Question

Show that the wave function ψ₀(x) = A exp(−x²/2a²) is a solution to the time- independent Schrodinger equation for a simple harmonic oscillator (SHO) potential.

with energy E₀ = (1/2)ℏω₀, and determine a in terms of m and ω₀. 

The corresponding dimensionless form of this equation is

Show that putting ψ(R) = AH (R) exp(−R²/2) into this equation leads to Hermite’s equation

H(R) is a polynomial of order n of the form a_nRⁿ + a_n−2 Rⁿ⁻² + a_n−4 Rⁿ⁻⁴+... 

Deduce that ε is a simple function of n and that the energy levels are equally spaced.

Answer 1

 By substituting ψ(R) = AH(R) exp (−R²/2) in the dimensionless form of the equation and simplifying we easily get the Hermite’s equation 

The problem is solved by the series method