Harmonic oscillator phase space

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

The microscopic state of the system defines coordinates, momenta, spins for every particle in the system. Each point in phase space corresponds to a microscopic state. There are, however, many microscopic states, in which the states of particular molecules or bonds are different, but values of the macroscopic observables are the same. For example, a very large number of molecular configurations and associated momenta in a fluid can correspond to the same number of molecules, volume, and energy. All points of the harmonic oscillator phase space that are on the same ellipse in Fig. 5 have the same total energy. 

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Figure 5 Trajectory of an element representing a simple harmonic oscillator in phase space.

As has been discussed in Section 111, the initial phase-space distribution pyj, for the nuclear DoF xj and pj may be chosen from the action-angle (18) or the Wigner (17) distribution of the initial state of the nuclear DoF. To specify the electronic phase-space distribution pgj, let us assume that the system is initially in the electronic state v i ). According to Eq. (80b), the electronic state vl/ ) is mapped onto Ne harmonic oscillators, whereby the nth oscillator is in its first excited state while the remaining Nei — 1 oscillators are in their ground state. The initial density operator is thus given by... 

The dynamics of the normal mode Hamiltonian is trivial, each stable mode evolves separately as a harmonic oscillator while the imstable mode evolves as a parabolic barrier. To find the time dependence of any function in the system phase space (q,pq) all one needs to do is rewrite the system phase space variables in terms of the normal modes and then average over the relevant thermal distribution. The continuum limit is introduced through use of the spectral density of the normal modes. The relationship between this microscopic view of the evolution... 

For e = 0, the quantities (10) and (17) become first integrals for the harmonic oscillator [141]. It is obvious from (15)—(16) that a trajectory in phase space (p, q) for the Kerr oscillator is analytically the same ellipse as for the harmonic oscillator... 

Consider a one-dimensional classical harmonic oscillator (Figure 3.1). Phase space in this case has only two dimensions, position and momentum, and we will define the origin of this phase space to correspond to the ball of mass m being at rest (i.e., zero momentum) with the spring at its equilibrium length. This phase point represents a stationary state of the system. Now consider the dynamical behavior of tlie system starting from some point other than the origin. To be specific, we consider release of the ball at time to from... 

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

For systems more complicated than the harmonic oscillator, it is almost never possible to write down analytical expressions for the position and momentum components of the phase space trajectory as a function of time. However, if we approximate Eqs. (3.10) and (3.12) as... 

Certain aspects of this phase space trajectory merit attention. We noted above that a phase space trajectory cannot cross itself. However, it can be periodic, which is to say it can trace out the same path again and again the harmonic oscillator example is periodic. Note that the complete set of all harmonic oscillator trajectories, which would completely fill the corresponding two-dimensional phase space, is composed of concentric ovals (concentric circles if we were to choose the momentum metric to be (mk) 1/2 times the position metric). Thus, as required, these (periodic) trajectories do not cross one another. 

D. J. Tannor To understand the role of dissipation in quantum mechanics, it is useful to consider the density operator in the Wigner phase-space representation. Energy relaxation in a harmonic oscillator looks as shown in Fig. 1, whereas phase relaxation looks as shown in Fig. 2 that is, in pure dephasing the density spreads out over the energy shell (i.e., spreads in angle) while not changing its radial distribution... 

Fig. 5.2. Contour plots of two representative Wigner distribution functions PW(R,P) for two harmonic oscillators in their ground vibrational states, Equation (5.15), in the two-dimensional phase-space (R,P). The widths in the R-and in the P-directions are inversely related.

Fig. 4.1.2 Harmonic oscillator with the energy E = p2/(2m) + (1/2)kq2 (which is the equation for an ellipse in the (q,p)-space). In the quasi-classical trajectory approach, E is chosen as one of the quantum energies, and all points on the ellipse may be chosen as initial conditions in a calculation, i.e., corresponding to all phases a [0, 27r].

Let us now lift this disk D2 into phase space. To do so, one must go back to the sphere equation, Eq. (37). There are several ways of depicting a 3-sphere one is particularly appropriate here [24]. The sphere is dynamically composed of two identical harmonic oscillators without explicit coupling, but whose total energy is a constant, hs3 > 0. Let us thus transform the Hamiltonian (37) in action angle variables, where N,Iy are the actions of the two oscillators and Q, 0, are the two associated angles. Since... 

Figure 4 The left-hand column shows the Poincare section v versus x of the phase space of a harmonic oscillator H = p llm + mw q l with m = 1, w = 1 for Nose-Hoover dynamics [top) (cf. Eqs. [65]) and for MTK dynamics [bottom) (cf. Eqs.

Although we have assumed in Eq. [209] that the velocity profile in the confined fluid is linear, it is not immediately obvious that this is technically possible in the absence of moving boundary conditions. A parallel to this situation is the comparison between Nose-Hoover (NH) thermostats and Nose-Hoover chain (NHC) thermostats. Although the Nose-Hoover equations of motion can be shown to generate the canonical phase space distribution function, for a pedagogical problem like the simple harmonic oscillator (SHO), the trajectory obtained from the NH equations of motion has been found not to fill up the phase space, whereas the NHC ones do. The SHO is a stiff system and thus to make it ergodic, one needs additional degrees of freedom in the form of an NHC.2 ... 

The virtue of this change of variables is that it allows us to visualize a phase space with trajectories frozen in it. Otherwise, if we allowed explicit time dependence, the vectors and the trajectories would always be wiggling—this would ruin the geometric picture we re trying to build. A more physical motivation is that the state of the forced harmonic oscillator is truly three-dimensional we need to know three numbers, x, x, and t, to predict the future, given the present. So a three-dimensional phase space is natural. 

This work is intended as an attempt to present two essentially different constructions of harmonic oscillator states in a FD Hilbert space. We propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, we also bring together several known FD quantum-optical states, thus making our exposition more self-contained. We shall discuss FD coherent states, FD phase coherent states, FD displaced number states, FD Schrodinger cats, and FD squeezed vacuum. We shall show some intriguing properties of the states with the help of the discrete Wigner function. 

Analogously to the generalized, CS in a FD Hilbert space, analyzed in Section IV. A, other states of the electromagnetic field can be defined by the action of the FD displacement or squeeze operators. In particular, FD displaced phase states and coherent phase states were discussed by Gangopadhyay [28]. Generalized displaced number states and Schrodinger cats were analyzed in Ref. 21 and generalized squeezed vacuum was studied in Ref. 34. A different approach to construction of FD states can be based on truncation of the Fock expansion of the well-known ID harmonic oscillator states. The same construction, as for the... 

Figure 9 Wigner distribution function of the n = 10 eigenfunction of the harmonic oscillator. The picture shows the extent of the wave function in phase space which has nearly optimal sampling due to the balance between the representation of the kinetic and potential energy.

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