Quantum
Harmonic Oscillator
☰
01 · Classical Limit
02 · Wave Functions
03 · Time Evolution
Quantum Number
▲
▼
E =
0.5 ℏω
Quick select:
0
1
5
10
20
50
100
Potential Strength
Angular frequency ω
1.00
Overlays
Potential V(x)
Classical P(x)
Turning points
Energy level Eₙ
About this State
Select n above to explore |ψₙ(x)|².
Probability density
|ψₙ(x)|²
|ψₙ(x)|²
Select State
n = 0 · E = 1/2 ℏω
ground state
n = 1 · E = 3/2 ℏω
first excited state
n = 2 · E = 5/2 ℏω
second excited state
n = 3 · E = 7/2 ℏω
third excited state
(1/√2)(ψ₂ + ψ₃)
equal superposition, t = 0
Uncertainty Principle
⟨ x ⟩
—
⟨ x² ⟩
—
Δx = √(⟨x²⟩−⟨x⟩²)
—
⟨ p ⟩
—
⟨ p² ⟩
—
Δp = √(⟨p²⟩−⟨p⟩²)
—
Display Options
Re[ψ], Re[φ]
Im[ψ], Im[φ]
Show ⟨·⟩ and Δ
Classical limits
State:
n = 1
Position space ψ(x)
Momentum space φ(p)
Position probability density |ψ(x)|²
Momentum probability density |φ(p)|²
Initial State
Ψ(x,0) = (1/√2)(ψ₀ + ψ₁)
STATE n₁
▲
▼
STATE n₂
▲
▼
ΔE =
1.00
Time
t =
0.000
·
0.000 τ
Time
0.00 τ
Play Speed
1.0×
Playback
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Overlays
Re[Ψ(x,t)]
Im[Ψ(x,t)]
⟨x⟩(t)
Classical turning points
Time evolution:
Ψ(x, t=0)
Wave function Ψ(x, t)
Probability density |Ψ(x,t)|²
ℏ=m=ω=
1
Eₙ =
n + ½
ψₙ(x) =
Nₙ H_n(x) e^(−x²/2)
φₙ(p) =
(−i)ⁿ ψₙ(p)
Δx·Δp =
n + ½ (eigenstates)