Quantization Of EM Field

Date:

1 August 2019

What is the problem ?

Derivation

Step 1 Vacuum Wave-Equation derived in section Maxwell-Equations

\[\begin{split}\begin{align*} & (\nabla^2 - {1\over c^2}{\partial^2 \over \partial t^2} ) A=0\\ \end{align*}\end{split}\]

Step 2 Plane-Wave-Expansion

\[\begin{split}\begin{align*} A(r,t) & = \sum\limits_M \underbrace{\hat{p}_M}_{\text{Polarization}}\; \underbrace{\mathcal{A}_M}_{\text{Amplitude}} e^{i(k_M \cdot r-\omega_M t)} + cc \\ & \equiv \sum\limits_M \hat{p}_M \mathcal{A}_M(t) e^{i(k_M \cdot r)} + cc \\ & where\\ & \mathcal{A}_M(t)=\mathcal{A}_M e^{-i\omega_M t}\\ & M = \{k_x,k_y,k_z,p \} \text{ is a set of parameters}\\ & \text{Each possible value of } M, \text{ eg:{1,2,3,1}, is a "Mode"}\\ & k_x \text{ is wavenumber along } x \text{ and } p \in \{1,2\}\\ & cc = \text{complex conjugate of its previous term, since }A \text{ must be real}\\ \end{align*}\end{split}\]

Step 3 From \(A\) we can find the Hamiltonian, refer to Maxwell-Equations

\[\begin{split}\begin{align*} & H = 2\epsilon_0 V \sum \omega_M^2 \; \mathcal{A}_M(t) \mathcal{A}_M^*(t)\\ & where \; V \text{ is volume} \end{align*}\end{split}\]

Step 4 As we did in Quantization-Of-LC-Circuit, the idea is to cast the Hamiltonian into form of Quantum-Harmonic-Oscillator

Using a common technique named ???

\[\begin{split}\begin{align*} & \text{Introduce } Q_M = \sqrt{\epsilon_0 V} (\mathcal{A}_M+\mathcal{A}_M^*) \text{ and } P_M = {1\over i} \sqrt{\epsilon_0 V \omega_M^2} (\mathcal{A}_M - \mathcal{A}_M^*)\\ & \Rightarrow H = \sum\limits_M {P_M^2\over 2} + {1\over 2}\omega_M^2 Q_M^2 \end{align*}\end{split}\]

Step 5 Results from Quantum-Harmonic-Oscillator then follows