Quantization three-state case

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

The simplest procedure is to take the origin of a global I-frame so that P = 0 and linear momentum conservation forces kj = —k2. At the antipodes, kd = k2 so that the common I-frame is restricted now. The particle model in this frame becomes strongly correlated. If spin quantum state for I-frame, one corresponds to the linear superposition (a /S)[CiC2]i and the other I-frame system should display the state (a P)[c, — Ci]2, namely, an orthogonal quantum state. The quantization of three axes is fixed. Spin and space are correlated in this manner. Now, the label states (a P)[c2 — cji and (a )[C C2]2 present another set of possibilities. This is because quantum states concern possibilities. All of them must be incorporated in a base state set. At this point, classical and quantum-physical descriptions differ radically. The former case handles objects that are characterized by properties, whereas the latter handle objects that are characterized by quantum states sustained by specific materiality. 

Some fundamental differences exist for the three types of quantization. In particular, the densities of electronic states (DOS) as a function of energy are quite different, as illustrated in Fig. 9.2. For quantum films the DOS is a step function, for quantum dots there is a series of discrete levels and in the case of quantum wires, the DOS distribution is intermediate between that of films and dots. According to the distribution of the density of electronic states, nanocrystals lie in between the atomic and molecular limits of a discrete density of states and the extended crystalline limit of continuous bands. With respect to electrochemical reactions or simply charge transfer reactions. 

Theories of chemical bonding based on the properties of degenerate states with fixed I assume independent behaviour of the electrons in these states. In particular, for three electrons in the three-fold degenerate /i-state with 1=1, they are assumed to have distinct values of m, without mutual interference. To make this distinction it is necessary to identify some preferred direction in which the components of angular momentum are quantized. By convention this direction is labeled as Cartesian Z. If the electrons share the degenerate p-state with parallel spins, they must share the same direction of quantization. This being the case, only one of the electrons can have the quantum number m = 0, characteristic of the real function (7). 

Fig. 4.1 Three types of reactive resonances near the transition state region in chemical reactions, adapted from [66]. Panel (a) illustrates the case associated with a deep potential well along the reaction coordinate. The resulting bound and pie-dissociative quasi-bound states can be characterized, for a three-atom system, by three vibrational modes, (b) Threshold resonance for which only the two motions orthogontil to the unbound reaction coordinate tire quantized and thus assignable by vibrational quantum numbers. The dynamical trapped-state resonance is schematically shown in panel (c). Despite the repulsive potential energy surface along the reaction coordinate, this metastable state can be assigned by three vibrational quantum numbers...

In the second quantization (SQ) one introduces quantum operators instead of quantum states. Thus, for instance, for a harmonic oscillator we would introduce the following set (1, a, a+). If we take the commutator between any two of these operators we get either zero or the third operator. Thus the set is closed with respect to commutations. The hamiltonian for the harmonic oscillator can be expressed in terms of these operators Hq = Tiw a i). If this is the case, i.e., that the hamiltonian for the system can be expressed in terms of operators, it is possible to solve also the TDSE using operator algebra. The advantage of using this technique is that the number of operators is smaller than the number of states. For the harmonic oscillator the number of operators is three but the number of states is infinite. Thus a certain class of problems can be solved analytically, i.e., formally. One such hamiltonian is... 

For Cooper pairs with S = 1,7 = 1 there are a very large number of possible states for the liquid as a whole. It has turned out that of the three possible spin projection values Sz = 0 or 1, on the quantization axis, only = 1 pairs are present in the A-phase all three types of pairs are present in the B-phase and only = 1 pairs are present in the Al-phase. In each case, the relative orientations of the spin and orbital angular momenta adjust themselves so as to minimize the free energy of the liquid as a whole, and they do so in quite different ways for the A- and B-phases. 

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