Mathematical Framework of Quantum Wave Interference: Wavefunctions and Phases
Introduction
Quantum interference is the phenomenon whereby probability amplitudes—complex-valued wavefunctions—superpose and produce probability distributions that depend on their relative phases. Below is a concise, self-contained mathematical presentation covering the core formalism, common examples, and key implications.
1. Wavefunction, probability amplitude, and Born’s rule
- A quantum state in position representation is described by a wavefunction ψ(x,t) ∈ C.
- Born’s rule: probability density for finding a particle at x at time t is
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P(x,t) = |ψ(x,t)|^2 - Linear evolution (nonrelativistic, single particle) is governed by the Schrödinger equation:
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iħ ∂ψ/∂t = Ĥ ψ
2. Superposition and interference term
- If two alternatives produce amplitudes ψ1(x) and ψ2(x), the total amplitude is
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ψ(x) = ψ1(x) + ψ2(x) - Probability contains cross term:
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P(x) = |ψ1|^2 + |ψ2|^2 + 2 Re[ψ1ψ2] - Writing ψj = |ψj| e^{iφj}, the interference piece is
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2 |ψ1||ψ2| cos(φ2 - φ1)showing that interference depends on relative phase Δφ ≡ φ2 – φ1.
3. Plane-wave example and fringe spacing
- For two plane waves with wavevectors k1, k2 and common frequency ω:
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ψj(x,t) = A_j e^{i(k_j·x - ω t + φj)} - Intensity (time-averaged probability density) along coordinate x:
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I(x) ∝ |A1|^2 + |A2|^2 + 2|A1||A2| cos[(k2 - k1)·x + Δφ] - Fringe spacing d along direction n̂ satisfying (k2 – k1)·n̂ = 2π/d.
4. Double-slit (far-field, scalar approximation)
- Two slits separated by distance s, observation at angle θ: path difference ≈ s sin θ gives phase difference
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Δφ = (2π/λ) s sin θ - Resulting intensity (equal amplitudes):
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I(θ) = I0 [1 + cos(Δφ)] = 2 I0 cos^2(Δφ/2)
5. Coherence, visibility, and mixed states
- For partially coherent sources or statistical mixtures, replace pure amplitudes with density operator ρ. Probability at x:
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P(x) = Tr[ |x⟩⟨x| ρ ] - Fringe visibility V quantifies contrast:
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V = (I_max - I_min)/(I_max + Imin)For two-mode pure state with intensities I1,I2:
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V = 2√(I1 I2)/(I1 + I2) · |γ|where |γ| ≤ 1 is degree of coherence (|γ|=1 pure, |γ|<1 decohered).
6. Path-integral viewpoint and phases from action
- Feynman path integral: amplitude from point a to b is sum over paths
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K(b,a) = ∫ D[path] e^{(i/ħ) S[path]} - Interference arises from relative phases e^{(i/ħ)S}; stationary-phase (classical path) contributions dominate when phases vary rapidly.
7. Multi-path interference and Sorkin parameter
- For N alternatives, amplitude is sum of N amplitudes; Born’s rule implies only pairwise cross-terms. Tests for higher-order interference define Sorkin parameter ε; standard QM predicts ε = 0 (no intrinsic third-order term beyond pairwise interference).
8. Phase shifts and operators
- A relative phase can be produced by a unitary phase operator U = e^{iφ(x)} acting on one path: ψ → e^{iφ}ψ. In interferometers phase differences arise from potentials (Aharonov–Bohm), path lengths, refractive index, or dynamical evolution under Ĥ:
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ψ(t) = e^{- (i/ħ)Ĥ t} ψ(0)yielding phases from energy eigenvalues: e^{-i E t/ħ}.
9. Observability and which-path information
- Any process that entangles path degree of freedom with environment (providing which-path information) reduces coherence. If system+env state is
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|Ψ⟩ = ψ1⊗|e1⟩ + ψ2⊗|e2⟩the reduced density matrix for the particle has off-diagonal term proportional to ⟨e1|e2⟩; loss of overlap (⟨e1|e2⟩→0) destroys interference.
10. Practical calculations — recipe
- Identify relevant alternatives and compute their complex amplitudes ψj(x) including dynamical phase factors.
- Sum amplitudes: ψ = Σj ψj.
- Compute probability: P = |ψ|^2 or, for mixed states, P = Tr(|x⟩⟨x| ρ).
- Extract interference term 2 Re[Σ_{i
- If decoherence present, include environmental overlaps ⟨ei|ej⟩ multiplying cross-terms.
Conclusion
Quantum interference is fully captured by linear superposition of complex amplitudes and their relative phases. Practical use requires careful accounting of phase sources (path length, potentials, dynamics) and coherence (environmental entanglement or mixtures). The formalism above suffices for analyzing textbook interferometers, double-slit patterns, and more advanced path-integral or multi-path settings.
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