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hw6 - b Find the exact energy eigenstates and expand them...

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HOMEWORK # 6 Physics 6572 Wednesday, 10/22/08; due 10/31/08 1. Problem 2, page 263, Chapter 5, Gottfried & Yan. There are several typos. The corrected equations are δV ( ~ r ) = Ze 2 4 πr - Ze 2 Z d ~ r 0 ρ N ( ~ r 0 ) 4 π | ~ r - ~ r 0 | Δ E n = 1 6 Ze 2 h r 2 i N | ψ ns (0) | 2 and the last expression for Δ E n is correct. The mean square radius of the nuclear charge distribution is defined as h r 2 i N = Z d ~ r r 2 ρ N ( ~ r ) and ρ N ( ~ r ) is normalized to 1, Z d ~ N ( ~ r ) = 1 . Replace part (c) by problem 1 in this chapter and use the result to estimate the value of Z beyond which the approximation in (a) fails for a muon. 2. a) Suppose the Hamiltonian of a rigid rotator in a magnetic field perpendicular to the axis is of the form A ~ L 2 + BL z + CL y if terms quadratic in the field are neglected. Assuming B C , use perturbation theory to lowest nonvanishing order to get approximate energy eigenvalues.
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Unformatted text preview: b) Find the exact energy eigenstates and expand them in a power series in C . Compare the results with those in part (a). 3. This problem concerns the so called Stark eﬀect. A hydrogen atom is placed in a constant electric ﬁeld in the positive z-direction. The perturbation is given by H 1 = e | E | z , where | E | is the magnitude of the electric ﬁeld and e is the electric charge of the electron. The electric ﬁeld is suﬃciently strong that the relativistic and spin-orbit eﬀects can be neglected. Use the degenerate state perturbation theory to determine the ﬁrst order changes in energy eigenvalues of the n = 2 states. What are the corresponding new eigenfunctions?...
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