Quantum Harmonic Oscillator Differential Equation help

cybla

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

JJacquelin

The solutions of this EDO are known in terms of Modified Bessel functions or alternately in terms of Parabolic Cylinder functions (in attachment)

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dextercioby

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

cybla

Okay i understand, thank you very much