Quantum discrete states momentum

In addition to the quantum approaches mentioned above, classical optimal control theories based on classical mechanics have also been developed [3-6], These methods control certain classical parameters of the system like the average nuclear coordinates and the momentum. The optimal laser held is given as an average of particular classical values with respect to the set of trajectories. The system of equations is solved iteratively using the gradient method. The classical OCT deals only with classical trajectories and thus incurs much lower computational costs compared to the quantum OCT. However, the effects of phase are not treated properly and the quantum mechanical states cannot be controlled appropriately. For instance, the selective excitation of coupled states cannot be controlled via the classical OCT and the spectrum of the controlling held does not contain the peaks that arise from one- and multiphoton transitions between quantum discrete states. 

Quantum mechanics tells us that only certain discrete values of E, the total electron energy, and J, the angular momentum of the electrons are allowed. These discrete states have been depicted in the familiar semiclassical picture of the atom (Fig. 1.1) as a tiny nucleus with electrons rotating about it in discrete orbits. In this book, we will examine nuclear structure and will develop a similar semiclassical picture of the nucleus that will allow us to understand and predict a large range of nuclear phenomena. 

The uncertainty principle, according to which either the position of a confined microscopic particle or its momentum, but not both, can be precisely measured, requires an increase in the carrier energy. In quantum wells having abmpt barriers (square wells) the carrier energy increases in inverse proportion to its effective mass (the mass of a carrier in a semiconductor is not the same as that of the free carrier) and the square of the well width. The confined carriers are allowed only a few discrete energy levels (confined states), each described by a quantum number, as is illustrated in Eigure 5. Stimulated emission is allowed to occur only as transitions between the confined electron and hole states described by the same quantum number. 

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... 

We use the parameters N = 128 and M = 1. Thus the quantum states are represented by 128 discrete points, and the range of momentum is from —7ti to 7ti. The value of i = 2nM/N is 0.3436. " When the control purpose is to steer a wavepacket in a torus to another place in another torus, OCT fails. This is because the wavepacket is trapped in one torus, and it is very hard to escape from the torus with a weak external field. 

With box normalisation the channel states d>,) are countable. For discrete target states the index i stands for the internal quantum numbers n,j,m,, v of the target and projectile and the box quantum numbers nix,nty,ni characterising the relative motion. When L —> 00 the box quantum number set is replaced by the momentum continuum k,. The limiting procedure is summarised as follows... 

The experimental foundation of the quantum theory of atomic structure as put forward by Bohr, lies in the stability of the atom and in the existence of discrete energy levels and the ability of the atom to absorb and emit energy only in quanta, as demonstrated by the discontinuous nature of atomic spectra and by the critical potential measurements of Franck and Hertz. Bohr postulated that the atom could only exist in a limited number of orbits or stationary states, which were defined by the quantum condition that the angular momentum can assume only certain limited values which are given by the expressioiT ... 

For any atomic multipole transition, the excited state can be described in terms of the dual representation of corresponding SU(2) algebra, describing the azimuthal quantum phase of the angular momentum. In particular, the exponential of the phase operator and phase states can be constructed. The quantum phase variable has a discrete spectrum with (2j + 1) different eigenvalues. 

A very important practical matter in MBPT calculations is the efficient and accurate evaluation of sums such as encountered in 2- As a first step the atom is considered as being at the center of a large sphere that confines electrons within a radius R this serves to discretize the continuum states. Care must be taken with the boundary conditions to avoid the Klein paradox the electron mass, a scalar, rather than the potential is chosen to go to infinity for r > R. 72 is chosen to be large compared to the atom, typically around 72 = 50 — 70 a.u.. For a given value of the angular momentum quantum number k, the Dirac equation for an electron of energy e with upper and lower components 72c(r) and respectively, can be obtained by requiring 5S = 0, where... 

Some years later, aided by considerably more rapid computers than available to Wall and co-workers, Karplus, Porter, and Sharma reinvestigated the exchange reaction between H2 and H [24]. As with the earlier work, the twelve classical equations of motion were solved. In addition, discrete quantum-mechanical vibrational and rotation states were included in the total energy so that the trajectories were examined as a function of the initial relative velocity of the atom and molecule and the rotational and vibrational quantum numbers j and v of the molecule. The more sophisticated potential energy surface of Porter and Karplus was used [7], and the impact parameter, orientation and momentum of the reactants, and vibration phase were selected at random from appropriate distribution functions. This Monte Carlo approach was used to examine 200-400 trajectories for each set of VyJ, and v. The reaction probability P can be written as... 

Owing to the closely spaced energy levels, quantum effects can often be neglected and the state distribution treated as continuous. This corresponds to replacing the discrete sum over energies by an integral over all coordinates (r) and momentum (p), called the phase space. 

It should be noted that the salient difference between the energy-momentum relation between bulk semiconductor and quantum well material is that the k vector associated with Eq takes on discrete, well-separated values. In the quantum well device, the density of states is obtained fi om the magnitude of the two-dimensional k vector associated with the y-z plane, as compared to the three-dimensional wavevector for the bulk semiconductor. As a result, the final density of states for the quantum well structure is given by... 

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