211A_W11_Homework_4

211A_W11_Homework_4 - these impurities to the surrounding...

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1 ECE/MAT 211A: Homework 4. Due February 2, 2011, at noon. Problem 1 : Particle in a Spherical Shell A particle of mass m is constrained to move between two concentric impermeable spheres of radii r = a and r = b . There is no other potential. Find the ground state energy and normalized wave function. Problem 2: Spherical Harmonic Oscillator in Different Coordinates Kroemer # 3.1-2 (p. 102). Problem 3: Radii in the Hydrogen Atom Calculate (a) The mean radius (b) the mean square radius (c) the most probable radius of the 1 s, 2 s, and 3 s orbitals of a hydrogenic one-electron atom with nuclear charge Ze . Hint: For the most probably radius look for the principal maximum of the radial distribution function. Problem 4 : Hyperfine parameters Electron paramagnetic resonance (EPR) is a powerful experimental technique that can yield information about the chemical identity of impurities in solids, and the bonding of
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Unformatted text preview: these impurities to the surrounding host atoms. EPR probes the interaction between the electronic wave function and specific nuclei. Results of EPR measurements are often expressed in terms of hyperfine parameters . (a) The isotropic hyperfine parameter (also referred to as the Fermi contact interaction ) is directly related to the probability of an electron being at the same location as the nucleus. Evaluate this probability density for an electron interacting with the nucleus (i.e., the proton) in the 1 s , 2 s , and 3 s orbitals of a hydrogen atom. (b) The anisotropic hyperfine parameter is directly related to the average value of 1/ r 3 . Evaluate <1/ r 3 > for an electron in the 2 p orbital of a hydrogen atom....
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