Fine Structure Perturbations - Quantum Mechanics - Exam, Exams of Quantum Mechanics

Download Fine Structure Perturbations - Quantum Mechanics - Exam and more Exams Quantum Mechanics in PDF only on Docsity! Introduction to Quantum Mechanics 171.304 Midterm Exam 3/30/06 1-2 Check the attached formula pages. Start each problem on a fresh page and give detailed reasoning. Please ask your proctor for clarification if the text is unclear. Problem 1 (30 points) Consider the n=4 level of the hydrogen atom. (a) Neglecting any perturbations how many degenerate states are there? (10 points) (b) In the presence of a strong magnetic field and neglecting fine structure, how many distinct energy levels exist and what is the degeneracy of each of these levels. (10 points) (c) Now consider the fine structure perturbations in zero external magnetic field. How many distinct energy levels exist and what is the degeneracy of each of these levels? (10 points) Problem 2 (20 points) Consider a three state system with unperturbed Hamiltonian ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = 100 020 002 ˆ 0H Add a small perturbation ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − −− = αα ααα αα i ii i H 0 0 ˆ 1 , where 1<<α . Compute the leading corrections to the three energy eigenvalues. Problem 3 (25 points) Consider the following variational wave function for the hydrogen atom: ( ) ( ) ⎩ ⎨ ⎧ ≤− = otherwise0 for arraC ra θφψ Determine the value of a that minimizes the total energy of the state and compare your result to the Bohr radius: 2 2 0 me a h= .(Hint: Use the Laplacian in spherical coordinates which is on the next page) Problem 4 (25 points) Consider two identical charged spin-1 particles in a central Coulomb potential. You can neglect spin orbit coupling and the Coulomb repulsion between the two spin-1 particles can be considered a weak perturbation. (a) Why is the total spin a conserved quantity for this system? (5 points) (b) Give detailed reasoning for why the ground state has a total spin quantum number of 0 or 2. (Hint: Consult the table of Clebsch-Gordan coefficients) (10 points) (c) Argue why the first excited two particle state must have a total spin quantum number of 1. (10 points) Formulae Laplace operator in spherical coordinates: 2 2 222 2 2 2 sin 1sin sin 11 φθθ θ θθ ∂ ∂ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ =∇ rrr r rr A table of Clebsch-Gordan coefficients is provided