Quantum Amplitude & Phase Interference

Interactive Complex Vector Superposition Sandbox
Interference Presets:
Argand Phasor Plane Complex Space
Relative Phase Δφ (Phase Shift) 180°
Amplitude |a₁| (Cyan Vector) 1.00
Amplitude |a₂| (Magenta Vector) 1.00
Spatial Wavefunction & Particle Detector Screen |Ψ(x)|² Density
Spatial Frequency k (Fringe Density) 2.50
Interference Regime Complete Destructive
Total Real Re(Ψ) 0.000
Total Imaginary Im(Ψ) 0.000
Center Magnitude |Ψ(0)| 0.000
Center Prob. Density |Ψ(0)|² 0.000
Detector Accumulation 0 Hits
ψ₁ = |a₁|e^(iφ₁) + ψ₂ = |a₂|e^(iφ₂) ⟹ Ψ_total = ψ₁ + ψ₂
Ψ = (1.00 + 0.00i) + (-1.00 + 0.00i) = 0.00 + 0.00i
P(x) = |Ψ(x)|² = |ψ₁(x) + ψ₂(x)|² ≠ |ψ₁|² + |ψ₂|² (Interference Term Active)
Core Quantum Principle: In quantum mechanics, probability amplitudes are complex vectors that add linearly before calculating probabilities. When relative phase Δφ = 180°, amplitudes cancel perfectly (Ψ = 0) at interference nodes. Destructive interference does not destroy quantum probability; it redistributes detection events to regions where amplitudes align constructively.

Complex sums and phase in the saved quantum sandbox

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The saved phase sandbox adds complex amplitudes before squaring their magnitude. With both amplitudes equal to one and zero relative phase, the vectors point together: the sum is two and its squared magnitude is four. At one hundred eighty degrees they point opposite ways and cancel. Four here is an unnormalized center intensity proxy. It is not a probability of four. The source later uses its spatial distribution to sample detector hits. A relative phase of ninety degrees makes the two unit vectors perpendicular. Their sum has real component one and imaginary component one. Magnitude is square root of two and the center proxy is two. The law is amplitude one squared plus amplitude two squared plus twice their product times cosine of relative phase. This curve evaluates the actual saved algebra across zero to three hundred sixty degrees. It is a phase relation, not a linear interpolation between endpoints. Destructive phase alone does not guarantee zero intensity. Set amplitude one to one and amplitude two to one half. At one hundred eighty degrees the residual vector is one half, so its squared magnitude is point two five. At zero phase the sum is one point five and the square is two point two five. The two bars show these actual squared sums. Resetting detector hits changes accumulated samples, while phase and amplitudes control the distribution from which new hits are drawn.

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