# Quantum Harmonic Oscillator Differential Equation help

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 P: 16 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
 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
 Sci Advisor HW Helper P: 11,915 For the ODE just use the Frobenius method. Series expansion.
 P: 16 Quantum Harmonic Oscillator Differential Equation help Okay i understand, thank you very much

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