Quantum Mechanics Midterm Exam for Phys 560, Fall 2007, Exams of Quantum Physics

Download Quantum Mechanics Midterm Exam for Phys 560, Fall 2007 and more Exams Quantum Physics in PDF only on Docsity! Phys 560 Professor Thompson Quantum Mechanics 1st In class Midterm Fall 2007 October 15, 2007 Name: [110 points total] Closed book, closed notes, no calculator. One page (both sides) notes allowed. Turn in everything: Blue Book, Exam Sheet, Note Page. Do all eight 10 point exercises in this section. Short answer or derivation 1. (10 pts) Express the following operator as it would appear in the momentum-basis. Ĥ = ( P̂ 2 2m + mω2 2 X̂2) 2. (10 pts) Consider ψ = Aeipx/~+Be−ipx/~ in one dimension. Show that the ’probability current’ j = (|A|2 − |B|2) p m . 3. (10 pts) Note that the unnormalized eigenfunction for a free particle Ĥ = P̂ 2/2m in the p-representation is ψ̃(p) = Aeipx/~, and its energy eigenvalue is E = p2/2m. Show how to normalize it and indicate how to find the normalization constant A = 1/ √ 2π~. 4. (10 pts) Show for system prepared in an eigenstates of the Hamiltonian, |E〉, the Probability of a measurement of any eigenstate of any observableΩ, P (ω), is time independent. 5. (10 pts) A property Ω is known from Classical physics to have a dependence on momentum and position as follows, ω = x2p. ‘Quantize‘ this operators, that is, suggest an appropriate operator (with P̂ and X̂) that describes this property in quantum mechanics and show that your operator is Hermitian. Of course, we already know that P̂ and X̂ are Hermitian. 6. Consider particle in square well potential centered about zero. (V (x) = 0 for |x| ≤ a and V (x) = V0 for |x| ≥ a). Bound states energies < V0. We solve and find even solutions satisfy transcendental equation κa = ka tan ka and the odd solutions satisfying κa = −ka cot ka. The k and κ are related to the wave numbers inside and outside the well, (k2 = 2mE/~2 and κ2 = 2m(V0−E)/~2) and k2a2 + κ2a2 = 2ma2V0/~2. Graphical solutions are shown in Figure for an electron m = 0.511meV/c2, in a box of length 2a = 1 nm, and depth V0 = 3eV. (a) (10 pts) How many solutions are obtained for these particular conditions? (b) (10 pts) Choose ONE of the solutions and circle it and state whether your solution is an even or odd state. Determine the Energy in eV. (i.e., leave in form that even a non-physicist could plug your result and get the answer in eV). (c) (10 pts) If the size of the well were increased from 1 nm to √ 2 = 1.41 nm, indicate roughly all changes on the graph that would be necessary to solve this problem, and discuss whether more or fewer bound states would be obtained, and whether those affected would be even or odd. Choose ONE of the following problems: Problem 7, 8, 9, or 10 7. A simple change to a free particle is to insert a potential step. Let V (x) = 0 for x < 0 and V (x) = V0 for x > 0. (otherwise expressed as V (x) = V0θ(x) with θ(x) as the usual Heaviside step function). Denote x < 0 as region I, and x > 0 as region II. There are two regions of space, and two ranges of energy to consider (0 < E < V0 and E > V0). Phys 560 in class Midterm (Continued) October 15, 2007 (a) (5 pts) Show that the the wave function ψI(x) = A exp(ipx/~) +B exp(−ipx/~) ψII(x) = C exp(iqx/~) describes a particle with energy E > V0, provided that certain conditions hold on p, q, A, B, C. State those conditions (it is not necessary to solve this completely but set up as far as possible) (b) (5 pts) Calculate probability current density in region I and II and discuss. (c) (10 pts) Now consider solutions when E < V0. Show that wavefunctions ψI(x) = A exp(ipx/~) +B exp(−ipx/~) ψII(x) = C exp(−qx/~) describes particle of energy E, provided certain conditions hold on A, B, C, p, and q. State those conditions. (d) (10 pts) What is the probability density current for region I and II and discuss. 8. Consider the Hermitian matrices M1, M2, M3, M4, that obey MiMj +MjMi = 2δijI (a) (10 pts) Show that the eigenvalues of Mi are ±1 (Hint, go to the eigenbasis of Mi and use the equations for i = j.) (b) (10 pts) By considering the relation MiMj = −MjMi for i 6= j, show that the M i are traceless. (Hint: Tr(ACB) = Tr(CBA)). (c) (10 pts) Show that they cannot be odd-dimensioned matrices. 9. A particle moves in a potential described by an attractive delta function at the origin. In following, if necessary to refer to different regions, please use notation that I refers to the region of the negative x-axis, and II to the positive x axis. The potential is; V (X̂) = −Wδ(X̂) (a) (5 pts) What are the x-representation of the Schrodinger equation valid in each region? (b) (5 pts) What are the general conditions on the energy eigenvalues for bound states in this system (i.e., E > 0,E = 0,E < 0,E =pink, blue, π?) (c) (5 pts) Set up problem to find the solutions of the bound state, that is, derive the general form of the (unormalized) valid wavefunctions that describe bound states. (d) (15 pts) Continue the solution (applying all the necessary boundary conditions and nor- malizing) to its completion to get the ψ(x) show there is only one bound solution (and give its energy). 10. The matrix elements of the operator K̂ in position representation are〈x|K̂|x′〉 = −iδ′(x− x′). This operator the same as the P̂ operator within a factor of ~. (a) (15 pts) Show that K̂ is Hermitian and what requirements this implies for the linear space of functions f(x) under which it is Hermitian. Page 2