Free particle wave equations

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... 

Spin Particles.—The covariant relativistic wave equation which describes a free spin particle of mass m is Dirac s equation ... 

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... 

Both photons and electrons are particle-waves, but different equations describe their properties. Table 7-1 summarizes the properties of photons and free electrons, and Example shows how to use these equations. 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

Following the theoretical scheme of Schrodinger, we associate a wave packet (jc, 0 with the motion in the jc-direction of this free particle. This wave packet is readily constructed from equation (1.11) by substituting (1.32) and (1.33) for CO and k, respectively... 

We now wish to derive the energy-time uncertainty principle, which is discussed in Section 1.5 and expressed in equation (1.45). We show in Section 1.5 that for a wave packet associated with a free particle moving in the x-direction the product A A/ is equal to the product AxApx if AE and At are defined appropriately. However, this derivation does not apply to a particle in a potential field. 

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... 

The function 4> k) is known as the wave function in momentum space. The Fourier integral represents the superposition of many waves of different wave vectors. This construct defines a wave packet, once considered as the theoretically most acceptable description of a wave-mechanical particle5. Schrodinger s dynamical equation (4) for a free particle... 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

But by de Broglie s principle of wave-particle duality, this free particle has associated with it a wave vector, = p/h, and angular frequency, = E/h (cf eqs (2.13) and (2.14)), which substituting into the above equation gives... 

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... 

Unlike molecular mechanics, the quantum mechanical approach to molecular modelling does not require the use of parameters similar to those used in molecular mechanics. It is based on the realization that electrons and all material particles exhibit wavelike properties. This allows the well defined, parameter free, mathematics of wave motions to be applied to electrons, atomic and molecular structure. The basis of these calculations is the Schrodinger wave equation, which in its simplest form may be stated as ... 

A Volkov state is obtained from the solution of the time-dependent Schrodinger equation for a free particle in an external plane-wave laser field. Such states were first derived by Volkov, in a relativistic context [21]. 

Although we have not introduced in detail the character of the wave function and the structure of the associated spaces we may, for example associate p — mo for a free particle. In general, however, one needs to take into account that p is an operator, which in its extended form may not be self-adjoint. Hence our secular equation yields well-known relations and by standard operator identification one may obtain a familiar Klein-Gordon type equation. We will, however, proceed by looking further at the secular equation. In an obvious notation one obtains for the... 

One can gain some insight into the nature of the Dirac wave equation and the spin angular momentum of the electron by considering the Foldy Wouthuysen transformation for a free particle. In the absence of electric and magnetic interactions, the Dirac Hamiltonian is... 

The simplest illustration of this argument is provided by a free particle in linear motion, correctly described by a wave function (T6.2.1) that satisfies the equation... 

For the case of an electron, or any other particle of spin 1/2 (a fermion) considered free in a one-dimensional (the z direction, say) potential well of infinite walls, the time-independent relativistic wave equation... 

The division of the Hamiltonian into an unperturbed and perturbed part is of course arbitrary. However in most cases jiPq is chosen to make the scattering particle Green s function simple to calculate. Typical choices would be the target plus a free particle or in the case of positive ions, target plus Coulomb wave. Equation... 

There is a well known expansion theorem to expand the plane wave solution (which are the solutions to the free particle equation) in terms of Bessel functions as well [39] ... 

The most important and difficult quantity to calculate in the above equation is the electronic coupling matrix element. The initial state wave function, TC, of a conducting electron in the metal electrode is not taken as a Bloch wave function because the periodicity of the metal is no longer effective at the interface (normal to the surface). The wave function of a free particle is taken as that given in [35], but allowance is made for the interaction with the metal using an internal effective mass of the conducting electron [39] ... 

In the region 0 < x < a the general solution of the wave equation is a sine function of arbitrary amplitude, frequency, and phase, as for the free particle. Several such functions are represented in Figure 14-1. All of these are not acceptable wave... 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

Problem 17-1. The equation for the free particle is separable in many coordinate systems. Using cylindrical polar coordinates, set up and separate the wave equation, obtain the solutions in

recursion formula for the coefficients in the series solution of the p equation. Hint In applying the polynomial method, omit the step of finding the asymptotic solution. 

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