A quantum theory corresponding to the classical theory of action-angle variables is developed and applied to the examples of a harmonic oscillator and an infinite potential well. The quantum state of the system which is canonically connected to the original system, of action-angle variables are obtained. Using those quantum states, we show that the quantum behavior and the expectation values of some physical quantities, including the energy eigenvalues of the original system, can be found without prior knowledge of the quantum state of the original system. From the uncertainties of the quantum operators of the action-angle variable system, a, quantum state is interpreted as ail amplitude state of the original quantum system.

A controlled quantum dense coding scheme is investigated with a four-particle non-maximal quantum channel. The amount of classical information is shown to be capable of being controlled by the controllers through adjustments of the local measurement angles and to depend on the coefficients of the quantum channel; in addition, the four particles are distributed in two inverse ways in such an quantum channel. A restricted condition for distributing the particles to realize quantum dense coding in an arbitrary (N + 2)-particle quantum channel is proposed.

We realized the four-body approximate W state by the interaction of two atoms and two mode vacuum cavity fields, no matter that the two atoms are initially in Einstein-Podolsky-Rosen entangled states or in disentangled states. The four-body entanglement changes with the initial state of the system and the detuning degree. What's important is that when the two atoms are initially in Einstein-Podolsky-Rosen, the entanglement of atom-atom and field-field can reach a maximal value of 1. Also if the two atoms are initially in disentangled states, the entanglement of atom-atom and field-field won't go beyond 1/2.

We propose a scheme for generating entangled states for two superconducting quantum interference devices in a thermal cavity with the assistance of a microwave pulse.

The quantum mechanical quantities, invariant operator and unitary transformation operator of a mesoscopic capacitance-coupled circuit are evaluated through the dynamical invariant method. The uncertainty relation between charges and current does not satisfy the minimum uncertainty, (Deltaq(i)Deltap(i))(min) = h/2, even though for two loops coupled via capacitance the resistances for each loop reach the limit of zero.

For a mesoscopic coupled circuit with mutual inductance, we calculate the quantum uncertainty relations for the electric charges and currents, and we investigate the squeezed effect using the methods of canonical transformation and unitary transformation. We confirm that the quantum fluctuations in two loops are connected with each other because of the existence of the mutual inductance. The existence of the mutual inductance cause the capacitances in the two loops to become important factors influencing the squeezed effect.

The zero-point energy of a mesoscopic nonlinear inductance-capacitance (L-C) circuit is obtained by making use of the variation method.This is somewhat less than that of simple harmonic oscillator.

Starting with the quantization of the Caldirola-Kanai Hamiltonian, we review in detail various phenomenological methods to treat the damped harmonic oscillator as a dissipative system. To show that the path integral method yields the exact quantum theory of the Caldirola-Kanai Hamiltonian without violating the Heisenberg's uncertainty principle. Through the dynamical invariant together with the path integral, we also present systematically the exact quantum theories fur various dissipative harmonic oscillators, as well as the relations between the canonical, and unitary transformations for the classical, and quantum dissipative systems.

Using the dynamical invariant operator method, we obtain the exact wave function, uncertainty relation, and energy eigenvalues for the harmonic oscillator with the classical equation of motion in the form of Mathieu functions. The probability density varies as a function of position, but is almost constant in time. The uncertainty relations satisfy the minimum uncertainty, and the energy eigenvalues oscillate slowly or rapidly depending on the frequency. The quantum and classical energies oscillate in a similar fashion with respect to frequency and time.

Quantum fluctuations and squeezing effect of charges and currents of mesoscopic capacitance–inductance–resistance coupled circuit are investigated using canonical transformation and unitary transformation method. Even if the resistance of the mesoscopic circuit is zero, the uncertainty relation between charges and those conjugate currents do not satisfy minimum uncertainty relation. We confirmed that the uncertainties of charge can be reduced by paying the cost that the uncertainties of currents becoming larger relatively, or vice versa.

