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

Download Introduction to Quantum Mechanics 1 - Questions for Final Exam | PHYS 5250 and more Exams Quantum Mechanics in PDF only on Docsity! Leo Radzihovsky, December 19, 2006 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. (25 points) Quickies (a) (10 points) Show that if two observables  and B̂ do not commute with each other but each commutes with a Hamiltonian Ĥ, i.e., [Â, B̂] 6= 0, [Ĥ, Â] = 0, [Ĥ, B̂] = 0, then this generally implies that the spectrum of the Hamiltonian is degenerate. (b) (7 points) In quantum mechanics, in terms of a unitary transformation operator Ûα what is a statement of the corresponding symmetry of a system governed by a Hamiltonian Ĥ, and what is its main consequence? After a general statement, illustrate with an example of e.g., translational symmetry. (c) (8 points) Show that a state |α〉 = eix̂α/h̄|p〉 (where |p〉 is an eigenstate of mo- mentum operator p̂ with eigenvalue p) is also an eigenstate of p̂ and derive its eigenvalue α in terms of p. 2. (50 points) A particle in a linear potential (e.g., electron in an electric or a gravitational field) (a) (5 points) Is the spectrum of a particle of mass m in a linear potential V (x) = fx discrete or continuous? (b) (10 points) Find the eigenstates |ψE〉 for a given energy E in momentum repre- sentation, not worrying too much about their normalization. Hint: Since p̂ and x̂ appear on equal footing in the canonical commutation relation, for this problem it is convenient to use a momentum representation directly. (c) (10 points) Write down a formal expression for the corresponding coordinate rep- resentation of |ψE〉, i.e., for the wavefunction ψE(x) and show that it is a function of a dimensionless variable ζ = (x − xE)/af . What are lengths xE and af and what is their significance? Hint: This wavefunction is proportional to the so-called Airy function, Ai(ζ). (d) (15 points) If in addition a particle experiences an impenetrable potential (hard wall) at x = 0, confining it to x > 0, what is the resulting spectrum? De- duce the characteristic length scale xn (expressing it in terms of a characteristic length found above) and the qualitative nature (upto dimensionless factors) of the corresponding spectrum via very simple and rough semi-classical (qualitative) ar- guments (of the type used in class and on the homework) of balancing kinetic and potential energies, i.e., by minimizing the total energy subject to the uncertainty relation constraint. (e) (10 points) Write down the spectrum in terms of zeros αn of the Airy function defined by Ai(αn) = 0. Sketch a few lowest eigen-wavefunctions ψE(x). 3. (25 points) Consider a diatomic molecule consisting of two identical, neutral, spin-1/2, fully spin-polarized fermionic atoms bound by a potential U(|r1− r2|). Treating atoms as point particles: 2