With the help of principle of superposition in quantum mechanics and the operator splitting method,we study the movement of the center position of the initial Gaussian wave packet in one-dimensional harmonic oscillator potential.The results show that its quantum evolution performs characteristics of a classical harmonic oscillator;the initial location and the initial momentum of the wave packet play a similar role as the classical oscillator.Our results should be helpful to understand the Gaussian wave packet method,which is commonly applied in the research on complex quantum system.
With the help of principle of superposition in quantum mechanics and the operator splitting method,we study the movement of the center position of the initial Gaussian wave packet in one-dimensional harmonic oscillator potential.The results show that its quantum evolution performs characteristics of a classical harmonic oscillator;the initial location and the initial momentum of the wave packet play a similar role as the classical oscillator.Our results should be helpful to understand the Gaussian wave packet method,which is commonly applied in the research on complex quantum system.
The one-dimensional classical forced oscillator is quantized.The exact solution of time evolution operator of the system is obtained.The quasi-probability distribution function of phase space is used to investigate the quantum characteristics of one-dimensional forced oscillator.The results show that the Wigner function is two-dimensional Gaussian wave packet from the initial vacuum state.The central coordinate and momentum of the Gaussian wave packet move in the same manner as one-dimensional classical harmonic oscillator.The amplitude and phase of the centre of the wave packet are modulated by the force.
In this paper,the propagation nature of the wave packet was discussed through the Green's function under three common potential fields.We had discovered some valuable conclusions.Firstly,when the wave packet of free particle varied with the time,the wave packet center of gravity moved by a speed 0/μ, but when the wave packet amplitude changed from the initial δ to the final δt,the Position variable r still deferred to the Gaussian distribution. Secondly,the wave packet amplitude in a homogeneous field changed by a power series with relative to the initial time.Finally,in simple harmonics oscillator potential field,the momentum mean value in random time is precisely the momentum form of classical simple harmonics oscillator.
A Gaussian wave packet of the inverted oscillator is investigated using the invariant operator method together with the unitary transformation method. A simple wave packet directly derived from the eigenstates of the invariant operator of the system corresponds to a plane wave that is fully delocalized. However, we can construct a weighted wave packet in terms of such plane waves, which corresponds to a Gaussian wave. This wave packet is associated with the generalized coherent state, which can be crucially utilized for investigating the classical limit of quantum wave mechanics. Various quantum properties of the system, such as fluctuations of the canonical variables, the uncertainty product, and the motion of the wave packet or quantum particle, are analyzed by means of this wave packet. We have confirmed that the time behavior of such a wave packet is very similar to the counterpart classical state. The wave packet runs away from the origin in the positive or negative direction in the 1D coordinate depending on the condition of the initial state. We have confirmed that this wave packet not only moves acceleratively but also spreads out during its propagation.
The dynamics of an initially excital Gaussian wave packet (harmonic oscillator coherent state, HOCS) in a general one-dimensional anharmonic system is studied analytically with the aid of a second quantization formalism for times shorter than a classical oscillation period. The initial state population probability PQ(t), its classical counterpart PC(t), and the overlap probability PQC(t) between a time evolved wave packet and a classically moving Gaussian with stationary shape are calculated in order to analyze the spreading of the wave packet. It was found that, in general, PQ(t) and PC(t) agree very well if the potential energy function has no strongly changing low order derivatives. This agreement is held even if the wave packet spreads drastically within the time scale considered as can be demonstrated by analyzing PQC(t). The analytical expressions for PQ(t), PC(t), and PQC(t) are applied to a dissociative third-order oscillator and a strongly anharmonic fourth-order oscillator. While for a harmonic system, the initial decay time of PQ(t) and PC(t) (which are identical in this case) are only dependent on the energy expectation value 〈H〉 with respect to the initial HOCS, differences up to an order of magnitude occur for HOCS with the same energy in the model systems.
The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks.