Exact Solutions of Schrödinger Equation for the Position-Dependent Effective Mass Harmonic Oscillator

A one-dimensional harmonic oscillator with position-dependent effective mass is studied. We quantize the oscillator to obtain a quantum Hamiltonian, which is manifestly Hermitian in configuration space, and the exact solutions to the corresponding Schrödinger equation are obtained analytically in terms of modified Hermite polynomials. It is shown that the obtained solutions reduce to those of simple harmonic oscillator as the position dependence of the mass vanishes.     

Classical formulation of quantum mechanics

The concept of quantum state is given in terms of classical probability for position in squeezed and rotated classical reference frames in phase space. Stationary states and energy levels of the quantum system are obtained in a classical formulation of quantum mechanics. The positive probability density of the harmonic oscillator position is obtained by solving a new eigenvalue equation of standard quantum mechanics instead of the Schrödinger equation. The orthogonality and completeness relations are found for the eigendistributions.     

Coherent States and Modified de Broglie-Bohm Complex Quantum Trajectories

This paper examines the nature of classical correspondence in the case of coherent states at the level of quantum trajectories. We first show that for a harmonic oscillator, the coherent state complex quantum trajectories and the complex classical trajectories are identical to each other. This congruence in the complex plane, not restricted to high quantum numbers alone, illustrates that the harmonic oscillator in a coherent state executes classical motion. The quantum trajectories we consider are those conceived in a modified de Broglie-Bohm scheme. Though quantum trajectory representations are widely discussed in recent years, identical classical and quantum trajectories for coherent states are obtained only in the present approach. We may note that this result for standard harmonic oscillator coherent states is not totally unexpected because of their holomorphic nature. The study is extended to coherent states of a particle in an infinite potential well and that in a symmetric Poschl-Teller potential by solving for the trajectories numerically. For the Gazeau-Klauder coherent state of the infinite potential well, almost identical classical and quantum trajectories are obtained whereas for the Poschl-Teller potential, though classical trajectories are not regained, a periodic motion results as t→∞. Similar features were found for the SUSY quantum mechanics-based coherent states of the Poschl-Teller potential too, but this time the pattern of complex trajectories is quite different from that of the previous case. Thus we find that the method is a potential tool in analyzing the properties of generalized coherent states.     

Nonstatistical quantum-classical correspondence in phase space

We study the properties of a causal quantum theory in phase space for which phase space classical mechanics is obtained as a limit. The causal quantum theory is obtained from a generalized coherent state representation. The behavior for the one particle case and the manyparticle case are illustrated for the harmonic oscillator. We also answer to the arguments against the possibility of constructing causal theories in phase space.     

The driven-oscillator evolution in the tomographic-probability representation

We consider the problem of the driven harmonic oscillator in the probability representation of quantum mechanics, where the oscillator states are described by fair nonnegative probability distributions of position measured in rotated and squeezed reference frames in the system??s phase space. For some specific oscillator states like coherent states and nth excited states, the tomographic-probability distributions (called the state tomograms) are found in an explicit form. The evolution equation for the tomograms is discussed for the classical and quantum driven oscillators, and the tomographic propagator for this equation is studied.     

Exact density matrix of an oscillator-bath system: Alternative derivation

Starting from a total Lagrangian describing an oscillator-bath system, an alternative derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The modified quantum propagator is obtained in the generalised case where the main oscillator is under the influence of a classical external force. By introducing auxiliary classical external fields, the generalised quantum propagator or generating functional of position correlation functions is obtained.     

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.     

Classical Interpretation of a Deformed Quantum Oscillator

Following the same procedure that allowed Shcrödinger to construct the (canonical) coherent states in the first place, we investigate on a possible classical interpretation of the deformed harmonic oscillator. We find that, these oscillator, also called q-oscillators, can be interpreted as quantum versions of classical forced oscillators with a modified q-dependant frequency.     