Starting with the quantization of the Caldirola–Kanai Hamiltonian, various phenomenological methods to treat the damped harmonic oscillator as a dissipative system are reviewed in detail. We show that the path integral method yields the exact quantum theory of the Caldirola–Kanai Hamiltonian without violation of Heisenberg's uncertainty principle. Through the dynamical invariant and second quantization methods together with the path integral, we also present systematically the exact quantum theories for the various dissipative harmonic oscillators, bound and unbound quadratic Hamiltonian systems, and the relation between the canonical and unitary transformations for the classical and quantum dissipative systems.

The quantum invariant operator and unitary transformation operator of mesoscopic capacitance coupled circuit are obtained. The uncertainty relation of the charges and the currents are calculated. The uncertainty relations between charges and currents do not satisfy minimum uncertainty relation, (ΔqiΔpi)min=ℏ/2, even though resistances R1 and R2 are zero.

We have studied a cascaded nonlinear interferometer having degenerate optical parametric amplifiers, one for each arm. The nonlinear interferometer generates supposed squeezed state. The photon statistics of the states of the output ports of the interferometer are analyzed in tcr ms of the photon number uncertainty and the power of the mean, as a function of the gain of the crystals and the amplified input source intensity. If the nonlinear cascaded interferometer is used, it is possible to reduce the photon number uncertainty enormously.

Temperature variations of the structure factor and the pair correlation function of liquid He-4 films are derived within the chain-diagram approximation. Using the heat capacity data of He-4 monolayers adsorbed on an evaporated gold substrate, we analyzed the structure factor and the pair correlation function as functions of the temperature and the density. The principal structure factor maximum and principal maximum of the pair correlation function increase with decreasing density and inceasing temperature. The position of the principal structure factor maximum shifts smoothly toward a small value of the wave number with decreasing density, but the position of the pair correlation function increases strongly with decreasing density. Neither of the positions show a significant temperature dependence. The behaviors of the structure factor and the pair correlation function of He-4 films are generally in agreement with those of bulk liquid He-4.

Using the temperature-dependent elementary excitation of liquid He-4 films derived within the chain-diagram approximation, the heat capacity data of He-4 films adsorbed on an evaporated gold substrate are successfully fitted for coverages between 0.007 and 0.096 Angstrom (-2) and temperatures between 0.4 and 2.5 K. In the long-wavelength limit, the spectra deduced from the data are anomalous and the upward bending becomes stronger as the density increases. With increasing density, the sound and roton energy gap increase respectively, while the roton energy gap decreases with increasing temperature. This reduction becomes larger as the coverage decreases. In general, the elementary excitation spectra deduced from the specific heat data are very similar to those of bulk liquid He-4.

The cases of under-, critical- and over-damping are treated for the quantum driven harmonic oscillator. Following a survey of the classical version, the quantum invariant operator for each case is constructed and their eigenvalues are evaluated. Using these eigenvalues and the Schrodinger equations, the wavefunctions are obtained for each case. From formal path integral theory, their propagators are evaluated and are checked with those obtained by the closed property formed by the complete set of wavefunctions.

Using the time-dependent dynamical invariant method, we have evaluated the exact wave function, the uncertainty relation. and the energy expectation values of a pendulum with a linearly decreasing mass. The energy eigenvalues increase as time goes by, and the minimum uncertainty is larger than h/2.

We study the phase fluctuations of superposed squeezed states. The superposed squeezed states are obtained by inserting two squeezers in a Mach-Zehnder interferometer, one for each arm. Also, the characteristics of the coherence of the photons generated in a nonlinear interferometer are analyzed. For the analysis, the mean-square phase fluctuation and the fringe visibility are considered as functions of the gain and the input phase when the amplified input is a coherent state, a number state, or a thermal state.

Using the Caldirola-Kanai Hamiltonian with a linear damping constant for the damped harmonic oscillator as a quantum dissipative system, we have obtained the exact wave function, the uncertainty relation, and the energy expectation values by using the dynamical invariant method.

The connection between wave functions of harmonic plus inverse harmonic potential with time-dependent mass and frequency and those of harmonic plus inverse harmonic potential with time-dependent frequency is investigated. Thus the correct wave function of the harmonic plus inverse harmonic potential with time-dependent mass and frequency is obtained.