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PHY 6645 Fall 2001 – Final Exam DO NOT TURN OVER THIS PAGE UNTIL YOU ARE TOLD TO DO SO Instructions: Please write your name in the space provided at the bottom of this page. At the end of the exam, you should hand in your solutions with this page on top. Attempt all five questions overleaf. The maximum score for each part of each question is shown in square brackets. (The maximum overall score on the exam is 100.) To gain full credit you should explain your reasoning and show all working. You may consult Shankar’s “Principles of Quantum Mechanics,” your PHY 6645 lecture notes, and/or solutions to exam or homework assignments from this semester’s course; you may not use any other written or printed material. You must not communicate with anyone other than the proctor. In accordance with the UF Honor Code, by turning in this exam to be graded, you affirm the following pledge: On my honor, I have neither given nor received unauthorized aid in doing this assignment. Name: PHY 6645 Fall 2001 – Final Exam Follow the instructions overleaf. 1. A certain particle has spin s = 12 . (a) A measurement is made of the instantaneous sum of the x and y components of the particle’s spin angular momentum. List all the possible results (i.e., measured values) that might be obtained from this measurement. [10] (b) Immediately after the measurement described in (a), the x component of the spin angular momentum is measured. For each of the possible results of the first measurement, list all the possible results of the second measurement and their respective probabilities. [20] 2. Suppose that two noninteracting spin- 12 particles are known to be in a spin-singlet state. Let Sz(1) measure the z component of the spin angular momentum of particle 1. Let Sn(2) measure the component of particle 2’s spin angular momentum along the direction n̂ described by polar angles (θ, φ), i.e., in Cartesian coordinates n̂ = (sin θ cos φ, sin θ sin φ, cos θ). Find the expectation value hSz(1) Sn(2) i. [15] 3. Construct a (normalized) total angular momentum eigenstate |j = 12 , m = 12 i from product states of the form |j1 = 1, m1 i ⊗ |j2 = 12 , m2 i. To gain full credit, you should construct the eigenstate from first principles, i.e., without quoting any Clebsch-Gordan coefficients. [15] 4. A point-like particle moves in a one-dimensional potential, 0 V (x) = V >0 0 +∞ for −a < x ≤ 0, for 0 < x < a, for |x| ≥ a. Plot y = |ψ(x)|2 over the range −a ≤ x ≤ a for the first excited state of this problem (i.e., the energy eigenstate of second-lowest energy), assuming that the state has an energy E > V0 . You should focus on getting the plot qualitatively (rather than quantitatively) correct. Make sure that it is clear on your plot at which points (if any) y is zero, and at which points (if any) dy/dx is zero. Also make clear the relative magnitudes of y in the regions x < 0 and x > 0. [20] 5. Consider a one-dimensional harmonic oscillator that at time t = 0 is in a state exp(−iP b/h̄)|0i, where |0i is the ground state of the harmonic oscillator, P is the momentum operator, and b is a constant length. Calculate the position expectation value at time t. [20] Hints: (1) Recall (from Exam 2) that aH (t) = aS e−iωt . (2) You may use without proof the formula (for operators A and B, and any scalar λ) eλA Be−λA = B + λ2 λ3 λ [A, B] + [A, [A, B]] + [A, [A, [A, B]]] + . . . 1! 2! 3!