Quantum LC circuit

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator.

A capacitor stores energy in the electric field between the plates, which can be written as follows: 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: 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 and enforcing the canonical commutation relation 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.

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).

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.

This is equivalent to a pair of harmonic oscillators with a kinetic coupling term.

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

we shall have oscillation frequency: and wave impedance of the LC circuit (without dissipation): Hamiltonian's equations solutions: At

we shall have the following values of charges, magnetic flux and energy: In the general case the wave amplitudes can be defined in the complex space where

Note that, at the equal area elements we shall have the following relationship for the wave impedance: Wave amplitude and energy could be defined as: In the quantum case we have the following definition for momentum operator: Momentum and charge operators produce the following commutator: Amplitude operator can be defined as: and phazor: Hamilton's operator will be: Amplitudes commutators: Heisenberg uncertainty principle: When wave impedance of quantum LC circuit takes the value of free space where

von Klitzing constant then "electric" and "magnetic" fluxes at zero time point will be: where

Furthermore, there are the following relationships between charges (electric or magnetic) and voltages or currents: Therefore, the maximal values of capacitance and inductance energies will be: Note that the resonance frequency

This energy problem produces the quantum LC circuit paradox (QLCCP).

Furthermore, this BWB could be "closed" (in Bohr atom or in the vacuum for photons), or "open" (as for QHE and Josephson junction).

The total energy balance should be calculated with considering of "input" and "output" devices.

Very close to this approach now are Devoret (2004),[2] which consider Josephson junctions with quantum inductance, Datta impedance of Schrödinger waves (2008) and Tsu (2008),[3] which consider quantum wave guides.

The scaling current for QHE will be: Therefore, the inductance energy will be: So for quantum magnetic flux

By analogy to the DOS LC circuit, we have two times lesser value due to the spin.

But here there is the new dimensionless fundamental constant: which considers topological properties of the quantum LC circuit.

In other words, charges "disappear" at the "input" and "generate" at the "output" of the wave LC circuit, adding energies to keep balance.

could be due to the "rest mass" of electron, energy gap for Bohr atom, etc.

Note, that this electron radius is consistent with the standard definition of the spin.

Compton wavelength of electron, first defined by Yakymakha (1994)[4] in the spectroscopic investigations of the silicon MOSFETs.

Since defined above quantum inductance is per unit area, therefore its absolute value will be in the QHE mode: where the carrier concentration is: and

dielectric constant, first defined by Yakymakha (1994)[4] in the spectroscopic investigations of the silicon MOSFETs.

The standard wave impedance definition for the QHE LC circuit could be presented as: where

The standard resonant frequency definition for the QHE LC circuit could be presented as: where

Combining equations for derivatives yields junction voltage: where is the Devoret (1997)[6] quantum inductance.

First FA was discovered by Yakymakha (1994)[4] as very low frequency resonance on the p-channel MOSFETs.

Contrary to the spherical Bohr atom, the FA has hyperbolic dependence on the number of energy level (n)[7]