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Unformatted text preview: Chem 120A The Hydrogen Atom, Part 1 03/08/06 Spring 2006 Lecture 20 READING: Engel: Ch 9 In the last lecture we saw how the separation of variables technique could be used to find solutions to the 2-D Particle in a Box. Now we will use this same technique to solve for the 3-D Hydrogen Atom. We will use separation of variables in two different ways. First we will separate the wavefunction into a component which determines the external energy of the particle and a component which determines the internal energy (the internal structure is what we are really interested in). Next we will take the wavefunction which de- scribes the internal structure and separate it into a part which depends on the radial distance of the electron from the nucleus (the radial part of the wavefunction) and a part which depends on the angular coordinates in space (the angular component of the wavefunction). Hydrogenic atoms consist of a nucleus at a point, r 1 which has charge +Ze (Z is the number of protons and e is the elementary charge = 1 . 602 10 19 Coulombs) and an electron of charge -e at a point r 2 (see Figure 1). Hydrogenic atoms have only one electron (e.g. He + , Li 2 + , O 7 + ). For more than one electron the results will be quite different. Hydrogenic atoms have spherical symmetry, so it will become natural for us to use spherical coordinates ( r , , ) to describe the system. The Hamiltonian for a Hydrogenic atom is: H =- h 2 2 m 1 2 1- h 2 2 m 2 2 2- Ze 2 4 r (1) where m 1 is the nuclear mass, m 2 is the electron mass and r is the magnitude of separation between the nucleus and the electron = | r 1- r 2 | Chem 120A, Spring 2006, Lecture 20 1...
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