PHY4604 Exam 2 in Physics: Problems on Quantum Mechanics - Prof. Richard D. Field, Exams of Physics

Download PHY4604 Exam 2 in Physics: Problems on Quantum Mechanics - Prof. Richard D. Field and more Exams Physics in PDF only on Docsity! PHY4604 Exam 2 Department of Physics Page 1 of 8 PHY 4604 Exam 2 Name:__________________________________ Wednesday November 16, 2005 (Total Points = 100) Problem 1 (10 points): Circle true or false for following (1 point each). (a) (True or False) Bohr’s model of the atom assumes the orbital angular momentum is quantized according to hnL = , with n = 1, 2, 3, …, which is also true quantum mechanically. (b) (True or False) The eigenvalues of hermitian operators are real numbers. (c) (True or False) Eigenfunctions of hermitian operators corresponding to different eigenvalues are orthogonal. (d) (True or False) If an observable, O, commutes with the Hamiltonian and if O does not depend explicitly on time, then <O> is zero. (e) (True or False) If [A, B] = 0, then ∆A∆B ≥ 0. (f) (True or False) All quantum operators can be expressed in terms of functions or differentials. (g) (True or False) If the three operators J1, J2, and J3, all commute with each other then they are said to form a “Lie Algebra”. (h) (True or False) SU(2) is the group of all unitary 2 × 2 matrices with determinant equal to one. (i) (True or False) The spin 2 (i.e. s = 2) gravaton has five possible spin states. (j) (True or False) In SU(2), 2 × 2 = 4. PHY4604 Exam 2 Department of Physics Page 2 of 8 Problem 2 Name:_________________________________________ Problem 2 (30 points): Consider the (one dimensional) wave function at t = 0 given by 22)( ax Ax + =ψ , where A and a are real constants. (a) (5 points) Find the normalization constant A such that 1|)(| 2 =∫ +∞ ∞− dxxψ and sketch the probability density .|)(|)( 2xx ψρ = (b) (5 points) Compute <x> and <x2> and ∆x using the position space wave function ψ(x). (c) (10 points). Find the momentum space wave function at t = 0, ∫ +∞ ∞− −= dxexp xixpx h/)()( ψφ and verify that it is properly normalized. Sketch the probability density .|)(|)( 2xx pp φρ = (d) (5 points). Compute <px> and <px2> and ∆px using the momentum space wave function φ(px). (e) (5 points). What is ∆x ∆px? Is it consistent with the uncertainty principle? x ρ(x) px ρ(px) PHY4604 Exam 2 Department of Physics Page 5 of 8 Problem 3 Name:_________________________________________ Scratch Paper PHY4604 Exam 2 Department of Physics Page 6 of 8 Problem 4 Name:_________________________________________ Problem 4 (30 points): Consider a spin ½ system described by the Hamiltonian: 222 0 )(4 zyx aSSSH ++= where a is a real constant and σr r h 2=S with       = 01 10 xσ       − = 0 0 i i yσ       − = 10 01 zσ (a) (10 points) Find the energy levels and the corresponding eigenkets of the system. How many energy levels are there? How many eigenkets are there? What is the ground state energy, E0? (b) (10 points) If the Hamiltonian in (a) is changed to xbSHH += 0 , where b is a positive real constant, what are the energy levels and the corresponding eigenkets of this new system. How many energy levels are there? How many eigenkets are there? What is the ground state energy? Express your answer in terms of the ground state energy, E0, from (a). How does adding this additional term shift the energy levels of H0? (c) (10 points) Suppose that at t = 0 the system described by the Hamiltonian in (b) is in the state       >=>= + 0 1 |)0(| χχ . What is >)(| tχ ? If I measure the energy of the state >)(| tχ , what values might I get and what is the probability of getting these values. What is the expectation value of Sz in the state >)(| tχ ? PHY4604 Exam 2 Department of Physics Page 7 of 8 Problem 4 Name:_________________________________________ Scratch Paper