Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. The horizontal axis shows the position x. The graphs are not normalised
Probability densities |ψn(x)|2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.
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).
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:
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.
In the classical case the energy of LC circuit will be:
where capacitance energy, and
inductance energy. 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 has nothing to do with the energy in the classical case. But it has the following relationship with energy in the quantum case:
So, in the quantum case, by filling capacitance with the one electron charge:
The relationship between capacitance energy and the ground state oscillator energy will then be:
where quantum impedance of LC circuit.
The quantum impedance of the quantum LC circuit could be in practice of the two types[clarification needed]:
So, the energy relationships will be:
and that is the main problem of the quantum LC circuit: energies stored on capacitance and inductance are not equal to the ground state energy of the quantum oscillator.
This energy problem produces the quantum LC circuit paradox (QLCCP).
Some simple solution of the QLCCP could be found in the following way. Yakymakha (1989) (eqn.30) proposed the following DOS quantum impedance definition:
where magnetic flux, and
So, there are no electric or magnetic charges in the quantum LC circuit, but electric and magnetic fluxes only. Therefore, not only in the DOS LC circuit, but in the other LC circuits too, there are only the electromagnetic waves.
Thus, the quantum LC circuit is the minimal geometrical-topological value of the quantum waveguide, in which there are no electric or magnetic charges, but electromagnetic waves only.
Now one should consider the quantum LC circuit as a "black wave box" (BWB), which has no electric or magnetic charges, but waves.
Furthermore, this BWB could be "closed" (in Bohr atom or in the vacuum for photons), or "open" (as for QHE and Josephson junction).
So, the quantum LC circuit should has BWB and "input - output" supplements. The total energy balance should be calculated with considering of "input" and "output" devices.
Without "input - output" devices, the energies "stored" on capacitances and inductances are virtual or "characteristics", as in the case of characteristic impedance (without dissipation).
Very close to this approach now are Devoret (2004), which consider Josephson junctions with quantum inductance, Datta impedance of Schrödinger waves (2008) and Tsu (2008), which consider quantum wave guides.
As presented below, the resonance frequency for QHE is:
where cyclotron frequency,
The scaling current for QHE will be:
Therefore, the inductance energy will be:
So for quantum magnetic flux , inductance energy is half as much as the ground state oscillation energy. This is due to the spin of electron (there are two electrons on Landau level on the same quantum area element). Therefore, the inductance/capacitance energy considers the total Landau level energy per spin.
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. This fundamental constant first appeared in the Bohr atom for Bohr radius:
where Compton wavelength of electron.
Thus, the wave quantum LC circuit has no charges in it, but electromagnetic waves only. So capacitance or inductance "characteristic energies" are
times less than the total energy of the oscillator. In other words, charges "disappear" at the "input" and "generate" at the "output" of the wave LC circuit, adding energies to keep balance.
Thus, the total energy of the quantum LC circuit should be:
In the general case, resonance energy could be due to the "rest mass" of electron, energy gap for Bohr atom, etc.
However, energy stored on capacitance is due to electric charge. Actually, for free electron and Bohr atom LC circuits we have quantized electric fluxes, equal to the electronic charge,
Furthermore, energy stored on inductance is due to magnetic momentum. Actually, for Bohr atom we have Bohr Magneton:
In the case of free electron, Bohr Magneton will be:
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 is the Planck constant.
By analogically, the absolute value of the quantum capacitance will be in the QHE mode:
is DOS definition of the quantum capacitance according to Luryi, - quantum capacitance ‘’ideal value’’ at , and other quantum capacitance:
where dielectric constant, first defined by Yakymakha (1994) > in the spectroscopic investigations of the silicon MOSFETs.
The standard wave impedance definition for the QHE LC circuit could be presented as:
where von Klitzing constant for resistance.
The standard resonant frequency definition for the QHE LC circuit could be presented as:
where standard cyclotron frequency in the magnetic field B.
Total electric charge on the first energy level of FA:
where Bohr quantum area element.
First FA was discovered by Yakymakha (1994)  as very low frequency resonance on the p- channel MOSFETs.
Contrary to the spherical Bohr atom, the FA has gyperbolic dependence on the number of energy level (n) 
^Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p. 91. ISBN5-11-002309-3. djvuArchived June 5, 2011, at the Wayback Machine</
^Devoret M.H., Martinis J.M. (2004). "Implementing Qubits with Superconducting Integrated Circuits". Quantum Information Processing, v.3, N1. Pdf
^Raphael Tsu and Timir Datta (2008) "Conductance and Wave Impedance of Electrons".
Progress In Electromagnetics Research Symposium, Hangzhou, China, March 24–28
^Yakymakha O.L.(1989). High Temperature Quantum Galvanomagnetic Effects in the Two- Dimensional Inversion Layers of MOSFET's (In Russian). Kyiv: Vyscha Shkola. p.91. ISBN5-11-002309-3. djvuArchived June 5, 2011, at the Wayback Machine
^ abcYakymakha O.L., Kalnibolotskij Y.M. (1994). "Very-low-frequency resonance of MOSFET amplifier parameters". Solid- State Electronics 37(10),1739-1751 Pdf
^Serge Luryi (1988). "Quantum capacitance device". Appl.Phys.Lett. 52(6). Pdf
^Devoret M.H. (1997). "Quantum Fluctuations". Amsterdam, Netherlands: Elsevier. pp.351-386. PdfArchived April 1, 2010, at the Wayback Machine
^Yakymakha O.L., Kalnibolotskij Y.M., Solid- State Electronics, vol.38, No.3,1995.,pp.661-671 pdf