One-particle problem

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

As a simple example to illustrate reciprocal-space solutions to the many-center one-particle problem, we can think of an electron moving in the Coulomb potential of two nuclei, with nuclear charges Zi and Z2, located respectively at positions Xi and X2. In the crude approximation where we use only a single Is orbital on each nucleus, we can represent the electronic wave function of this system by ... 

Equation (4.1) is a two-particle problem. In Section 1.12 we showed that we can separate a two-particle problem into two one-particle problems, 142... 

In addition to the individual and uncorrelated particle motions, we also have collective ones. In a strict sense, the hopping of an individual vacancy is already coupled to the correlated phonon motions. Harmonic lattice vibrations are the obvious example for a collective particle motion. Fixed phase relations exist between the vibrating particles. The harmonic case can be transformed to become a one-particle problem [A. Weiss, H. Witte (1983)]. The anharmonic collective motion is much more difficult to treat theoretically. Correlated many-particle displacements, such as those which occur during phase transformations, are further non-trivial examples of collective motions. 

The most serious limitation of wave mechanics is the complexity of any wave equation that describes interacting particles and prevents application to atoms other than hydrogen. To separate the equation and solve for any situation of interest it is necessary that it be reduced to a one dimensional one-particle problem. In the case of the hydrogen atom this is done by assuming the proton to be of infinite mass and therefore stationary. The only molecular system that can be treated in the same way is H, if the two protons are clamped to remain at a fixed distance apart. 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

The X-ray singularity problem was originally solved in the asymptotic limit and the complicated many-body problem was turned into an effective one-particle problem (219). For the X-ray photon frequency threshold frequency (o0, the absorption spectrum g(m) for the process in which a deep, structureless core electron is excited to the conduction band by the absorption of an X-ray of frequency w is expressed by the power law... 

Figure 1.1 Reduction of a two-body elastic collision to the equivalent one-particle problem.

The constant p, the chemical potential, is a Lagrange parameter that is introduced to ensure proper normalization, as in Hartree-Fock theory. At this stage, Kohn and Sham noted that Eq. (3.36) is the Euler equation for noninteracting electrons in the external potential V ff. Thus, finding the total energy and the density of the system of electrons subject to the external potential V is equivalent to finding these quantities for a noninteracting system in the potential Veir- Such a problem can in principle be solved exactly, but we have to know E and the potential V c-The one-particle problem can be solved as ... 

The only systems that can be solved exactly are those composed of only one or two particles. The latter can be separated into two pseudo one-particle problems by... 

At first sight, it may look like the two particle problem of describing the motion of a molecule 2 relative to molecule 1 is equivalent to the one particle problem of describing the motion of a single molecule 2 with reduced mass /i acted on by the force F21. With respect to a fixed particle 1 the scattering process appears as shown in Fig. 2.2. 

Equation 5 encompasses the complex scaling variational principle of Equation 3 as a special case, and this can be seen, for example. In a one-particle problem from the following Identity for radial matrix elements with respect to square-lntegrable basis functions. 

Boci round. The dissolution of a salt into a surrounding solution is easiest to think of as taking place in a closed vessel, that is, in a "batch" with no flows in or out of the vessel. But remember there ate "flows" between the solid and the liquid phases. We take a particle of the solid as one control volume and the volume of solvent as the other. We can solve this one particle problem and then handle many particles. The process is taken to occur at constant temperature. The physical situation looks like that shown in Figure 3. 

For the present, we confine ourselves to one-particle problems. In this section we consider the three-dimensional case of the problem solved in Section 2.2, the particle in a box. 

We have shown that, for my one-particle problem with a spherically symmetric potential-energy function Vif), the stationary-state wave functions are = R r)Yf(6, ), where the radial factor R r) satisfies (6.17). By using a specific form for V(r) in (6.17), we can solve it for a particular problem. 

Thus, when the system is composed of two noninteracting particles, we can reduce the two-particle problem to two separate one-particle problems by solving... 

Section 6.3 Reduction of the Two-Particle Problem to Two One-Particle Problems 127... 

The hydrogen atom contains two particles, the proton and the electron. For a system of two particles 1 and 2 with coordinates Zi) and X2,y2, z-, the potential energy of interaction between the particles is usually a function of only the relative coordinates X2 Xi,y2 y, and Z2 Zi of the particles. In this case the two-particle problem can be simplified to two separate one-particle problems, as we now prove. 

Since the potential energy of this two-particle system depends only on the relative coordinates of the particles, we can apply the results of Section 6.3 to reduce the problem to two one-particle problems. The translational motion of the atom as a whole simply adds some constant to the total energy, and we shall not concern ourselves with it. To deal with the internal motion of the system, we introduce a fictitious particle of mass... 

Figure 13.3 shows H. The nuclei are at a and b R is the intemuclear distance and Tf, are the distances from the electron to nuclei a and b. Since the nuclei are fixed, we have a one-particle problem whose purely electronic Hamiltonian is [Eq. (13.5)]... 

The parameter A will occur in the kinetic-energy operator as part of the factor multiplying one or more of the derivatives with respect to the coordinates. For example, taking A as the mass of the particle, we have for a one-particle problem... 

A popular trick to avoid this problem rests on the following reasoning If the equation is linear, one might first want to look for a solution of the one-particle problem and then simply create the two-particle solution by superposition. We can then apply Eq. (14) to calculate the force, which in the present example would yield the interaction potential [224] ... 

True or false (a) For a one-particle problem with V = bP, where Z> is a positive constant, the stationary-state wave functions have the form if/ = f r)Yf 0, ). (b) Every one-particle Hamiltonian operator commutes with and with L. ... 

This corresponds to a one-particle problem in a noncentral potential. Astute calculations could make use of elliptic or bipolar coordinates. In fact, one may simply use an ordinary expansion into partial... 

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