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Course: PHYSICS 3220, Fall 2008School: Colorado
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3220, PHYS Fall 09, Homework #1 due Wed, Aug 26, at 2PM (bring in to class) On all homework assignments this term, please show your work and explain your reasoning. We try to grade for clarity of explanation as much as we do for mere "correctness of final answer". This first homework is meant to be a review of math (and a little physics). Feel free to use any notes or texts, or talk to other...
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3220, PHYS Fall 09, Homework #1
due Wed, Aug 26, at 2PM (bring in to class) On all homework assignments this term, please show your work and explain your reasoning. We try to grade for clarity of explanation as much as we do for mere "correctness of final answer". This first homework is meant to be a review of math (and a little physics). Feel free to use any notes or texts, or talk to other students, or come to our homework session (Tuesday 2-4 this week, G2B60). In the end, all your work should be your own. To help us better understand where everyone is starting On every question this week, please put a number in a circle at the end of each question: (i) - I knew this material, it was fairly trivial for me. (ii) - I knew this material, and didn't need to look up anything or get help, but it was not what I would call "trivial" for me. (iii) - I knew this material, but still needed to look something up in a book/notes/web. (iv) - I knew this material, but still needed to get help from a (live) person. (v) - I did NOT know this, and had to learn it for this homework! __________________________________________________________________________ 2 1) Consider the complex number z = -4 + 4i . What is the value of z = z * z ? Plot this number in the complex plane (y-axis = Im, x-axis = Re). If we rewrite z in the form Aei z , what is the value of A? What is the value of ? 2 3 1 0 2) Given matrices A = and B = , what is their product: A B = ? 4 5 0 4 3) Evaluate
-7
( x - 7) ( x - 5) dx , explaining clearly (use some words!) your work.
2 2
7
(Here,"" is a Dirac delta function) 4) Sketch a graph of y = 3e - ( x / a ) vs. x for a=10. (Be sure to indicate the x and y scales you choose). Does the function show any symmetries? What is the limit for x ? What is the significance of a in the exponent? 5) What is the relation between the energy, E, and the frequency, f, of a photon? What is the approximate energy of a photon of yellow light? (both in Joules and eV) 6) What is the relation between the wavelength, , and the momentum, p, of a particle? Roughly what kinetic energy should an electron have in order that its wavelength be of roughly an "atomic" distance scale? Then re-express this as the number of volts needed to accelerate that electron from rest to this energy. Would you call this a high-voltage device? 7) Write down an equation describing a sinusoidal traveling wave (in 1-D). Tell us (words and/or equations) what in your equation tells us the speed and direction of the wave? 8) What does it mean for two functions to be "orthogonal"? Then, give a specific example of two orthogonal functions.
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Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#2 due Wed, Sep 2, 2PM at start of class 1. Using the formula for four-momentum conservation (special relativity) to derive the dependence of the wavelength of the scattered radiation on t
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #2Due Wednesday, September 3 at 2pm Problem 2.1: Fun with statistics. (20 points) To do this problem, while you may use a calculator, spreadsheet or mathematical software to do simple algebraic opera
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#3 due Wed, Sep 9, 2PM at start of class 1. Probabilities and expectation (average) values (total: 10 pts) A box contains 18 small items, of various lenghts. The distribution of lengths in
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #3Due Wednesday, September 10 at 2pm Problem 3.1: Practice with complex numbers. (20 points) Every complex number z can be written in the form z = x + iy where x and y are real; we call x the real pa
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#4 due Wed, Sep 16, 2PM at start of class 1. (Griffiths, problem 1.7, 15 pts) Prove the Ehrenfest theorem d < px > V = - dt x (1)where the potential V is a real quantity. (This theorem te
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #4Due Wednesday, September 17 at 2pm Problem 4.1: Stationary state in the infinite square well. (20 points) The infinite square well has the potential V (x) = 0 , = 0 x a, otherwise , (1) (2)and the
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#5 due Wed, Sep 23, 2PM at start of class 1. A few more properties of Hermitian operators (total: 10pts) a) Show that the sum of two Hermitian operators is a Hermitian operator. ^ b) Suppo
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #5Due Wednesday, September 24 at 2pm Problem 5.1: Properties of the simple harmonic oscillator. (20 points) Some of these results are discussed in class or in the book, but it's quite useful to work
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #6 Due Wednesday, Oct 8 at 2pm Problem 6.1 Gaussian wave packets: part 1 [20 pts] Free particles are often modeled as "wave packets", meaning we start with an initial Gaussian 2 wave function, (x,t =
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#6 due Wed, Sep 30, 2PM at start of class 1. Momentum space (Total: 15 pts) In class we have defined the momentum space wave function (p, t) as 1 (p, t) = 2 where 1 (x, t) = 2(x, t) exp(-
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #7 Due Wednesday, Oct 15 at 2pm Problem 7. 1 Qualitative methods. [20 pts] 0 V 0 V (x) = V 0 /2 0 V for x < 0 for 0 < x < a for a < x < 2a for 2a < xA) The potential energy V(x) for a particle is giv
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#7 due Wed, