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An exploration of the other half of Quantum Mechanics. Most Quantum mechanical wave functions are obtained as functions of the positions of particles. An equivalent, but less widely utilized representation is in terms of the momentum of the particles. The usual description states that the momentum representation is the Fourier Transform of the position representation. However, this holds only for Cartesian coordinate systems. For curvilinear coordinates we must use the "DeWitt" transform. This allows an operator formalism in which the important 1/r operator has a simple integral form in momentum space.
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First we derive the wave functions
for the hydrogen atom by transforming the Schrodinger equation into an appropriate integral equation in momentum space. The results are extended to He, and a two electron function is derived in which 90% of the correlation energy is obtained with only one parameter.A relativistic formulation for the Hydrogen atom is obtained, and the effect of the Yukawa potential is explored. Finally, we examine ways to find molecular functions with a study of the hydrogen molecule ion. |
1. "The Hydrogen Atom in the Momentum Representation," Phys. Rev. A, 22, 797 (1980).
2. "The Helium Atom in the Momentum Representation," J. Phys. Chem., 86, 3513 (1982).
3. "A Correlated One-Parameter Momentum Space Wave Function for
Helium," J. Chem.
Phys., 78, 2476 (1983).
4. "Relativistic Hydrogen Atom in the Momentum Representation,"
Phys. Rev. A, 27, 1275
(1983).
5. "The Yukawa Potential in the Momentum Representation," J.
Chem. Phys. 85, 949
(1986).
6. "The Hydrogen Molecule-ion in the Momentum Representation",
C.Abrams, Ph.D.
Thesis submitted to the City University
of New York,1992.
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