Regular Singular Point - Hermite Differential Equation

Hermite Differential Equation

One encounters this ordinary second order differential equation in solving the one dimensional time independent Schrödinger equation

for a harmonic oscillator. In this case the potential energy V(x) is:

This leads to the following ordinary second order differential equation:

$\frac {d^2 f} {dx^2} - 2 x \frac {df} {dx} + \lambda f = 0.$

This differential equation has an irregular singularity at ∞. Its solutions are Hermite polynomials.

Read more about this topic: Regular Singular Point

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