Isolated coupled-harmonic-oscillator here is the system of two distinguishable particles coupled with a harmonic oscillator interaction potential. Each particle stays in a mixed state due to entanglement. However, in center-of-mass reference frame, we obtain quasi wavefunction of the first particle expressing quasi pure state by replacing the second coordinate in the total wavefunction. We discuss the similar systems with the first particle and the potential being same and the second mass changing from micro to macro one. Measured by fidelity and coherence, the quasi pure state approaches to the pure state of a usual harmonic oscillator with same mass and similar potential. It conversely shows that the latter purely superposed state in position representation and its coherence originate from those of the first particle, which are related with some neglected macro object and the interaction between them. The current results provide a possible clue to new insights into quantum states.
Isolated coupled-harmonic-oscillator here is the system of two distinguishable particles coupled with a harmonic oscillator interaction potential. Each particle stays in a mixed state due to entanglement. However, in center-of-mass reference frame, we obtain quasi wavefunction of the first particle expressing quasi pure state by replacing the second coordinate in the total wavefunction. We discuss the similar systems with the first particle and the potential being same and the second mass changing from micro to macro one. Measured by fidelity and coherence, the quasi pure state approaches to the pure state of a usual harmonic oscillator with same mass and similar potential. It conversely shows that the latter purely superposed state in position representation and its coherence originate from those of the first particle, which are related with some neglected macro object and the interaction between them. The current results provide a possible clue to new insights into quantum states.
We investigate the entanglement dynamics of several models of coupled harmonic oscillators, whereby a number of properties concerning entanglement have been scrutinized, such as how the environment affects entanglement of a system, and death and revival of entanglement. Among them, there are two models for which we are able to vary their particle numbers easily by assuming identicalness, thereby examining how the particle number affects entanglement. We have found that the upper bound of entanglement between identical oscillators is approximately inversely proportional to the particle number.
A field theoretical formulation of the rather easy problem of N identical, one-dimensional bosons, interacting pairwise via harmonic oscillator potentials of equal force constants is given, in a way similar to the way the nonlinear Schrodinger model provides a field theoretical formulation of a one-dimensional, delta -function Bose gas. The nonlinear integro-differential equation obtained for the complex scalar field of the model is treated classically, and solved in the semiclassical limit of a large number of particles. Working in the centre of mass frame of the field, a sequence of operators that would raise or lower the energy of a state by definite amounts without changing its particle number are constructed from certain bilinear expressions in the field. These operators have, in the limit of a large number of particles, commutation relations similar to those of the Fourier modes of a compact free field, except that the first mode is missing as a result of the fact that the field in the centre of mass frame is subject to the obvious constraint that its centre of mass always coincides with the coordinate origin. The quantum states can be labelled by the total centre of mass momentum, and the quasi-particle numbers in each of the independent oscillator modes. The energy spectrum of the system is calculated, giving the same results as the exact, quantum-mechanical N-body analysis.
The coupled quantum harmonic oscillator is one of the most researched and important model systems in quantum optics and quantum informatics. This system is often investigated for quantum entanglement in the environment. As a result, such studies are complex and can only be carried out using numerical methods that do not reveal the general pattern of such systems. In this work, the external environment is considered to be two independent particles interacting with coupled harmonic oscillators. It is shown that such a system has an exact analytical solution to the dynamic Schrödinger equation. The analysis of this solution is carried out, and the main parameters of this system are revealed. The solutions obtained can be used to study more complex systems and their quantum entanglement.
In an isolated two-body composite system, the discussion of how the change of one body affects the state of the other will help to know the relation of a single particle's mixed and pure states. Given 5 one-dimensional hydrogen-like atoms models, each Coulomb interaction potential keeps invariant, while the masses of the nuclei are different. These two-body composite systems stay in their respective entangled state, each electron stays in a mixed state. If we suppose a one-dimensional hydrogen atom model having infinite nuclear mass, the electron stays in a pure state. We select position representation. The wave function of the ground state of the atom has the form of the square root of a δ function. To avoid calculation trouble of δ function, we select the first excited state and the superposed state of the first and the second excited states. We compare the two pure states, the first excited state and the superposed state, with those corresponding mixed states by fidelity and l1 norm coherence, and calculate the purities of the mixed states. The summations become integrations due to the position variable having a continuous eigenvalue spectrum. We investigate the changes in these quantities with the increase of the nuclear mass. The results show that the purities of the mixed states and the fidelities increase with the nuclear mass increasing. However, the coherences of the mixed states decrease with the nuclear mass increasing. This can be explained as that a mixed state with non-zero coherence may approach to a pure eigenstate, while the latter has zero coherence in the eigenspace. These mean that the greater a nuclear mass is, the closer the mixed state approaches to the corresponding pure state. Therefore, the two pure states are the approximations of the corresponding mixed states. The entangled state of the electron and the nucleus is related with the nuclear mass and the Coulomb interaction potential. So, each electron mixed state and its coherence are related with the nucleus and their Coulomb interaction potential. If the nuclear mass is great, the nucleus is approximately motionless or its state is approximately unchanged, and the Coulomb interaction potential approximates to the external Coulomb potential of the electron. The electron approximately stays in a pure state. The state and its coherence are related with the nucleus and the Coulomb interaction. From other point of view, the entangled state of the nucleus and the electron approximates to the product state of the unchanged nucleus state and the electron state. In this case, an electron mixed state approximates to its corresponding pure state, and then these states and their coherences are all related with the nucleus and the Coulomb interaction.
Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.
A Hookean model of a three-body problem for particles with arbitrary masses and charges where two of them interact with each other through a Coulomb potential and with the third through a harmonic potential is presented. It is shown that a condition relating the masses to the harmonic coupling constants must be satisfied in order to render this problem separable. A general exact analytic solution written in terms of the relative interparticle coordinates is given as well as general expressions for the total and binding energies of this three-body system. We apply these results to examine electronic, muonic, antiprotonic, and pionic families of non-Born-Oppenheimer Hookean systems. The first contains the atoms or atomic ions: ${\mathrm{Ps}}^{\ensuremath{-}}({e}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{H}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{D}}^{\ensuremath{-}}({d}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, ${\mathrm{T}}^{\ensuremath{-}}({p}^{+}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, $^{4}\mathrm{He}(h{e}^{+2}{e}^{\ensuremath{-}}{e}^{\ensuremath{-}})$, and the following molecular ions: ${\mathrm{Ps}}_{2}^{+}({e}^{\ensuremath{-}}{e}^{+}{e}^{+})$, ${\mathrm{H}}_{2}^{+}({e}^{\ensuremath{-}}{p}^{+}{p}^{+})$, $\mathrm{H}{\mathrm{D}}^{+}({e}^{\ensuremath{-}}{d}^{+}{p}^{+})$, ${\mathrm{HT}}^{+}({e}^{\ensuremath{-}}{t}^{+}{p}^{+})$, ${\mathrm{DT}}^{+}({e}^{\ensuremath{-}}{d}^{+}{t}^{+})$, ${\mathrm{D}}_{2}^{+}({e}^{\ensuremath{-}}{d}^{+}{d}^{+})$, ${\mathrm{T}}_{2}^{+}({e}^{\ensuremath{-}}{t}^{+}{t}^{+})$. The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing ${e}^{\ensuremath{-}}$ by ${\ensuremath{\mu}}^{\ensuremath{-}}$ or ${p}^{\ensuremath{-}}$. The pionic family comprises exotic atoms containing at least one pion. We also apply these results to two-electron three-dimensional spherical quantum dots and for these systems we examine the effect of electronic correlation, particularly on the singlet-triplet transitions and on the collective motion of the electrons and center of mass leading to ``floppy''dynamics.