Particle Phase

From Particle Energy, we know E=ℏωE=\hbar\omega so that particle energies ∣Ek⟩\left\lvert E_k \right\rangle will oscillate at frequency

ωk=Ekℏ\begin{gather*} \omega_k=\frac{E_k}{\hbar} \end{gather*}

Oscillating means the phase of the particle changes so we can represent the phase as a complex number.

Let’s say there is a two-level Quantum System in some superposition. This just means

∣ψ⟩=α∣0⟩+β∣1⟩=(αβ)\begin{gather*} \left\lvert \psi \right\rangle=\alpha\left\lvert 0 \right\rangle+\beta\left\lvert 1 \right\rangle=\begin{pmatrix} \alpha\\\beta\end{pmatrix} \end{gather*}

Because α,β\alpha,\beta can be complex, we proved experimentally that a glass polarizer, like a polarized sunglass, can change the phase of the state ∣1⟩\left\lvert 1 \right\rangle for example.

Let

β′=βeiδ\begin{gather*} \beta'=\beta e^{i\delta} \end{gather*}

Where δ\delta can be any phase angle from 0 to 2π2\pi.

This is the same phase as which destructively interfere in the double slit experiment. That’s why phases exist. Because light is a wave with a frequency (and also a particle!)