Quantum mechanics analogies

The total energy, for example, may be written as a sum of atomic energies, and these atomic energies are again almost constant for the same structural units in different molecules. The atomic basins are probably the closest quantum mechanical analogy to the chemical concepts of atoms within a molecule. 

Equation (2.23) is the quantum-mechanical analog of the classical definition of momentum, p = mv = m(Ax/At). This derivation also shows that the association in quantum mechanics of the operator (h/i)(d/dx) with the momentum is consistent with the correspondence principle. 

This approach to defining the Lagrangian density with the aid of both forward and backward Euler densities ip and ip uses the neat construct that ip ip is time-invariant. This is as true in the quantum mechanical analogy. 

This formalism is the quantum-mechanical analog of the Klimontovich phase-space density formalism, and is particularly effective in the theory of long-range correlation and fluctuation in the kinetic theory of plasmas. 

Figure 14 shows some examples. An interesting point is that all modes are localized [105], as one expects from the quantum-mechanical analog of a one-dimensional electron gas in a random potential [157]. Alternatively, from a localized tight-binding point of view, micromagnetic delocalization... 

It is a constant of the motion for the classical Kepler problem (Saletan and Cromer, 1971). The magnitude of X is proportional to the eccentricity of the orbit and X points in the direction of the major axis. Using the correspondence principle Pauli first showed that its quantum mechanical analog,... 

The quantum mechanical analog of the equilibrium correlation function (6.6) is... 

What is the quantum mechanical analog of this approach Consider the simple example that describes the decay of a single level coupled to a continuum. Fig. 9.1 and Eq. (9.2). The time-dependent wavefunction for this model is (Z) = Ci (t) 11)+ Ci(t ) l, where the time-dependent coefficients satisfy (cf. Eqs (9.6) and (9.7))... 

Our approach to the study of the departure from equilibrium in chemical reactions and of the "microscopic theory of chemical kinetics is a discrete quantum-mechanical analog of the Kramers-Brownian-motion model. It is most specifically applicable to a study of the energy-level distribution function and of the rate of activation in unimolecular (dissociation Reactions. Our model is an extension of one which we used in a discussion of the relaxation of vibrational nonequilibrium distributions.14 18 20... 

This is the quantum mechanical analog of the classical expression Eq. (11), with the canonical momentum replaced by the expectation value of the momentum operator (P)t. It is then clear that the choice... 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

This is the quantum-mechanical analog of the virial, i. e. it is the average (expectation) value of Xj Px) because J y y[Pg.227]

Fig. 2.8. A quantum-mechanical analogy of the stable positions of a chair on the floor. A stiff molecule CioHlo with the shape shown here, when interacting with a crystal surface, would acquire several stable positions similar to those of the chair on the floor. They would correspond to some vibrational states (the molecule would vibrate about these positions) of a given electronic state ( the same bond pattern" ), which in this analogy would correspond to the fixed structirre of the chair.

Schrodinger equation for the collision dynamics, which is the quantum mechanical analog of what KPS did with classical mechanics, and thus should provide rigorously correct results. 

In summary, the plasmon can be taken as a quantum-mechanical analog of the classical resonance of sufficiently high electron density. Such a description of excitations can be appropriate for energy-loss spectroscopy and related methods. For dipole interaction with the electromagnetic field it seems that the existence conditions for plasmons cannot be fulfilled at all because the Brillouin theorem does not account for electron-electron... 

Eq. (4.44) allows us to make the connection to classical physics. Comparing this equation with the classical equation of motion as written in Eq. (2.86) we note that they are of similar form. They are true analogs if we consider the commutator / (ih) to be the quantum mechanical analog of the... 

The second term in Eq. (3.20) is a quantum mechanical analog of the classic electrostatic energy of interaction ... 

The Lorentz and Drude models can be explained rigorously in relation to electronic band structure. Indeed, both models have quantum-mechanical analogs intraband transitions for the Drude model and direct interband transitions for the Lorentz model. To see the role of both models in describing real metals, consider the schematic band diagram as shown in Figure 149. TWo typical transitions are illustrated in Figure 149. The first of these, called an intraband transition, corresponds to the optical excitation of an electron from below the Fermi level (Ep) to another state above the Ef within the same band. There is no threshold energr for such transitions, and they can occur only in metals. 

The use of the correspondence principle to define a quantum mechanical analog of the stress tensor in classical mechanics goes back to Schrodinger and Pauli [33, 34]. The interest of electronic structure theorists in the stress tensor has been episodic, starting with the rise of computational density functional theory in the... 

Experimental fall-off data and theoretical extrapolations to high temperature are given for pressure-dependent reactions. These extrapolations were done with the aid of a quantum mechanical analog of the simple Kassel formula (Robinson and Holbrook, 1972), which was fitted to low-temperature experimental results by variation of the deactivation collision efficiency. 

---