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Quantum Harmonic Oscillator Differential Equation help 
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#1
Dec311, 11:59 PM

P: 16

Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation...
ψ''+ (2εy[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
Dec411, 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)



#3
Dec411, 09:51 AM

Sci Advisor
HW Helper
P: 11,915

For the ODE just use the Frobenius method. Series expansion.



#4
Dec411, 09:59 AM

P: 16

Quantum Harmonic Oscillator Differential Equation help
Okay i understand, thank you very much



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