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Wave-Particle Duality

Light exhibits properties of both waves and particles, described by concepts like photons and interference patterns, central to quantum optics.
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The statement of the theorem

Let H\mathcal{H} be a separable Hilbert space representing the quantum state ∣ψ⟩|\psi\rangle. The wave aspect is described by the continuous field ψ(r,t)\psi(\mathbf{r}, t), satisfying the time-dependent Schrödinger equation: iℏ∂∂tψ(r,t)=H^ψ(r,t)i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r}, t) = \hat{H}\psi(\mathbf{r}, t). The particle aspect is characterized by the expectation values of position ⟨x^⟩\langle\hat{x}\rangle and momentum ⟨p^⟩\langle\hat{p}\rangle. The duality is formally constrained by the generalized uncertainty principle, which mandates that for any observable pair (A^,B^)(\hat{A}, \hat{B}) with commutator [A^,B^]=iℏC^[\hat{A}, \hat{B}] = i\hbar\hat{C}, the following inequality must hold for any state ∣ψ⟩|\psi\rangle: ΔAΔB≥12∣⟨[A^,B^]/iℏ⟩∣\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}]/i\hbar \rangle| Specifically, for position X^\hat{X} and momentum P^\hat{P}, this yields: ΔXΔP≥ℏ2\Delta X \Delta P \ge \frac{\hbar}{2}. Furthermore, the energy-momentum relation, linking the wave frequency ω\omega and wave number kk, is quantized via the Planck relation: E=ℏω=ℏckE = \hbar\omega = \hbar c k, where EE is the energy operator H^\hat{H} and kk is the wave vector magnitude, confirming the particle energy EE derived from the wave properties.