Introduction to Quantum Mechanics 1 - Final Exam | PHYS 5250, Exams of Quantum Mechanics

Download Introduction to Quantum Mechanics 1 - Final Exam | PHYS 5250 and more Exams Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, December 17, 2007 PHYS 5250: Quantum Mechanics - I FINAL EXAM Note: Please be as explicit as possible; points will be deducted for answers that are correct but do not show any work. Generous partial credit will be given for a correct approach, even if the final answer is not quite correct, if you can convince me that you know what you are doing. Unless explicitly requested, you are not required to derive things from scratch (particularly when I say, “What is...?”, as opposed to “Derive...”). However, if you are indeed not deriving your answer from scratch (e.g., because you simply know it), please, at least say some brief words about how you are (maybe mentally) obtaining the answer. For example “This Schrodinger’s equation can be solved by separation of variables introducing ..., which then reduces it to a problem that is identical to the one that we studied in class and gives Laguerre polynomials as eigenstates.”. Of course at the same time please keep in mind, that the fewer in-between steps and explanations you give, the more difficult it will be for me to give you partial credit for conceptual understanding, if you make a mistake. Please check your answers carefully! Total Points: 100. Good Luck! 1 1. (40 points) Quickies (with brief explanations) (a) (9 points) A system’s J2 and Jx operators are measured and found to have eigen- values 3h̄2/4 and +h̄/2, respectively, (i) if Jz is subsequently measured once, what value(s) can be found? (ii) what is the probability of finding Jz = +h̄/2, right after the above Jx measurement? (iii) what is the expectation value of Jz, right after the above Jx measurement? (b) (5 points) What is the unitary operator Uφ0 acting in Hilbert space that executes rotation in 2D by angle φ0? Write its coordinate representation and demonstrate explicitly that when acting on a wavefunction ψ(φ) it produces a state of a system rotated by φ0. (c) (8 points) (i) Write down a (formal) path-integral representation for a wavefunc- tion ψ(x, t) at time t for a Hamiltonian H = p2/2m+V (x) and t = 0 wavefunction ψ0(x), clearly indicating all the limits. (ii) Give the explicit coordinate-based an- swer in terms of ψ0(x) (based on whatever you like, memory, hints from the physically correct form and units, quick derivation using (i), etc...) for the case V (x) = 0. Comment: I am looking for an explicit and detailed expression (with everything clearly labelled and defined), but of course without any attempt to evaluate the path integral and the formal expression in (i) for V (x) 6= 0. (d) (10 points) A harmonic oscillator (defined by frequency ω0) at time t = 0 is in a coherent state |z〉. (i) What state does it evolve into at a subsequent time t? (ii) What is the expectation value of â2 at time t in this state (â is annihilation operator.)? (iii) Answer (ii) via Heisenberg representation instead. Comment: For convenience, you can ignore the vacuum energy, i.e., define the zero of the energy to be the ground state energy. (e) (8 points) A spin 1/2 system is described by a density matrix ρ = ( 1/2 0 0 1/2 ) . (1) (i) Does this density matrix represent a pure or a mixed state? How do you know? (ii) Use ρ to compute the expectation value of ~S. 2. (30 points) Bose-Einstein condensate trapped in a harmonic potential (a) (5 points) Write down the corresponding (i) Hamiltonian, H (ii) ground state N -particle wavefunction, ΨGS(x1, x2, ..., xN) and (iii) the total energy EGS of N noninteracting bosonic atoms trapped in a harmonic potential characterized by frequency ω0? Comment: Although you need not do any technical calculations for this part, please do provide brief comments as you take steps to deduce your answer. 2