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P246_Hmwrk8_Sol - Physics 246 Eighth Homework Set Due April...

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Unformatted text preview: Physics 246 Eighth Homework Set Due: April 3, 2009 La) Consider the Hamiltonian H = .481 '82 where S] - 82 are spin operators of two spin-1/2 particles. Express H in terms of 32. S22 and (S; + Sg)2. Use this to compute the energy spectrum. What is the degeneracy of each level? b) Now consider H=Asl'SQ+B(Sg'S3+Sl-S3) for three spin-‘1/2 particles. Extend the technique of a) to determine the energy levels and their degeneracies. 2. For the two dimensional oscillator 1 V = 3mw2(:i:2 + 3/2) the energy levels are given by Emm = hw[2n +1+ lml]. a) Derive the radial Schrodinger equation in terms of p where p2 = (moo/M13. b) Eliminate the p2 term by writing R = exp[— p2 / 2] f ( p) and eliminate the p“2 term by writing f (p) = p""'9(p). c) Now set p2 = :c and show that one obtains the radial equation for the hydrogen atom. Verify the result for Em", and determine the wave functions. 3. Consider the Runge—Lenz vector 1 K— [pr—pr]+%r. 2me2 a) Consider first the classical case. Show that K is a constant (i.e., conserved) vector. 1)) Now show that in the quantum mechanical case [K, H] = 0 where H=£-2--i. r 8v %va+f%j§f_ b) R: 51%? f R’—» 12-60%? R” : «$4, “R42 5% _V.q: I: _m" WWI“ ” @wk 311% f )9 “WV 8113" '* ”’ 0&1” 9113‘ ‘V ”km-ED ,, , mi: “4&1ch -%BKV .j wig/17‘ «1»; Egflfl’ E&)o‘”fiv[(4v$)+5g + ifiag «%(A~B)w%8] 5+ 1; :. 3*Q'Cln+1~+l-u~1> i f, ...
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