QUANTUM MECHANICS IPHYS 516Problem Set # 2: 3-Dimensional OscillatorsDistributed: Jan. 19, 2011Due: Jan. 28, 20111. Harmonic Motion in 3D: A particle of mass m is placed in apotential of the form V (x, y, z) =12kxx2+12kyy2+12kzz2.a. Show that Schr¨odinger’s equation separates in Cartesian coordinates.b. Write down the energy eigenvalue and the wavefunction when there is1 excitation in the x direction, 2 in the y direction, and 3 in the z direction.c. En1,n2,n3= ?d. Suppose ω1= ω2= 1.0 and ω3= 1.1. Determine the energy spec-trum when 3 or fewer excitations are present. If (when) there is an energydegeneracy, identify the number of states in the degenerate multiplet.e. If all spring constants are the same, what is the degeneracy of the statecontaining N quanta? (Hint: this is the same as the Bose-Einstein countingproblem.)2. Molecules: The linear molecule ABBA oscillates in one dimension.The masses are MA= M, MB= 2M. The interaction is between onlyadjacent atoms and is represented by linear springs with spring constantskAB= kBA= k and 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.3. Lattices: A simple, very short linear lattice consists of only threeatoms. Each has mass m. The atoms are connected to each other withsprings of spring constant k. The two atoms at the ends of this chain are1connected to brick walls with springs of spring constant k (there is a total of4 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 spectrum.4. Nuclear Physics: Fill in the table:Harmonic 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).)5. 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 (Seehttp://en.wiki- pedia.org/wiki/Baryon, Table JP=32+Baryons):2Particle 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