Particle Energy

We just said photons have discrete energy levels. Let’s say they have kk energy levels.

We know experimentally, that any state ∣ψ(t)⟩\left\lvert \psi(t) \right\rangle (e.g., spin) of a quantum particle can be expanded in the energy basis. Let the energy basis be just a Set of Basis State E1,E2,…,EkE_1,E_2, …, E_k. This means a photon can be at those energies.

A state is a superposition of all those energies. Recall that a superposition is just a sum ∑\sum) of those energies. I put kk under the sum showing we must add all the energies and their corresponding ckc_k values (aka. coefficients) together.

∣ψ(0)⟩=∑kck∣Ek⟩\begin{gather*} \left\lvert \psi(0) \right\rangle=\sum_kc_k\left\lvert E_k \right\rangle \end{gather*}

Note that ckc_k are probability amplitudes