LectureQ9

LectureQ9 - Chapter 9: Atoms Although we won't solve the...

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4/20/2004 H133 Spring 2004 1 Chapter 9: Atoms Although we won’t solve the equations which lead to the energy levels of hydrogen (or any other atom) we can use some of what we have learned to make some qualitative statements about the wave functions for energy levels and the behavior of atom. Warnings: ± This chapter is very qualitative ± In many cases we will state a principle without justification…maybe we will make a plausible argument but by no means will we prove anything. Much of this will need to wait for a later quantum mechanics course.
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4/20/2004 H133 Spring 2004 2 Radial and Angular Wave Functions One of the obvious shortcomings of the Bohr model is that it demands the electrons are in circular orbits ± It does not allow for any radial motion of the electron ± If we perform a fully quantum mechanical treatment of hydrogen we find that electrons are not in circular orbits and can spread out in the radial direction. ± The hydrogen atom is clearly a 3-dimensional problem. Let’s break it down into spherical coordinates ² (r, φ, θ ) Consider only the radial direction ± This is clearly suspect but it turns out that there are some energy eigenfunctions which do not depend of φ and θ and therefore only depend on r. ± Plausible Statement: The lowest energy state that depends only on r will fit a “half a wavelength” between the classical turning points. ± First three radial wavefunctions: a b c V a b c r r r r x z y r θ φ
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4/20/2004 H133 Spring 2004 3 Radial Wavefunctions Properties of the radial wave functions ± At the classical turning point at large r, the wavefunction is an exponential decay in the classically forbidden region. (More on this in Chapter 11) ± R=0 is the other “classical turning point”…the wave function goes to zero at r=0. This implies that the electron won’t be found in near the nucleus. If we take the absolute square of the radial wave functions we get: Notice that for these energy levels “n” is the number of “radial bumps”. ..not the number of full wavelengths around the atom as in the case in the Bohr Model. ± To distinguish this let’s label these “n r ± Also remember we were considering wavefunctions that had no dependence on φ and θ …no for these there are NO bumps in the wavefunction “around” the atom.
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This note was uploaded on 06/03/2011 for the course H 133 taught by Professor Furnstahl during the Spring '11 term at Ohio State.

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LectureQ9 - Chapter 9: Atoms Although we won't solve the...

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