cybla
Dec3-11, 11:59 PM
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
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