrt{2}}(\psi_1 (x)-\psi_3 (x)) ##

    The system is undisturbed, obtain an expression for ##\psi (x,t)## that is valid for all t ≥ 0. Express in terms of the functions ##\psi_1 (x)##, ##\psi_3 (x)## and ##ω_0##, the classical angular frequency of the oscillator.

    I am trying to approach this by simply inputting the eigenfunctions for

    ##\psi _1(x) = (\frac {1}{2\sqrt \pi a})^ \frac{1}{2}\frac{2x}{a} e^\frac{-x^2}{2a^2}##

    And then for

    ##\psi _3(x)## (which as yet I haven't understood)

    And

    ##a = \sqrt{\frac {\hbar}{ω_0}}##

    Would this be the correct approach to express in the terms as stated?

    Thank you for assisting me with my problem.
     
  6. Dec 24, 2017 #5

    kuruman

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  7. Jan 1, 2018 #6
    So would i use the fact that ## E_1 = \frac {3}{2} \hbar ω_0 ## which would give ## e^ \frac {- 3iω_0t}{2} ##
    And ## E_3 = \frac {7}{2} \hbar ω_0 ## which would give ## e^ \frac {- 7iω_0t}{2} ##

    Am I on the right track?
     
  8. Jan 1, 2018 #7

    PeroK

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    Yes.
     
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