⚛️ Quantum Harmonic Oscillator
The quantum harmonic oscillator — a mass on a spring in quantum mechanics — is one of the most important exactly-solvable models in physics. Its energy levels are E_n = ℏω(n + ½), quantized in steps of ℏω with a non-zero zero-point energy ½ℏω at n = 0. The wavefunctions involve Hermite polynomials multiplied by a Gaussian envelope. The probability of finding the particle tunnels beyond the classical turning points ±√(2n+1) (shown as dashed lines). 🇺🇦 Українська
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Energy Levels
Key Quantum Features
At n = 0 (ground state) the wavefunction is a pure Gaussian — the minimum uncertainty state. For large n, the probability density peaks near the classical turning points (where classical motion is slowest), recovering the correspondence principle. The n-th eigenstate has exactly n nodes. Tunneling probability decreases rapidly with n as the wavefunction becomes more classical. Applications: molecular vibrations (IR spectroscopy), phonons in solids, vacuum fluctuations in quantum optics, Casimir effect, squeezed states in quantum information.