Deriving hermite differential equation from schrødinger harm oscillator

georg gill · Jan 21, 2016

Homework Statement

I am trying to obtain the hermite polynomial from the schrødinger equation for a har monic oscillator. My attempt is shown below. Thank you! The derivation is based on this site:

http://www.physicspages.com/2011/02/08/harmonic-oscillator-series-solution/

The Attempt at a Solution

Two typeset equations: (1) -(ħ²/2m) d²ψ/dx² + (1/2)kx²ψ = Eψ, and (2) y ≡ sqrt(mω/ħ) x.

Handwritten chain-rule derivation rewriting partial derivatives of psi with respect to y as products of x and y derivatives scaled by sqrt(hbar/(m omega)), concluding hence the mixed second derivative identity.

Handwritten-style derivation of the Hermite equation from the harmonic oscillator Schrödinger equation, showing substitution psi = e^(-y^2/2) f(y) and resulting equation f'' - 2y f' + (y^2 - k/(m w^2) y^2 + 2E/(h w)) f = 0, with omega = sqrt(k/m).

however the book says:

blue_leaf77 · Jan 21, 2016

Seems like your mistake lies in your calculation of ##\frac{\partial^2 \psi}{\partial y^2}##. I suggest that you check again on this point to see whether you had left out one term.

georg gill · Jan 21, 2016

thanks!

Deriving hermite differential equation from schrødinger harm oscillator

Homework Statement

The Attempt at a Solution

Discussion

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