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Source - QHORenderer
- Render QHO energy levels and wavefunctions.
- coherent_state_evolution
- Time evolution of coherent state parameter: alpha(t) = alpha_0 * exp(-iomegat)
- hermite_polynomial
- Physicist’s Hermite polynomial H_n(x) via recurrence.
H_0(x) = 1, H_1(x) = 2x, H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)
- number_state_probability
- Probability of finding n photons in a coherent state |alpha>: P(n) = |alpha|^{2n} e^{-|alpha|^2} / n!
- qho_energy
- Energy of the nth level of a quantum harmonic oscillator.
- qho_ladder_down
- Annihilation (lowering) operator: a |n> = sqrt(n) |n-1>
a = (alpha/sqrt(2)) * x + (1/(alpha*sqrt(2))) * d/dx
- qho_ladder_up
- Creation (raising) operator: a+ |n> = sqrt(n+1) |n+1>
Applied numerically: a+ psi(x) = (1/sqrt(2)) * (xi - d/dxi) psi(x) in scaled coords
- qho_wavefunction
- QHO wave function psi_n(x) = N_n * H_n(xi) * exp(-xi^2/2)
where xi = sqrt(m*omega/hbar) * x