PRELIMINARY EXAMINATION Department of Physics University of Florida Part A, January, 2016, 09:00 12:00. Instructions

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Horace Booker
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1 Studet ID Number: PRELIMINRY EXMINTION Part, Jauary, 6, 9: : Istructios. You may use a calculator ad CRC Math tables or equivalet. No other tables or aids are allowed or required. You may NOT use programmable calculators to store formulae. a ll of the problems will be graded ad will be tabulated to geerate a fial score. Therefore, you should submit work for all of the problems. b For coveiece i gradig please write legibly, use oly oe side of each sheet of paper, ad work differet problems o separate sheets of paper. The sheets for each problem will be stapled together but separately from the other two problems. c Your assiged studet ID Number, the Problem Number, ad the Page Number should appear i the upper right had corer of each sheet. Do NOT use your ame aywhere o the Eam. d ll work must be show to receive full credit. Work must be clear ad uambiguous. e sure that you had your completed work to the Proctor. e Each problem is worth poits. f Followig the UF Hoor Code, your work o this eamiatio must reflect your ow idepedet effort, ad you must ot have give, or received, ay uauthorized help or assistace. If you have ay questios, ask the Proctor. Hoor Code: We, the members of the commuity, pledge to hold ourselves ad our peers to the highest stadards of hoesty ad itegrity. O all work submitted for credit by studets at the, the followig pledge is either required or implied: O my hoor, I have either give or received uauthorized aid i doig this assigmet. DO NOT OPEN EXM UNTIL INSTRUCTED

2 PRELIMINRY EXMINTION Part, Jauary, 6, 9: :. Koigsberg spi / electro with magetic momet µ is placed i a uiform magetic field ˆk i the positive z-directio. The itrisic spi of the electro is poited alog the positive -directio at t. a [ poits] Write dow the Schrödiger equatio for the two compoet wave fuctio for the electro at rest. b [3 poits] Fid the correspodig time-depedet wave fuctio. c [3 poits] Calculate the epectatio values < S t >,< S y t >,< S z t >. d Fid the probability as a fuctio of time that the itrisic spi will be poited alog: [ poit] the positive z-directio [ poit] the positive -directio This maybe useful: σ i σ y i σ z S z S

3 PRELIMINRY EXMINTION Part, Jauary, 6, 9: :. Field Cosider a electro with mass m e cofied withi a oe-dimesioal ifiite square well defied by V for < < L V otherwise. Ifiite Square Well V ifiity V ifiity L a [ poits] Ug Schrödiger s equatio, calculate the allowed statioary state eigefuctios Ψ, where the complete wavefuctios are give by Ψ,t Ψ e iet/. Normalize the eigefuctios so that the probability of fidig the electro somewhere i the bo is oe. b [ poits] Show that the wavefuctios Ψ,t correspod to states with defiite eergy i.e., show that E σ E. c [ poits] Calculate the allowed eergy levels, E, of the system. Epress your aswer i terms of the Compto wavelegth of the electro, λ e /m e c, ad the rest mass eergy of the electro, m e c. What is the groud state eergy i MeV for the case L λ e? Note that m e c.5 MeV. d [ poits] Suppose the electro i this ifiite square well has a wave fuctio at t which is give by Ψ, L /L/L. If you measure the eergy of this particle, what are the possible values you might get, ad what is the probability of gettig each of them? What is the epectatio value of the eergy for this state i.e., average eergy? What is Ψ,t ad what is the epectatio value of,, for the state Ψ,t? Does deped o time?

4 Useful Math Trigoometric Relatios: m Idefiite Itegrals: d d d d d d d d d Defiite Itegrals: d e d d m d m m m Γ δ δ Γ Γ Γ

5 PRELIMINRY EXMINTION Part, Jauary, 6, 9: : 3. Cheg Two idetical particles, each with mass M ad spi, are described by the Hamiltoia, Ĥ M Mω. The two particles are i a state described by the wave fuctio, ψ, e /a χ. 3 a Here, χ is a vector i a four-dimesioal spi vector space, where m,m is a basis state for this space satisfyig Ŝjz m,m m j m,m for j ad, ad a /Mω. a [ poits] What are the three mutually orthogoal ad physically acceptable spi vectors χ i the state described by Eq. 3? Epress your ormalized aswers i the basis { m,m }. b [3 poits] Fid the total eergy of the system i the state described by Eq. 3. c [3 poits] Write dow aother statioary state of the system that is orthogoal to, but degeerate i eergy with, the three states i part a. Specify both the spatial ad spi parts.

Quantum Mechanics I. 21 April, x=0. , α = A + B = C. ik 1 A ik 1 B = αc.

Quantum Mechanics I. 21 April, x=0. , α = A + B = C. ik 1 A ik 1 B = αc. Quatum Mechaics I 1 April, 14 Assigmet 5: Solutio 1 For a particle icidet o a potetial step with E < V, show that the magitudes of the amplitudes of the icidet ad reflected waves fuctios are the same Fid