Quantum Mechanics — Exactly Solvable Models
Quantum Harmonic Oscillator
1-D potential well — Hermite polynomials — coherent statesĤ = p̂²/2m + ½mω²x̂²
Parameters
State coefficients
c₀ (n=0)1
c₁ (n=1)0
c₂ (n=2)0
State coefficients
c₃ (n=3)0
c₄ (n=4)0
c₅ (n=5)0
Time speed1 ×
Animation speed factor
X-range5 a
Horizontal axis limit (× characteristic length)
E-range4.5 ℏω
Energy axis limit
Levels shown5
Number of energy level lines
Calculated
⟨E⟩0.000 ℏω
⟨x⟩0.000 a
Δx0.000 a
⟨p⟩0.000 ℏ/a
Δp0.000 ℏ/a
Δx·Δp0.000 ℏ
x_turn (left)0.00 a
x_turn (right)0.00 a
max |ψ|²0.000
Curriculum
WAECNECOIGCSESATJUPEB
📋 Teacher notes
- •The quantum harmonic oscillator is the most important exactly solvable model in quantum mechanics. Its energy spectrum and wavefunctions appear in every sub-field, from molecular vibrations to quantum field theory.
- •Energy levels are equally spaced: E_n = ℏω(n + ½). The ground-state energy ℏω/2 is called zero-point energy — a purely quantum effect with no classical analogue.
- •Wavefunctions are Hermite polynomials multiplied by a Gaussian envelope. The number of nodes equals the quantum number n.
- •Stationary states (pure n) have time-independent probability densities |ψ_n|². Only superpositions of different n produce time-dependent "motion".
- •A coherent state is a special superposition that mimics classical oscillation: a Gaussian wave packet that swings back and forth without spreading. It is the quantum state closest to a classical particle.
- •The classical turning points mark where a classical particle with the same energy would reverse direction. In quantum mechanics, there is non-zero probability of finding the particle beyond these points — quantum tunnelling.
- •The expectation value ⟨x⟩ (white dot) shows the "average position". For a stationary state it is zero; for superpositions it oscillates, tracking the classical motion.
✏️ Exercises