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phy246s11_hw3sol

phy246s11_hw3sol - Physics 246 Third Homework o1 Set Due...

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Physics 246 Third Homework Set Due: Febuary 1 1, 201 1 L. Assume that the eigenfunctions o1 fI . ! ^' t 't^2 = 2*P- * 5mw-r are of the form r,. lbn:\a&i"-o". i=0 We know this to be true for n -- 0 and 1. a) Use the fact that, $2 satisfies I ,h6,tthzdn :0 to determine $2. Show that it satisfies the Schrodinger equation and determine the eigenvalue' b) Repeat this procedure to get ds and the corresponding eigenvalue. 2. Two particles are in the ground state of the system described in problem L. a ) What is the normalized probability function P(r1 ,rz)? Express the result also in terms of the center of mass coordinate X : Ll2(rt * sz) and relative coordinate nt - rz. b ) What is the probabitrity that X is between X and X + dX (with no regard to r)? c ) What is the probability that r is between r and x * dn (with no regard to X)? d ) Show that each of the probabilities is normalized to one' 3, A harmonic osciilator in two-dimensions can be written H - Ht*Hz where Ht : l- P? +***'*?' 2m"' ' 2 The angular momentum L_ r x P ib r"i L : frtpz-rzpt. Define A_
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