Free-particle solutions

In order to demonstrate the ambiguity of the X-operator, explicit expressions for both X+ and X are given for the case of a free particle, defined by V = 0, 

The square-root operator is difficult to evaluate in position space because of the square root to be taken of a differential operator that would represent p. We have already discussed this issue in the context of the Klein-Gordon equation in section 5.1.1. Hence, the action of the X-operator is most conveniently studied in momentum space, where the inverse operator may be applied in closed form without expanding the square root. 

The four normalized free-particle Dirac eigenspinors of section 5.3 with shifted eigenvalues e —nteC may now be written as 

Closed-Form Unitary Transfonnation of the Dirac Hamiitonian 

Another possibility for the reduction of the 4-spinor to two-component Pauli form is to decouple the Dirac equation by a xmitary transformation U to block- 

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It is of interest also to consider the x -component of linear momentum for the free-particle solutions (3.4). According to Eq (2.24), the eigenvalue equation for momentum should read... 

The importance of each partial wave can be estimated by comparing Li with the logarithmic derivative for the free particle, i.e. the solution of eq. (3.9) with V(r) = 0. For sufficiently large / the centrifugal potential becomes the dominant term, so the radial function becomes close to the free particle solution. As a result one needs to include only the lower / values in the sum in eq. (3.8). For rare earth band calculations it is sufficient to go up to / = 4. 

If we apply the free particle solution in a one-dimensional domain (0, L), which will be shown in (D.47) to obtain the ground state energy e = iJi = the... 

We start by considering the one-particle Dirac equation and proceed in a manner analogous to the free-particle solutions of chapter 7. The equation to solve is... 

The solution to this is a Gaussian function, which spreads out in time. Hence the solution to the Bloch equation for a free particle is also a Gaussian ... 

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics... 

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

The solution of this equation requires that = i k + G /2m with the wavefunctions heiiig of the form oc exp[i(k + G) r]. Although cast in a slightly different form, this i.-, et[uivalent to our earlier expression for the wavefunction of a free particle. Equation (3.92). 

In a solution of molecules of uniform molecular weight, all particles settle with the same value of v. If diffusion is ignored, a sharp boundary forms between the top portion of the cell, which has been swept free of solute, and the bottom, which still contains solute. Figure 9.13a shows schematically how the concentration profile varies with time under these conditions. It is apparent that the Schlieren optical system described in the last section is ideally suited for measuring the displacement of this boundary with time. Since the velocity of the boundary and that of the particles are the same, the sedimentation coefficient is readily measured. 

Freeing a solution from extremely small particles [e.g. for optical rotatory dispersion (ORD) or circular dichroism (CD) measurements] requires filters with very small pore size. Commercially available (Millipore, Gelman, Nucleopore) filters other than cellulose or glass include nylon, Teflon, and polyvinyl chloride, and the pore diameter may be as small as 0.01 micron (see Table 6). Special containers are used to hold the filters, through which the solution is pressed by applying pressure, e.g. from a syringe. Some of these filters can be used to clear strong sulfuric acid solutions. 

Next we investigate the physical content of the Dirac equation. To that end we inquire as to the solutions of the Dirac equation corresponding to free particles moving with definite energy and momentum. One easily checks that the Dirac equation admits of plane wave solutions of the form... 

When the silver nanocrystals are organized in a 2D superlattice, the plasmon peak is shifted toward an energy lower than that obtained in solution (Fig. 6). The covered support is washed with hexane, and the nanoparticles are dispersed again in the solvent. The absorption spectrum of the latter solution is similar to that used to cover the support (free particles in hexane). This clearly indicates that the shift in the absorption spectrum of nanosized silver particles is due to their self-organization on the support. The bandwidth of the plasmon peak (1.3 eV) obtained after deposition is larger than that in solution (0.9 eV). This can be attributed to a change in the dielectric constant of the composite medium. Similar behavior is observed for various nanocrystal sizes (from 3 to 8 nm). 

Equation (6.11) is the Schrodinger equation for the translational motion of a free particle of mass M, while equation (6.12) is the Schrodinger equation for a hypothetical particle of mass fi. moving in a potential field F(r). Since the energy Er of the translational motion is a positive constant (Er > 0), the solutions of equation (6.11) are not relevant to the structure of the two-particle system and we do not consider this equation any further. 

We should also mention that basis sets which do not actually comply with the LCAO scheme are employed under certain circumstances in density functional calculations, i. e., plane waves. These are the solutions of the Schrodinger equation of a free particle and are simple exponential functions of the general form... 

In general, the excess Helmholtz free energy AF of a hard-particle solution over that of the solvent is written in the form... 

Another possible approach to obtain some information on the solutions of the nonlinear equation, for a free particle, consists in rewriting Eqs. (11) for the one-dimensional case ... 

A free particle is a particle subject to no forces, so that V = 0 everywhere. For a free particle moving in one dimension, the Schrodinger equation is (1.122) and its solution is [Eq. (1.98)]... 

The particle moves freely between 0 and L but is excluded from x < 0 and x > L. Inside the box, the Schrodinger equation has the form of Equation (E.19) (the free particle). The time-independent solution can be written... 

Of the three models that have been proposed to explain the properties of excess electrons in liquid helium, two are considered in detail (1) The electron is localized in a cavity in the liquid (2) The electron is a quasi-free particle. The pseudopotential method is helpful in studying both of these models. The most useful treatment of electron binding in polar solvents is based on a model with the solution as a continuous dielectric medium in which the additional electron induces a polarization field. This model can be used for studies with the hydrated electron. 

In order to construct a collision integral for a bound-state kinetic equation (kinetic equation for atoms, consisting of elementary particles), which accounts for the scattering between atoms and between atoms and free particles, it is necessary to determine the three-particle density operator in four-particle approximation. Four-particle collision approximation means that in the formal solution, for example, (1.30), for F 234 the integral term is neglected. Then we obtain the expression... 

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