Fermion Labs

Parameters

Quantum Number (n)0

Energy Spacing (ħω)1.00 eV

Particle Mass1.0 m_e

Show |ψ|²

Coherent State

Energy (E)

— eV

Time (t)

— fs

⟨x⟩ Position

— nm

Δx · Δp / ħ

—

Awaiting Telemetry

A particle of mass m in a parabolic potential $V(x) = \frac{1}{2}m\omega^2 x^2$ represents a quantum harmonic oscillator.

-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi = E\psi

The energy levels are evenly spaced by $\hbar\omega$ , starting from a non-zero zero-point energy:

E_n = \hbar\omega\left(n + \frac{1}{2}\right)

The stationary eigenstates involve Hermite polynomials $H_n(x)$ :

\psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} e^{-\frac{m\omega x^2}{2\hbar}} H_n\left(\sqrt{\frac{m\omega}{\hbar}} x\right)

Coherent states are quantum superpositions that perfectly mimic classical oscillatory motion without wavepacket spreading.