Momentum harmonic oscillation

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

We now solve equation (6.24) by means of ladder operators, analogous to the method used in Chapter 4 for the harmonic oscillator and in Chapter 5 for the angular momentum. We define the operators Ax and Bx as... 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

The residual energy (designated of a harmonic oscillator in the ground state. The Heisenberg Uncertainty Principle does not permit any state of completely defined position and momentum. A one-dimensional harmonic oscillator has energy levels corresponding to ... 

The total vibrational energy is a sum of energies of 3N-6 distinct harmonic oscillators. Indeed, 3N-6 is the final number of coordinates in Equation 8. Namely, the number of 3N+3 coordinates of the initial equation has been reduced by three through the elimination of internal rotations. Furthermore, the equation of nuclear motion (mainly its potential) has to be invariant under rotations and translations of a molecule as a whole (which is equivalent to the momentum and angular momentum preservation laws). The latter requirement leads to a further reduction of the number of coordinates by six (five in the case of linear molecules for which there are only two possible rotations). 

What is the lowest possible energy for the harmonic oscillator defined in Eq. (5.10) Using classical mechanics, the answer is quite simple it is the equilibrium state with x 0, zero kinetic energy and potential energy E0. The quantum mechanical answer cannot be quite so simple because of the Heisenberg uncertainty principle, which says (roughly) that the position and momentum of a particle cannot both be known with arbitrary precision. Because the classical minimum energy state specifies both the momentum and position of the oscillator exactly (as zero), it is not a valid quantum... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

Electrons in metals at the top of the energy distribution (near the Fermi level) can be excited into other energy and momentum states by photons with very small energies thus, they are essentially free electrons. The optical response of a collection of free electrons can be obtained from the Lorentz harmonic oscillator model by simply clipping the springs, that is, by setting the spring constant K in (9.3) equal to zero. Therefore, it follows from (9.7) with co0 = 0 that the dielectric function for free electrons is... 

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... 

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... 

If the duration of the experiment is measured not from the onset of the UV pulse but from t0, the time spent in boosting the momentum in the lower electronic state should be less than the time saved evolving on V2. For the two harmonic oscillators displaced the distance D, this requirement gives in the limit of T small (a more general case is analyzed in [7]) the following condition for the strength E0 of the IR field (m is the mass and qel is the equilibrium distance of oscillator Vi) ... 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

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. 

Figure 9. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state O2 with ions (q = ionic charge, Q = O2 quadrupole moment N,J,M= free-rotor quantum numbers M, k, v = harmonic oscillator quantum numbers N = electronic angular momentum quantum number see Ref. 33).

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

In order to elucidate the general behavior we also show in Figure 10.3 what we will call the harmonic oscillator (HO) approximation, i.e., expression (10.7) without the sinusoidal factor. It represents the momentum distribution of the harmonic oscillator in the fcth bending vibrational state. Suppression of the fast oscillations has the advantage of elucidating more clearly the wide oscillations for k > 0, which reflect the nodal structure of the excited bending wavefunctions. The superimposed fast oscillations, on the other hand, reflect the shift of the equilibrium angle 7e away from zero. They are absent for a linear molecule and most pronounced for 7e 7t/2, as for H2S and H2O, for example. 

Fig. 10.5. Measured rotational state distributions of OH following the dissociation of the three lowest bending states of H2O (open circles). In addition to the bending quanta H20(X) also contains 4 respectively 3 quanta of OH stretching excitation. The local mode nomenclature nm k) is explained in Section 13.2. The total angular momentum is zero in all cases. The filled circles represent the harmonic oscillator approximation defined in the text. Reproduced from Schinke, Vander Wal, Scott, and Crim (1991).

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

Now, consider the normalized density operator pa of a system of equivalent quantum harmonic oscillators embedded in a thermal bath at temperature T owing to the fact that the average values of the Hamiltonian //, of the coordinate Q and of the conjugate momentum P, of these oscillators (with [Q, P] = ih) are known. The equations governing the statistical entropy S,... 

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