Quantum LC circuit
Encyclopedia
An LC circuit can be quantized using the same methods as Quantum harmonic oscillator. An LC circuit is a variety of resonant circuit or tuned 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: Since an LC circuit is nothing more than a harmonic oscillator where charge is the conjugate variable, the quantum mechanically solution is known exactly.
where L is the inductance
in henries, and C is the capacitance
in farad
s. The angular frequency
has units of radian
s per second. A capacitor stores energy in the electric field between the plates, which can be written as follows:
since 
Likewise, an inductor stores energy in the magnetic field depending on the current, which can be written as follows:
since 
Using charge as the conjugate variable, and magnetic flux as the conjugate "momentum", one can develop a Schrödinger equation to describe the stored energy in the system.
where 
is a function of charge
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).
Using Kirchoff'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
The Lagrangian for coupled LC circuits is as follows:
The Hamiltonian is found using a Legendre transform of the above 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 "Center-of-Mass" coordinate 
(for lack of a better term), 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.
Hamiltonian's equations:
,
where
stored capacitor charge (or electric flux) and 
magnetic momentum (magnetic flux),
capacitor voltage and 
inductance current, 
time variable.
Nonzero initial conditions:
At
we shall have oscillation frequency:
,
and wave impedance of the LC circuit (whithout dissipation):
Hamiltonian's equations solutions:
At
we shall have the following values of charges, magnetic flux and energy:
where
.
,
where
- electric charge at zero time, 
capacitance area.
,
where
- magnetic flux at zero time,
inductance area.
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:
.
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:
.
,
where
electron charge and 
fine structure constant,
then "electric" and "magnetic" fluxes at zero time point will be:
,
where
magnetic flux quantum.
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 we should start filling capacitance with the one electron charge:
and 
The relationship between capacitance energy and the ground state oscillator energy will then be:
where
quantum impedance of LC circuit.
As we know, the quantum impedance of the quantum LC circuit could be in practice of the two types:
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).
where
magnetic flux, and
electric flux, 
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 the electromagnetic waves only.
Thus, the quantum LC circuit is the minimal geometrical/topological value of the quantum waveguide, in which there no electric or magnetic charges, but electromagnetic waves only.
Now we should consider the quantum LC circuit as an "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.
Whithout "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 and Tsu (2008), which consider quantum wave guides.
where
cyclotron frequency,
and
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.
Energy stored on the quantum inductance:
Resonance energy of the quantum LC circuit:
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:
the same, as for Bohr atom.
where
electron radius and 
Compton wavelength.
Note, that this electron radius is consistent with the standard definition of the spin. Actually, rotating momentum of electron is:
where
is considered.
Spherical inductance of electron:
Characterictic impedance of electron:
Resonance frequency of electron LC circuit:
Induced electric flux on electron capacitance:
Energy, stored on electron capacitance:
where
is the "rest energy" of electron. So, induced electric flux will be:
Thus, through electron capacitance we have quantized electric flux, equal to the electron charge.
Magnetic flux through inductance:
Magnetic energy, stored on inductance:
So, induced magnetic flux will be:
where
magnetic flux quantum. Thus, through electron inductance there are no quantization of magnetic flux.
where
Compton wavelength of electron,
fine structure constant.
Bohr atomic surface:
.
Bohr inductance:
.
Bohr capacitance:
.
Bohr wave impedance:
Bohr angular frequency:
where
Bohr wavelength for the first energy level.
Induced electric flux of the Bohr first energy level:
Energy, stored on the Bohr capacitance:
where
is the Bohr energy. So, induced electric flux will be:
Thus, through the Bohr capacitance we have quatized electric flux, equal to the electron charge.
Magnetic flux through the Bohr inductance:
So, induced magnetic flux will be:
Thus, through the Bohr inductance there are no quantization of magnetic flux.
Photon "wave impedance":
Photon "wave inductance":
Photon "wave capacitance":
Photon "magnetic flux quantum":
Photon "wave current":
,
where
current carriers effective mass in a solid, 
electron mass, and 
dimensionless parameter, which considers band structure of a solid. So, the quantum inductance can be defined as follows:
,
where
- the ‘’ideal value’’ of quantum inductance at 
and another ideal quantum inductance:
, (3)
where
magnetic constant
,
magnetic “fine structure constant”(p. 62), 
fine structure constant and 
Compton wave length
of electron, first defined by Yakymakha (1994) 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
is the Planck constant.
By analogically, the absolute value of the quantum capacitance will be in the QHE mode:
,
where
,
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.
