Momentum-space Schrodinger

Fock was then able to show that the momentum-space Schrodinger equation for the hydrogen atom has properly normalized solutions of the form... 

If f satisfies the momentum-space Schrodinger equation then f satisfies the position-space Schrodinger equation. 

Thus, the momentum-space Schrodinger equation for hydrogenlike atoms becomes... 

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... 

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... 

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 ... 

Fock s argument rests on the theory of the Fourier transform. In particular, he uses the momentum-space version of the Schrodinger equation. We let f denote the Fourier transform of / e... 

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

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

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) ... 

Spin-coupled wavefunctions have proved to be very useful in studies of momentum-space properties " . Except for very simple systems, it is rather difficult to solve the Schrodinger equation directly in the momentum representation fortunately, the momentum-space wavefunction is also given by the Fourier transform of that in position space and this indirect approach proves to be much more tractable. The momentum-space formalism is particularly convenient for the interpretation of various scattering techniques such as Compton scattering and binary (e, 2e) spectroscopy. 

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... 

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... 

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 technology for solving the Schrodinger equation is so much more advanced in r-space than in p-space that it is most practical to obtain the momentum-space 4) from its position-space counterpart P. The transformation theories of Dirac [13,14] and Jordan [15,16] provide the link between these representations ... 

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