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:
ω = 1 L C {\displaystyle \omega ={\sqrt {1 \over LC}}}
where L is the inductance in henries, and C is the capacitance in farads. The angular frequency ω {\displaystyle \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 = 1 2 C V 2 = Q 2 2 C {\displaystyle U_{C}={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}}
Where Q is the net charge on the capacitor, calculated as
Q ( t ) = ∫ − ∞ t I ( τ ) d τ {\displaystyle 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 = 1 2 L I 2 = ϕ 2 2 L {\displaystyle U_{L}={\frac {1}{2}}LI^{2}={\frac {\phi ^{2}}{2L}}}
Where ϕ {\displaystyle \phi } is the branch flux, defined as
ϕ ( t ) ≡ ∫ − ∞ t V ( τ ) d τ {\displaystyle \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
ϕ → ϕ ^ {\displaystyle \phi \rightarrow {\hat {\phi }}}
q → q ^ {\displaystyle q\rightarrow {\hat {q}}}
H → H ^ = ϕ ^ 2 2 L + q ^ 2 2 C {\displaystyle H\rightarrow {\hat {H}}={\frac {{\hat {\phi }}^{2}}{2L}}+{\frac {{\hat {q}}^{2}}{2C}}}
and enforcing the canonical commutation relation
[ ϕ ^ , q ^ ] = i ℏ {\displaystyle \left[{\hat {\phi }},{\hat {q}}\right]=i\hbar }
One-dimensional harmonic oscillator
Hamiltonian and energy eigenstates
Like the one-dimensional harmonic oscillator problem, an LC circuit can be quantized by either solving the Schrödinger equation or using creation and annihilation operators. The energy stored in the inductor can be looked at as a "kinetic energy term" and the energy stored in the capacitor can be looked at as a "potential energy term". The Hamiltonian of such a system is:
H = ϕ 2 2 L + 1 2 L ω 2 Q 2 {\displaystyle H={\frac {\phi ^{2}}{2L}}+{\frac {1}{2}}L\omega ^{2}Q^{2}}
where Q is the charge operator, and ϕ {\displaystyle \phi } is the magnetic flux operator. The first term represents the energy stored in an inductor, and the second term represents the energy stored in a capacitor. In order to find the energy levels and the corresponding energy eigenstates, we must solve the time-independent Schrödinger equation,
H | ψ ⟩ = E | ψ ⟩ {\displaystyle H|\psi \rangle =E|\psi \rangle \ }
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