Hall scaling current quantum will be:
,
where
Hall angular frequency.
where
magnetic flux, 
Josephson junction quantum inductance and
Josephson junction current.
DC Josephson equation for current:
where
Josephson scale for current,
phase difference between superconductors.
Current derivative on time variable will be:
AC Josephson equation:
where
reduced Plank constant, 
Josephson magnetic flux quantum,
and 
electron charge.
Combining equations for derivatives yields junction voltage:
where
is the Devoret (1997) quantum inductance.
AC Josephson equation for angular frequency:
Resonance frequency for Josephson LC circuit:
where
is the Devoret quantum capacitance, that can be defined as:
Quantum wave impedance of Josephson junction:
For
mV and 
A wave impedance will be 
F.
Quantum inductance of FA:
H.
Quantum area element of FA:
.
Resonace frequency of FA:
rad/s.
Characteristic impedunce of FA:
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)
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
, represented by the letter L, and a capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, represented by the letter C. When connected together, an electric current
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
can alternate between them at the circuit's resonant frequency: Since an LC circuit is nothing more than a harmonic oscillator where charge is the conjugate variable, the quantum mechanically solution is known exactly.
where L is the inductance
Inductance
In electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
in henries, and C is the capacitance
Capacitance
In electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
in farad
Farad
The farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...
s. The angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
has units of radianRadian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s per second. A capacitor stores energy in the electric field between the plates, which can be written as follows:
since Likewise, an inductor stores energy in the magnetic field depending on the current, which can be written as follows:
since Using charge as the conjugate variable, and magnetic flux as the conjugate "momentum", one can develop a Schrödinger equation to describe the stored energy in the system.
where is a function of chargeHamiltonian and energy eigenstates
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 Kirchoff'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 fluxQuantization of coupled LC circuits
A coupled LC circuit, as shown in the Figure, consists of two LC circuits coupled by a mutual inductance between the inductors. This is equivalent to a pair of harmonic oscillators with a kinetic coupling term.The Lagrangian for coupled LC circuits is as follows:
The Hamiltonian is found using a Legendre transform of the above 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 "Center-of-Mass" coordinate (for lack of a better term), 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.
Classical case
Stored energy (Hamiltonian) for classical LC circuit:Hamiltonian's equations:
,where
stored capacitor charge (or electric flux) and magnetic momentum (magnetic flux), capacitor voltage and inductance current, time variable.Nonzero initial conditions:
At
we shall have oscillation frequency:,and wave impedance of the LC circuit (whithout dissipation):
Hamiltonian's equations solutions:
At
we shall have the following values of charges, magnetic flux and energy:Definition of the Phasor
In the general case the wave amplitudes can be defined in the complex spacewhere
.,where
- electric charge at zero time, capacitance area.,where
- magnetic flux at zero time, inductance area.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:
.Quantum case
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:
.Wave impedance of free space
When wave impedance of quantum LC circuit takes the value of free spece,where
electron charge and fine structure constant,then "electric" and "magnetic" fluxes at zero time point will be:
,where
magnetic flux quantum.General formulation
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 we should start filling capacitance with the one electron charge:
and The relationship between capacitance energy and the ground state oscillator energy will then be:
where
quantum impedance of LC circuit.As we know, the quantum impedance of the quantum LC circuit could be in practice of the two types:
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).
Possible solution
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 electric flux, 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 the electromagnetic waves only.
Thus, the quantum LC circuit is the minimal geometrical/topological value of the quantum waveguide, in which there no electric or magnetic charges, but electromagnetic waves only.
Now we should consider the quantum LC circuit as an "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.
Whithout "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 and Tsu (2008), which consider quantum wave guides.
Explanation for DOS quantum LC circuit
As presented below, the resonance frequency for QHE is:where
cyclotron frequency, andThe 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.Explanation for "wave" quantum LC circuit
By analogically to the DOS LC circuit, we havetwo 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.Total energy of quantum LC circuit
Energy stored on the quantum capacitance:Energy stored on the quantum inductance:
Resonance energy of the quantum LC circuit:
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:
the same, as for Bohr atom.
Electron as LC circuit
Electron capacitance could be presented as the spherical capacitor:where
electron radius and Compton wavelength.Note, that this electron radius is consistent with the standard definition of the spin. Actually, rotating momentum of electron is:
where
is considered.Spherical inductance of electron:
Characterictic impedance of electron:
Resonance frequency of electron LC circuit:
Induced electric flux on electron capacitance:
Energy, stored on electron capacitance:
where
is the "rest energy" of electron. So, induced electric flux will be:Thus, through electron capacitance we have quantized electric flux, equal to the electron charge.
