QUANTUM MECHANICS IPHYS 516Problem Set # 3: 3-Dimensional OscillatorsDistributed: Feb 17, 2010Due: Feb. 26, 20101. Particle Physics: The 10 states in the decuplet multiplet, |n1, n2, n3i,with ni≥ 0, n1+n2+n3= 3, can be assigned to the particles (See http://en.wiki-pedia.org/wiki/Baryon, Table JP=32+Baryons):Particle Charge Mass(MeV) n1n2n3Ω−− 1672.45 ± 0.29 003Ξ∗−− 1535.0 ± 0.6 102Ξ∗00 1531.80 ± 0.32 012Σ∗−− 1387.2 ± 0.5 201Σ∗00 1383.7 ± 1.0 111Σ∗++ 1382.8 ± 0.4 021∆−− 1232 ± 1 300∆00 1232 ± 1 210∆++ 1232 ± 1 120∆++++ 1232 ± 1 030a. Propose a simple model (linear in n1, n2, n3) to describe the masses.b. Carry out a χ2test on this model. Reject or “Accept” (i.e., Fail toReject) your fit to the data. Give me a “story”.c. How well does your model describe “the three fundamental quarks”?2. Nuclear Physics: Fill in the table:1Harmonic Oscillator Angular MomentumN = n1+ n2+ n3Degeneracy L Notation Degeneracy0 1 0 S 11 3 1 P 32 6 2 D 50 S 13 10 3 F 71 P 3456b. Plot the energy level spectrum generated by the perturbed harmonicoscillator hamiltonianH = (N +32)E0+ αL · Lwith E0= 1.0MeV and α = 0.1 MeV. (The spectrum of L · L is L(L + 1).)3. Molecules: The linear molecule ABBA oscillates in one dimension.The masses are MA= M, MB= 2M. The interaction is between only adjacentatoms and is represented by linear springs with spring constants kAB= kBA= kand kBB= 3k.a. Describe the classical normal modes.b. For each, what is the energy?c. Quantize the vibrations of this molecule.d. Write down the expression for the quantum vibrational energies.4. Lattices: A simple, very short linear lattice consists of only three atoms.Each has mass m. The atoms are connected to each other with springs of springconstant k. The two atoms at the ends of this chain are connected to brick wallswith springs of spring constant k (there is a total of 4 identical springs).a. Determine the resonance frequencies.b. Describe the three normal modes.c. Write down the quantum mechanical hamiltonian.d. Write down the energy