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Closed 21 hours ago.

I know that the Hermite differential equation is:

y′′−2xy′+2ny=0

and the solutions of this differential equations are Hn(x)
.

However, the quantum harmonic oscillator is of the form:

−ay′′+(x2−λ)y=0

I cannot see why the solutions of this differential equation involve hermite polynomials? Is there a way to convert the second ode to the first ode?

Answer 1

The quantum harmonic oscillator equation can be transformed into the Hermite differential equation by introducing appropriate variables and scaling factors. The solutions of the transformed equation yield Hermite polynomials, linking the two seemingly different differential equations.

Step-by-step explanation:

The connection between the quantum harmonic oscillator equation and the Hermite differential equation can be established through a coordinate transformation and appropriate scaling.

In the quantum harmonic oscillator equation, introducing a new variable \
`(y = e^{-(x^2)/(2)}u\)` helps in simplifying the equation. This substitution aims to remove the exponential term and facilitate the comparison with the Hermite differential equation.

Upon making this substitution, the differential equation transforms into a form resembling the Hermite equation.

After further scaling and manipulation, the equation takes on the structure of the Hermite differential equation, (y'' - 2xy' + 2ny = 0), where \(n\) corresponds to the quantum number. The term
`\(e^{-(x^2)/(2)}\)` serves as a weight function that connects the quantum harmonic oscillator solutions to the Hermite polynomials.

This transformation is a powerful mathematical tool that allows physicists to leverage known solutions of the Hermite differential equation (Hermite polynomials) to find solutions for the quantum harmonic oscillator.

The Hermite polynomials naturally emerge in this context, providing a bridge between different physical systems. This mathematical relationship is foundational in quantum mechanics, demonstrating the elegant interplay between seemingly distinct differential equations and revealing the underlying unity in the mathematical description of physical phenomena.