Quantization, group contraction and zero point energy

We study algebraic structures underlying 't Hooft's construction relating classical systems with the quantum harmonic oscillator. The role of group contraction is discussed. We propose the use of SU(1,1) for two reasons: because of the isomorphism between its representation Hilbert space and that of the harmonic oscillator and because zero point energy is implied by the representation structure. Finally, we also comment on the relation between dissipation and quantization.     

The simplex structure of the classical states of the quantum harmonic oscillator

It is shown that the convex set of classical states of the quantum harmonic oscillator is a simplex generated as the closed convex hull of the coherent states in the weak topology of the Banach space of trace class operators.     

Geometry of the Parameter Space of a Quantum System: Classical Point of View

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. Classical integrable systems are considered and a new approach is reported to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.     

Analytic study of a sequence of path integral approximations for simple quantum systems at low temperature

The sequence of Feynman-Trotter approximations to the thermal Feynman path integral for the simple harmonic oscillator is obtained in an easily analyzable closed form. While it converges pointwise at every non-zero temperature to the quantum thermal propagator, the sequence manifests a highly non-uniform behaviour in the zero temperature limit—every one of its elements tends toward theclassical ground state (static equilibrium). For high order elements of the sequence, there is an abrupt “collapse” from the quantum to the classical ground state with falling temperature, a phenomenon which bears a possibly misleading resemblance to a phase transition. It is shown that Feynman-Trotter sequences for many simple systems other than the harmonic oscillator also have all their elements tending to the classical static equilibrium state in the zero temperature limit.     

Forced Time-Dependent Harmonic Oscillators in Non-Commutative Space

For the time-dependent harmonic oscillator and generalized harmonic oscillator with or without external forces in non-commutative space, wave functions, and geometric phases are derived using the Lewis-Riesenfeld invariant. Coherent states are obtained as the ground state of the forced system. Quantum fluctuations are calculated too. It is seen that geometric phases and quantum fluctuations are greatly affected by the non-commutativity of the space.     

一维谐振子量子运动中的经典特性

利用量子力学的态叠加原理和算符劈裂法,对处于一维谐振子势中的初始态为高斯波包的中心位置的量子运动进行了研究.结果表明:其中心位置的量子运动呈现出经典谐振子的运动特性;波包的初始位置和初始时刻所加动量对波包中心位置量子动力学的影响与经典谐振子类似条件对运动的影响有相同的性质.本结果对理解复杂量子运动中的高斯波方法有一定的启示作用.     

Coherent presentation of density operator of the harmonic oscillator in thermostat

Based on basis of the coherent states the density matrix of harmonic oscillator in thermostat is obtained. This method is mathematically refined and physically transparent for the interpretation of quantum phenomena in classical language. Such an approach gives an opportunity to easily find the density matrix in the multi-dimensional case.     

On the Integrability of q-Oscillators Based on Invariants of Discrete Fourier Transforms

A discretization of the quantum mechanical phase space is presented in the context of q-noncommutative structures. We give two generalizations of the Heisenberg algebra in the arising lattice phase space. In contrast to ordinary quantum mechanics, there is, a priori, no systematic approach to an integrable oscillator Hamiltonian in lattice quantum mechanics. This is the central obstacle to deal with in this Letter. To do so, we show how in general the integrability of the harmonic oscillator is related to the Fourier transform between momentum and space variables. This will be done in both cases, the continuous and the discrete one. As an application, we finally obtain an integrable lattice Hamiltonian for the harmonic oscillator with generalized Hermite eigenfunctions.     

Extending Newton's law from nonlocal-in-time kinetic energy

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics.     

Exact propagator of the harmonic oscillator with a time-dependent mass

By finding a space and time transformation, the exact evaluation of the propagator for the harmonic oscillator with a time-dependent mass by the path integral method becomes possible. We then derive the wavefunctions from the propagator obtained. Finally, the propagator beyond and at caustics is evaluated by using its modified semi-group property and is confirmed by investigating the classical paths with two fixed end-positions.