Quantum LC circuit information

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

\omega ={\sqrt {1 \over LC}}

where L is the inductance in henries, and C is the capacitance in farads. The angular frequency $\omega \,$ has units of radians per second. A capacitor stores energy in the electric field between the plates, which can be written as follows:

U_{C}={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}

Where Q is the net charge on the capacitor, calculated as

Q(t)=\int _{-\infty }^{t}I(\tau )d\tau

Likewise, an inductor stores energy in the magnetic field depending on the current, which can be written as follows:

U_{L}={\frac {1}{2}}LI^{2}={\frac {\phi ^{2}}{2L}}

Where $\phi$ is the branch flux, defined as

\phi (t)\equiv \int _{-\infty }^{t}V(\tau )d\tau

Since charge and flux are canonically conjugate variables, one can use canonical quantization to rewrite the classical hamiltonian in the quantum formalism, by identifying

\phi \rightarrow {\hat {\phi }}

q\rightarrow {\hat {q}}

H\rightarrow {\hat {H}}={\frac {{\hat {\phi }}^{2}}{2L}}+{\frac {{\hat {q}}^{2}}{2C}}

and enforcing the canonical commutation relation

\left[{\hat {\phi }},{\hat {q}}\right]=i\hbar

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