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$^{2}$)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

 

[JJacquelin]

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

 

[dextercioby]

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

 

[cybla]

Okay i understand, thank you very much

 

[blue_raver22]

for your question. The solution to this differential equation is known as the Hermite function and it is a well-known solution to the quantum harmonic oscillator problem. The reason for guessing this particular form of the solution is based on the properties of the harmonic oscillator potential, which is a parabolic potential. The Hermite function is a solution to the Schrödinger equation for this type of potential and it is also a well-behaved and physically meaningful function. Therefore, it is a logical and reasonable choice for a solution to this differential equation. Additionally, the exponential term in the solution accounts for the wave-like behavior of the system. If you are interested in learning more about the derivation of this solution, I recommend studying the Schrödinger equation for the quantum harmonic oscillator in more detail.