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

Download Final Exam for Introduction to Quantum Mechanics 1 | PHYS 5250 and more Exams Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, December 12, 2005 PHYS 5250: Quantum Mechanics - I FINAL EXAM Note: Please be as explicit as possible. Generous partial credit will be given for correct approach, even for wrong final answers, 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. (30 points) Harmonic Oscillator Consider a particle free to move in three dimensions, of mass m and charge q and subjected to a 1D harmonic potential U = 1 2 mω2x2, and sitting in its ground state. At t = 0+, an uniform, constant electric field E = Ex̂ is suddently turned on. Answer the following clearly and explicitly, indicating all the relevant quantum numbers and coordinate dependences: (a) (4 pts) Write down the ground-state wavefunction for this 3D problem right before the E-field is turned on. (b) (4 pts) Write down the Hamiltonian for the particle after the electric field is turned on, and give corresponding eigenstates and the energy spectrum for this 3D problem. (c) (5 pts) Compute the probability P0 of finding the system in its ground state at time t = 0+. (d) (7 pts) Compute the probability Pe of finding the system in one of its (particular) excited states at time t = 0+, clearly indicating the dependence on all the quantum numbers characterizing the excited states. If you are having difficulties to compute the final answer explicitly, at least write down the formal expression in terms of the answers to questions above. (e) (10 pts) Find the wavefunction ψ(r, t), for t > 0. You might find the following useful:∫+∞ −∞ dxHn(x)Hm(x)e −x2 = π1/22nn!δn,m, where the Hermite polynomials have a convenient generating function e−s 2+2xs = ∑ n sn n! Hn(x). Hint: Note that this last generating function is actually equivalent to a decompo- sition of a coherent state in terms of the oscillator eigenstates. In order to take advantage of this (or related) relation you will find it useful to make a couple of simple variable manipulations. 2. (25 points) One-dimensional Molecule Consider an electron in a double square-well potential, modelling the attractive po- tential of two ions of a diatomic molecule (e.g., singly ionized H+2 ). Let’s for sim- plicity model the wells by δ-functions, so the total attractive potential is U(x) = −U0(δ(x+ a) + δ(x− a)). (a) (10 pts) Find the molecule’s ground state, determining the spatial wavefunction completely, except for the overall normalization (that you do not need to find) and one other parameter κ, that is related to the ground state energy E0. (b) (10 pts) Write down (but of course do not solve) the transcendental equation satisfied by this one parameter κ, and use its general form (e.g., graphically) to show that a bound state solution κ∗ always exists, independent of how weak the strength of the attractive potential U0 is. 2