Quantum LC circuit-Cedar Herald Index

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:

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

Quantum LC circuit

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

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

Quantum LC circuit

Where is the branch flux, defined as

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

Quantum LC circuit

and enforcing the canonical commutation relation

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:

where Q is the charge operator, and 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,

Since an LC circuit really is an electrical analog to the harmonic oscillator, solving the Schrödinger equation yields a family of solutions (the Hermite polynomials).

Magnetic Flux as a Conjugate Variable

A completely equivalent solution can be found using magnetic flux as the conjugate variable where the conjugate "momentum" is equal to capacitance times the time derivative of magnetic flux. The conjugate "momentum" is really the charge.

Using Kirchhoff's Junction Rule, the following relationship can be obtained:

Since , the above equation can be written as follows:

Converting this into a Hamiltonian, one can develop a Schrödinger equation as follows:

where is a function of magnetic flux

Quantization of coupled LC circuits

Two inductively coupled LC circuits have a non-zero mutual inductance. This is equivalent to a pair of harmonic oscillators with a kinetic coupling term.

The Lagrangian for an inductively coupled pair of LC circuits is as follows:

As usual, the Hamiltonian is obtained by a Legendre transform of the Lagrangian.

Promoting the observables to quantum mechanical operators yields the following Schrödinger equation.

One cannot proceed further using the above coordinates because of the coupled term. However, a coordinate transformation from the wave function as a function of both charges to the wave function as a function of the charge difference , where and a coordinate (somewhat analogous to a "Center-of-Mass"), the above Hamiltonian can be solved using the Separation of Variables technique.

The CM coordinate is as seen below:

The Hamiltonian under the new coordinate system is as follows:

In the above equation is equal to and equals the reduced inductance.

The separation of variables technique yields two equations, one for the "CM" coordinate that is the differential equation of a free particle, and the other for the charge difference coordinate, which is the Schrödinger equation for a harmonic oscillator.

The solution for the first differential equation once the time dependence is appended resembles a plane wave, while the solution of the second differential equation is seen above.