Quantum Harmonic oscillator
The eigenfunction of the quantum harmonic oscillator can be expressed in terms of the Hermite polynomials. The n-th eigenfunction Ψn(x) is given by:
where:
- n is the quantum number, which takes on values of 0, 1, 2, \ldots
- m is the mass of the particle,
- ω is the angular frequency of the oscillator,
- ℏ (h-bar) is the reduced Planck's constant (ℏ=h/2π, where h is Planck's constant),
- Hn(x) is the n-th Hermite polynomial,
- The factor 1√2nn!(mωπℏ)1/4 normalizes the wave function,
- The factor exp(−mωx22ℏ) is a Gaussian envelope,
- The function √mωℏx is a scaling of the x coordinate.
The Hermite polynomials are a set of orthogonal polynomials that appear frequently in quantum mechanics. They are defined through the generating function:
where the sum is over all nonnegative integers n. The n-th Hermite polynomial Hn(x) is then given by the coefficient of tn in this series expansion. These polynomials can be computed explicitly using the recurrence relation:
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