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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
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+ iﬁag «%(A~B)w%8] 5+ 1; :. 3*Q'Cln+1~+l-u~1> i f, ...

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