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Linear Harmonic Oscillator Quantum Mechanics

  1. Dec 13, 2015 #1
    1. The problem statement, all variables and given/known data
    A linear harmonic oscillator with frequency ω = hbar / M is at time t = 0 in the state described by the wave-function:
    Ψ(x,0) = C( 1 + √2x) e-x2/2

    Determine the values of energy which can be measured in this state.

    I'm not really sure where to start this question and was wondering if someone could help me get the ball rolling.
    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Dec 13, 2015 #2

    blue_leaf77

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    The question asks you to first recognize the eigenfunctions of linear harmonic oscillator. Go look up for it either in your notes or online.
     
  4. Dec 13, 2015 #3
    Is there a different between the harmonic oscillator and the linear harmonic oscillator? I have a pretty good understanding of the harmonic oscillator but I can't really find any literature to see if there's a difference between the two.
     
  5. Dec 13, 2015 #4

    blue_leaf77

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    Linear harmonic oscillator is the one whose potential energy is given by ##\frac{1}{2}m\omega^2x^2##.
     
  6. Dec 13, 2015 #5
    Do I have to work with ladder operators for this problem as well?

    We know the quantum system has energy, E, which means it is in an energy eigenstate, Ψ. We don't know for certain what energy the system has, but we know the probabilities for various states. Energy must be an eigenvalue of the energy operator, so we say the system is in a superposition of eigenstates:

    Ψ = Σ ciΨi

    |c|2 is the probability the system has energy Ei
     
  7. Dec 14, 2015 #6

    blue_leaf77

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    You don't need to.
    Yes, you are in the right track. The idea is to expand to given wavefunction into the eigenfunctions of harmonic oscillator. The expansion generally spans all possible eigenfunctions,.i.e. infinity. However, in the particular problem at hand, there should be only a few of them whose expansion coefficients are not vanishing. You task is to find the eigenfunctions which corresponds to these non-vanishing coefficients.
     
  8. Dec 14, 2015 #7
    I can't seem to find any reference material on the linear harmonic oscillator. It's all just harmonic oscillator stuff.
     
  9. Dec 14, 2015 #8
    Like I know how to derive E1 = 1/2 Hbarw and E2 = 3/2 Hbarw but I don't know if this is what the question is asking.

    We are given the solution that at the state at time t = 0 is a superposition of energy eigenstates
    Ψ(0) = 1/√2 (Ψ0 + Ψ1)

    I don't understand how to get from the original definition of the wavefunction to that.
     
  10. Dec 14, 2015 #9

    blue_leaf77

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    Check in those references if the potential being discussed satisifes the form I wrote in post #4. Your aim is to find the expression for the first few eigenfunctions.
     
  11. Dec 14, 2015 #10
    Hhat = phat2 / 2 + 1/2 ω2 x2

    The position operator xhat2 is just = to x so I just wrote x above. I've also set m=1 to get rid of it in the equation and make it a little simpler for now.

    Hhat = (-ihbar d/dx)2 + 1/2 ω2 x2

    set hbar = 1 because it clutters up the equation.

    so (-ibar d/dx)2 = -d2/dx2

    Hhat = -1/2 d2/dx2 + 1/2 ω2 x2

    H | Ψ > = -1/2 d2 | Ψ > / dx2 + 1/2 ω2 x2 | Ψ>

    = E | Ψ>

    | Ψ> = e-ωx2/2

    Differentiating this twice we get:

    d2 | ψ> / dx2 = -ω | Ψ> + ω2x2 | ψ>

    H | ψ> = -1/2 ( -ω | ψ > + ω2 x2 | ψ>) + 1/2 ω2 x2 | ψ>)

    H | ψ> = ω/2 | ψ> = E | ψ>

    E = ω/2
    (subbing hbar back in
    E=hbarω/2
    E=hf
     
  12. Dec 14, 2015 #11
    From here I know if we use the a and a+ operators we can find 3/2 hbar/2, 5/2, etc...

    I don't understand why the question has defined the wavefunction Ψ(x,0) = C( 1 + √2x) e-x2/2. Could you explain this to me conceptually?

    i.e. What is C?
     
  13. Dec 14, 2015 #12

    blue_leaf77

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    As I said, you only need to know the functional forms of the first few eigenfunctions of (linear) harmonic oscillator, like those given in http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html. Analyze the form of the asked wavefunction ##\psi(x,0)## an compare it to the form of eigenfunctions of the linear harmonic oscillator to find out which of the expansion coefficients, ##c_n##, in the infinite series ##\psi(x,0) = \sum_n c_n u_n(x)##, where ##u_n(x)## is the eigenfunction of harmonic oscillator, have non-zero value.
     
  14. Dec 14, 2015 #13
    Just looking at them now! I can see a lot of similarities between the first 4 normalised wavefunctions and the function I have been given in the question.
     
  15. Dec 14, 2015 #14

    blue_leaf77

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    Now then check which ##c_n##'s for those first few eigenfunctions have nonzero value.
     
  16. Dec 14, 2015 #15
    I'm still a bit new to the expansion coefficients. Still trying to understand it all.
    .
    I'm a bit lost at this point
     
    Last edited: Dec 14, 2015
  17. Dec 14, 2015 #16
    I don't understand what alpha or y are in the normalised wavefunctions
     
  18. Dec 14, 2015 #17
    or Ci -- it was in my text. I understand the two wave functions are a superposition of each other, but there is no definition in my text as to what Ci is, or any example.
     
  19. Dec 14, 2015 #18

    blue_leaf77

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    How ##\alpha## and ##y## are defined is already stated in that page, read more thoroughly.
    Which wavefunctions are superposition of each other?
    If you don't find how to calculate ##c_n## in your text, this means either you haven't read through it scrupulously or the level of the text is too advanced to be considered as introductory level quantum mechanics.
     
  20. Dec 14, 2015 #19
    What is Cn called? I could look it up!
     
  21. Dec 14, 2015 #20

    blue_leaf77

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    Expansion coefficient, sometimes it is also referred to as projection coefficient.
     
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