# Quantum Harmonic Oscillator Differential Equation help

1. Dec 3, 2011

### cybla

Hi, so i am looking at the quantization of the harmonic oscillator and i have the following equation...

ψ''+ (2ε-y$^{2}$)ψ=0

I am letting y$\rightarrow$ $\infty$ to get...

ψ''- y$^{2}$ψ=0

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

ψ= Ay$^{m}$e$^{\pm y^{2}/2}$

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. Dec 4, 2011

### 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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3. Dec 4, 2011

### dextercioby

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

4. Dec 4, 2011

### cybla

Okay i understand, thank you very much