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user3

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I have been studying Introduction to Quantum Mechanics by Griffith and in a section he solves the Schrodinger equation for a harmonic oscillator potential using the power series method. First he rewrites the shroedinger equation in the form d^2ψ/dε^2 = (ε^2 - K)ψ , where ε= x√(mw/hbar) and K=2E/(ω hbar ) .

Then he says that at large ε, we can approximate the equation to be d^2ψ/dε^2 ≈ ε^2ψ . So ψ≈Ae^(-ε^2 /2 ) + Be^(ε^2 /2 ) . But at large ε, we have to remove the Be^(ε^2 /2 ) term because otherwise, the wave function wouldn't be normalizable: ψ≈Ae^(-ε^2 /2 )

and then he does something that I don't understand :

" ψ(ε) → ( ) e^(-ε^2 /2 ) at large ε

This suggests we "peel off" the exponential part,

ψ(ε) = h(ε)e^(-ε^2 /2 )

"

and then he goes on to solve the h(ε) rather than the ψ(ε) : h(ε) = ∑aj ε^j

Why did he do that? Why not from the very beginning assume that ψ(ε)= ∑aj ε^j and find a recursion formula for that ?