PHY4221Fall05_Prob4

PHY4221Fall05_Prob4 - PHY4221 Quantum Mechanics I Fall term...

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Quantum Mechanics I Fall term of 2005 Problem Set No.4 (Due on November 19, 2005) 1. Consider a spinless particle of charge q moving in an electromagnetic ﬁeld de- scribed by a vector potential ~ A ( ~ r,t ) and a scalar potential φ ( ~ r,t ). The Hamil- tonian operator is given by H = ~ p - q c ~ A · 2 2 m + qφ . Show that the form of the time-dependent Schr¨odinger equation is invariant ( i.e. remains unchanged) under the gauge transformation ~ A ( ~ r,t ) = ~ A 0 ( ~ r,t ) + ~ χ ( ~ r,t ) φ ( ~ r,t ) = φ 0 ( ~ r,t ) - ∂t χ ( ~ r,t ) ψ ( ~ r,t ) = ψ 0 ( ~ r,t )exp ( iqχ/ ~ ) where χ ( ~ r,t ) is an arbitrary function of ~ r and t . 2. Using the generating function for the associated Laguerre polynomials U p ( ρ,s ) = ( - s ) p exp[ - ρs/ (1 - s )] (1 - s ) p +1 = X q = p L p q ( ρ ) q ! s q , | s | < 1 show that for the hydrogen atom the average values h r k i nlm are given respectively for k = 1 , - 1 , - 2 , - 3 by h r i nlm = a 0 n 2 1 + 1 2 1 - l ( l

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PHY4221Fall05_Prob4 - PHY4221 Quantum Mechanics I Fall term...

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