Quantum Harmonic Oscillator Differential Equation help

  1. Dec 3, 2011 #1
    Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation...

    ψ''+ (2ε-y[itex]^{2}[/itex])ψ=0

    I am letting y[itex]\rightarrow[/itex] [itex]\infty[/itex] to get...

    ψ''- y[itex]^{2}[/itex]ψ=0

    It says the solution to this equation in the same limit is...

    ψ= Ay[itex]^{m}[/itex]e[itex]^{\pm y^{2}/2}[/itex]

    The positive possibility in the exponential is ignored since it is not in the physical Hilbert space. My question is how did they solve this differential equation? I have read a couple websites and it says that you just have to "guess" it... however, is there a logical way to why you would guess this? Thank you
     
  2. jcsd
  3. Dec 4, 2011 #2
    The solutions of this EDO are known in terms of Modified Bessel functions or alternately in terms of Parabolic Cylinder functions (in attachment)
     

    Attached Files:

  4. Dec 4, 2011 #3

    dextercioby

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    For the ODE just use the Frobenius method. Series expansion.
     
  5. Dec 4, 2011 #4
    Okay i understand, thank you very much
     
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