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Quantum Harmonic Oscillator Differential Equation help

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cybla
#1
Dec3-11, 11:59 PM
P: 16
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
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JJacquelin
#2
Dec4-11, 04:42 AM
P: 759
The solutions of this EDO are known in terms of Modified Bessel functions or alternately in terms of Parabolic Cylinder functions (in attachment)
Attached Thumbnails
EDO 1.JPG  
dextercioby
#3
Dec4-11, 09:51 AM
Sci Advisor
HW Helper
P: 11,915
For the ODE just use the Frobenius method. Series expansion.

cybla
#4
Dec4-11, 09:59 AM
P: 16
Quantum Harmonic Oscillator Differential Equation help

Okay i understand, thank you very much


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