Momentum-space Schrodinger equation

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... 

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 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 ... 

The standard Schrodinger equation for an electron is solved by complex functions which cannot account for the experimentally observed phenomenon of electron spin. Part of the problem is that the wave equation 8.4 mixes a linear time parameter with a squared space parameter, whereas relativity theory demands that these parameters be of the same degree. In order to linearize both space and time parameters it is necessary to replace their complex coefficients by square matrices. The effect is that the eigenfunction solutions of the wave equation, modified in this way, are no longer complex numbers, but two-dimensinal vectors, known as spinors. This formulation implies that an electron carries intrinsic angular momentum, or spin, of h/2, in line with spectroscopic observation. 

The theory for a particle having a wavelength is represented by the Schrodinger equation, which, for the particle confined to a small region of space (such as an electron in an atom or molecule) can be solved only for certain energies, ie the energy of such particles is quantized or confined to discrete values. Moreover, some other properties, eg spin or orbital angular momentum, are also quantized. 

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