The free particle

Consider a particle of mass m which moves in the absence of external forces along the x-axis only the absence of forces implies that the potential energy is constant, so for convenience we may choose V = 0. The components of momentum along they-and z-axes are zero that along the x-axis is p. The total energy is a constant and is equal to the kinetic energy the classical description is 

Replacing p by p, as in Section 20.4, we obtain the Schrodinger equation for this system, 

Since is a constant, this differential equation has two solutions  

These equations are typical eigenvalue relations the constant ImE appearing on the right is an eigenvalue of the momentum operator. 

The interpretation is that in a state described by xj/ the momentum of the particle has a fixed precise value, j2mE. The classical values of momentum according to Eq. (21.4) are Px = 2mE. Thus ij/ describes a particle moving in the + x direction (p is positive) with 

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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... 

We can recover the free-particle result (i.e. zero potential) from Equation (3.98) by setting all i)f Ihe Fourier coefficients Uq to zero, in which case the equation reduces to ... 

Fig. 12-2. Plot of aa Against K for the Paramagnetic (2) and Antiferromagnetic (3) One-Dimensional Kronig-Penny Potentials. The free particle energy E m is included for the purpose of comparison. Note the discontinuity...

By way of contrast, recall that in treating the free particle as a wave packet in Chapter 1, we required that the weighting factor A(p) be independent of time and we needed to specify a functional form for A(p) in order to study some of the properties of the wave packet. 

Nuclear binding energy is the energy equivalent (in E = mc2) of the difference between the mass of the nucleus of an atom and the sum of the masses of its uncombined protons and neutrons. For example, the mass of a He nucleus is 4.0026 amu. The mass of a free proton is 1.00728 amu, and that of a free neutron is 1.00866 amu. The free particles exceed the nucleus in mass by... 

This mass has an energy equivalent of 4.54 x 10"12 J for each He nucleus. You would have to put in that much energy into the combined nucleus to get the free particles that is why that energy is called the binding energy. 

Multiply the free-particle equation (4) on the left by k and the conjugate equation by [Pg.199]

In the electromagnetic field the free particle operators therefore become p p — eA, E E — e... 

Despite its classical energy spectrum it would be wrong to simply treat the free particle as a classical entity. As soon as its motion is restricted, for instance, when confined to a fixed line segment 0 < x < L something remarkable happens to the spectrum. 

In many cases it is convenient to start from the free-particle formulation, V = 0, and introduce suitable potential barriers to simulate specific interactions. 

Fluorescent particles that have been opsonized with serum or specific IgG are mixed with phagocytic leukocytes at 37°C and continuously mixed to optimize the cell-particle interaction. The reaction is stopped by the addition of ice-cold medium, and the free particles are washed away from the leukocytes by centrifugation. The cells are resuspended in cold medium and analyzed... 

A simple method is given for obtaining root-mean-square electron-cloud radii yr. m. i. from diamagnetism. Comparison of these with structural radii r shows that for ionic and van der Waals bonds, r is about twice yr. m. s.for localised covalent bonds they are about equal. Comparison with theory shows that electron-cloud radii are about the same for localised-bond atoms, or for ions in crystals, as for the free particles with a notable exception in the hydride anion. 

Free single particle partition sum. The one-particle sum over states of the ideal gas is easily evaluated if the free-particle Hamiltonian (with V (R) = 0) is used,... 

Under conditions of low dimer concentrations, i.e., at low pressures and/or high temperatures, especially for the lighter gases, the free-particle expressions, Eq. 2.27, may be substituted, Zr v/f. ... 

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... 

It is conventional to divide the space into three regions I, II, and III, shown in Figure E.5. In regions I and III, we have the free-particle problem heated in Section E.4. In region I, we have particles moving to the left (the incident particles) and particles moving to the right (reflected particles). So we expect a wave function of the form of Equation (E.24), whose time-independent part can be written... 

Following in analogy with the theory of weak interactions we let / be a doublet that describes an electron according to the 1 field and the 3 field. Here we illustrate the sort of physics that would occur with a chiral theory. We start with the free particle Dirac Lagrangian and let the differential become gauge covariant,... 

VK is the interaction in the channel /q, which describes the interaction between the free particle and a bound pair. Thus we have... 

Therefore, we have now a two-component system consisting of free particles and atoms. For the description of this system it is necessary to introduce distribution functions for the free particles and atoms. It seems to be obvious that the distribution function of the atoms is connected with the bound-state contribution of Fbu,... 

Now we interpret the first term as a contribution corresponding to the free particles... 

Therefore, we can give the following definition of the distribution functions. For the free particles the distribution function is given by... 

Now we return to Eq. (3.15). Using the definition of F we can eliminate the bound-state part F 2 and obtain the Boltzmann equation for the free particles in the form... 

As in Section II. 1, we get from (3.25) the usual shape of the Boltzmann equation for free particles. Obviously, such equations are of no interest for systems with bound states. Equation (3.15a) for atoms is collisionless, and the equation for the free particles does not contain contributions that account for the formation and the decay of bound states. In order to derive such equations we must take into account three-particle and higher-order collisions. Subsequently, this problem will be dealt with. 

The quantity (3.115) may be interpreted as effective interaction between the free particle a and the bound pair (be). The expression (3.116) characterizes the static charge distribution in the atom (be). 

In Eq. (32), we have included the full spectrum of a second-order ordinary differential equation with negative bound state eigenvalues and the continuum being the positive real axis. The free-particle background is mbee = i /k. In this case, the full m-function becomes [in the equation below we have introduced a natural generalization of Js, the "jump" or imaginary part, see Eq. (36) for the general case]... 

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