Fall 2019 Homework Assignments

**Homework 1: (Due Thursday, October 10)**1) Consider the wavefunction { A (1-|x|/a) for |x| < a Ψ(x) = { { 0 elsewhere where A is the normalization constant and a is a constant with units of length. (a) Normalize the wavefunction. (b) Find the expectation values <x>, <x

^{2}>, <p>, <p^{2}> and the uncertainties Δx and Δp. What is Δx Δp ? (c) Find the probability currents at x = ± a/2. Interpret your result. 2) Repeat problem 1 with the wavefunction { A (1-|x|/a) exp(ik_{o}x) for |x| < a Ψ(x) = { { 0 elsewhere where k_{o}is a constant wave-number.**Homework 2: (Due Tuesday, October 22)**A one-dimensional quantum well has the potential { 0 for 0 < x < a V(x) = { { ∞ elsewhere. At time t=0, the wavefunction of the system is given as { A for 0 < x < a/2 Ψ(x,t=0) = { { 0 elsewhere. (1) Find the expansion coefficients of Ψ(x,t=0) in terms of the eigenfunctions u

t=0: n=15:_{n}(x) corresponding to V(x). (2) Make three plots for the series expansion for Ψ(x,t=0)/A as a function of x/a, keeping 15, 5, and 3 terms in the expansion. (3) Make three plots for |Ψ(x,t)/A|^{2}as a function of x/a with 15 terms in the expansion, at times t=0, t=0.1(ℏ/E_{1}), t=ℏ/E_{1}, and t=3(ℏ/E_{1}). (4) What time interval does ℏ/E_{1}correspond to for an electron in a 10nm wide well?**Homework 3: (Due Thursday, November 21)**Find the time-dependent expectation values of the following quantities (1) x (2) xP + Px when the initial (t=0) state of the system is (a) A ( u

_{n}+ u_{n+1}) (b) A exp(a^{+}) u_{o}with exp(a^{+}) = 1 + a^{+}+ (a^{+})^{2}/2! + (a^{+})^{3}/3! + (a^{+})^{4}/4! + . . . where u_{n}are the eigenstates of the harmonic oscillator. Use the a^{+}and a^{-}operators to solve this problem. (i.e. do NOT use the Hermite polynomials.) Determine the normalization constant A for each case. Obtain the general expression for the terms in series in part (b), but do not spend too much time to obtain a functional form for the sum. (Bonus points if you do!)**Homework 4: (Due Tuesday, December 3)**Suppose you were given the opportunity to design your potential barriers for an experiment: You can produce 1eV high barriers which can have widths which are multiples of 1nm. (i.e., you can fabricate barriers which are 1eV high and can have a width of 1nm, 2nm, 3nm, etc.) You can then treat these very thin barriers as delta-function potentials αδ(x-x

_{o}) where α can be 1eV-nm, 2eV-nm or 3eV-nm, etc. x_{o}is the position of the barrier. Construct a two-barrier potential that will "filter" a certain energy of electrons: Plot the transmission coefficient of the structure as a function of energy. Try to maximize the ratio of the first maximum of the transmission coefficient (T_{max}) to its value at the first minimum (T_{min1}) after the maximum. Give values for the positions and widths of the barriers (one of the barriers must be at x=0), T_{max}, T_{min1}and T_{max}/T_{min1}. The student who obtains the largest T_{max}/T_{min1}value will be the class champion.**Homework 5: (Due Thursday, December 19)**Evaluate the following matrix operations: (1) Show that [A,B]=0 , find the common eigenvector set. Expand V in terms of these eigenvectors. (2) Show that [A,C]≠0 . Expand V in terms of eigenvectors of C. (3) exp(A) V (4) exp(B) V (5) exp(A+B) V (6) exp(C) V (7) exp(A+C) V [ 1 0 3 ] [ 2 0 0 ] [ 1 1 0 ] [ 1 ] Use the quantities A = [ 0 2 0 ] B = [ 0 1 0 ] C= [ 1 1 0 ] V = [ 1 ] [ 3 0 1 ] [ 0 0 2 ] [ 0 0 1 ] [ 1 ]

Check the Stars system for your homework grade. In some cases, you may be able to re-submit your homework to the assistant with corrections and improve your grade.