Schrodinger equation in momentum space

The Schrodinger equation in momentum space for a single particle system is obtained on taking an FT of its position space counterpart in the form... 

It is known that the Schrodinger equation in momentum space takes the form of an integral equation ... 

We have dropped the index i because for the moment we are dealing with a single electron). The use of Coulomb Sturmian basis functions located on the different atoms of a molecule to solve (63) was pioneered by C.E. Wulfman, B. Judd, T. Koga, V. Aquilanti, and others [30-37]. These authors solved the Schrodinger equation in momentum space, but here we will use a direct-space treatment to reach the same results. Our basis functions will be labeled by the set of indices... 

In Section 2.1. we examined solution to the Schrodinger equation in free space and in a periodic potential and foimd families of wave functions having a simple, quadratic relationship between wave energy and momentum over wide ranges of energy values. Because there are a very large number of such states in a solid, it is normally impossible to distinguish one individual state from another. In this Section, we will examine, briefly, the impact of artificial potential structures on the wave solutions. For a complete discussion of such effects, the reader is referred to books on quantum mechanics. 

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... 

The two Schrodinger equations, in coordinate and momentum space respectively... 

At this point it is stressed that the Schrodinger equation can also be formulated and solved in momentum space instead of coordinate space. 

As briefly mentioned in Chapter 2, the Schrodinger equation can be pressed and solved in momentum space yielding co p) functions instead of jinic orbitals ip r). The formal equivalence between these two kinds of -actions is expressed by a Fourier transform, which allows one to generate p I from ip r) and vice versa (refs. 20 and 29) ... 

Direct solution of the Schrodinger equation is possible in momentum-space for simple systems, but it is usually more convenient to start from r-space quantities. We begin here with molecular orbitals (MOs) formed in the usual way by the overlap of atomic basis functions < t(r — if ), centred on nuclei A at positions... 

Here, (pi,p2, Ps) are the coordinates of momentum space, while (iii, u2, M3. m4) are unit vectors characterizing points on the surface of the hypersphere. (In (74), and throughout this paper, we indicate a unit vector by means of a hat ). He then transformed the Schrodinger equation for hydrogenlike atom in momentum space to a problem involving the unit vector u on the surface of the 4-dimensional hyper sphere. 

The discrete variable method can be interpreted as a kind of hybrid method Localized space but still a globally defined basis function. In the finite element methods not only the space will be discretized into local elements, the approximation polynomials are in addition only defined on this local element. Therefore we are able to change not only the size of the finite elements but in addition the locally selected basis in type and order. Usually only the size of the finite elements are changed but not the order or type of the polynomial interpolation function. Finite element techniques can be applied to any differential equation, not necessarily of Schrodinger-type. In the coordinate frame the kinetic energy is a simple differential operator and the potential operator a multiplication operator. In the momentum frame the coordinate operator would become a differential operator and hence due to the potential function it is not simple to find an alternative description in momentum space. Therefore finite element techniques are usually formulated in coordinate space. As bound states x xp) = tp x) are normalizable we could always find a left and right border, (x , Xb), in space beyond which the wave-functions effectively vanishes ... 

In 1935, V.A. Fock [27,28] solved the Schrodinger equation for hydrogen in momentum space by a remarkable and beautiful method He was able to show that when momentmn space is mapped onto the surface of a 4-dimensional hypersphere by a suitable transformation, the hydrogen wave functions are proportional to 4-dimensional... 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

In this case we have a one dimensional potential and only one quantum number, n, and dipole selection rules dictate An odd. An example of the photon spectrum observed in the forward direction for the injection of 54 Mey electrons along the (110) planar direction in Si is shown in Fig. 16.xhe bound state An = 1 transitions are evident up to n = 5>4 and at higher energies the An = 3 transitions are also evident. One can now compare the observed spectrum with calculations based on e.g. Hartree-Fock descriptions of the Si atom. This can be done directly through the solution for the onedimensional Schrodinger equation or one may work in momentum space and use the many-beam formulation of the Schrodinger equation for the transverse motion. The results of the many-beam calculations which use Doyle-Turner scattering factors derived from Hartree-Fock wave functions are compared with experimental results in Table II. 

From the fact that f/conmuites with the operators Pj) h is possible to show that the linear momentum of a molecule in free space must be conserved. First we note that the time-dependent wavefiinction V(t) of a molecule fulfills the time-dependent Schrodinger equation... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Momentum-space methods, pioneered by McWeeny, Fock, Shibuya, Wulfman, Judd, Koga, Aquilanti and others [4,17-26] provide us with an easy and accurate method for constructing solutions to the Schrodinger equation of a single electron moving in a many-center Coulomb potential... 

The first few 4-dimensional hyperspherical harmonics K i, ,m(u) are shown in Table 5. Shibuya and Wulfman [19] extended Fock s momentum-space method to the many-center one-particle Schrodinger equation, and from their work it follows that the solutions can be found by solving the secular equation (63). If Fock s relationship, equation (67), is substituted into (65), we obtain ... 

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