Magnetic flux through inductance:
Magnetic energy, stored on inductance:
So, induced magnetic flux will be:
where
magnetic flux quantum. Thus, through electron inductance there are no quantization of magnetic flux.Bohr atom as LC circuit
Bohr radius:where
Compton wavelength of electron, fine structure constant.Bohr atomic surface:
.Bohr inductance:
.Bohr capacitance:
.Bohr wave impedance:
Bohr angular frequency:
where
Bohr wavelength for the first energy level.Induced electric flux of the Bohr first energy level:
Energy, stored on the Bohr capacitance:
where
is the Bohr energy. So, induced electric flux will be:Thus, through the Bohr capacitance we have quatized electric flux, equal to the electron charge.
Magnetic flux through the Bohr inductance:
So, induced magnetic flux will be:
Thus, through the Bohr inductance there are no quantization of magnetic flux.
Photon as LC circuit
Photon "resonant angular frequency":Photon "wave impedance":
Photon "wave inductance":
Photon "wave capacitance":
Photon "magnetic flux quantum":
Photon "wave current":
Quantum Hall effect as LC circuit
In the general case 2D- density of states (DOS) in a solid could be defined by the following:,where
current carriers effective mass in a solid, electron mass, and dimensionless parameter, which considers band structure of a solid. So, the quantum inductance can be defined as follows:,where
- the ‘’ideal value’’ of quantum inductance at and another ideal quantum inductance:, (3)where
magnetic constantVacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...
,
magnetic “fine structure constant”(p. 62), fine structure constant and Compton wave lengthCompton wavelength
The Compton wavelength is a quantum mechanical property of a particle. It was introduced by Arthur Compton in his explanation of the scattering of photons by electrons...
of electron, first defined by Yakymakha (1994) 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
is the Planck constant.By analogically, the absolute value of the quantum capacitance will be in the QHE mode:
,where
,is DOS definition of the quantum capacitance according to Luryi,
- quantum capacitance ‘’ideal value’’ at , and other quantum capacitance:,where
dielectric constantDielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
, 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.Hall scaling current quantum will be:
,where
Hall angular frequency.Josephson junction as LC circuit
Electromagnetic induction (Faraday) low:where
magnetic flux, Josephson junction quantum inductance and Josephson junction current.DC Josephson equation for current:
where
Josephson scale for current, phase difference between superconductors.Current derivative on time variable will be:
AC Josephson equation:
where
reduced Plank constant, Josephson magnetic flux quantum, and electron charge.Combining equations for derivatives yields junction voltage:
where
is the Devoret (1997) quantum inductance.
AC Josephson equation for angular frequency:
Resonance frequency for Josephson LC circuit:
where
is the Devoret quantum capacitance, that can be defined as:Quantum wave impedance of Josephson junction:
For
mV and A wave impedance will be Flat Atom as LC circuit
Quantum capacitance of Flat Atom (FA):F.Quantum inductance of FA:
H.Quantum area element of FA:
.Resonace frequency of FA:
rad/s.Characteristic impedunce of FA:
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)
See also
- LC circuitLC circuitAn LC circuit, also called a resonant circuit or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C...
- Harmonic oscillatorHarmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
- Quantum harmonic oscillatorQuantum harmonic oscillatorThe quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...
- Quantum Electromagnetic Resonator
Sources
- W. H. Louisell, “Quantum Statistical Properties of Radiation” (Wiley, New York, 1973)
- Michel H.Devoret. Quantum Fluctuation in Electric Circuit.PDF
- Fan Hong-yi, Pan Xiao-yin. Chin.Phys.Lett. No9(1998)625.PDF
- Xu, Xing-Lei; Li, Hong-Qi; Wang, Ji-Suo Quantum fluctuations of mesoscopic damped double resonance RLC circuit with mutual capacitance inductance coupling in thermal excitation state. Chinese Physics, Volume 16, Issue 8, pp. 2462–2470 (2007).http://adsabs.harvard.edu/abs/2007ChPhy..16.2462X
- Hong-Qi Li, Xing-Lei Xu and Ji-Suo Wang. Quantum Fluctuations of the Current and Voltage in Thermal Vacuum State for Mesoscopic Quartz Piezoelectric Crystal. http://www.springerlink.com/content/t56269h3773x5163/
- Boris Ya. Zel’dovich. Impedance and parametric excitation of oscillators. UFN, 2008, v. 178, No 5 PDF
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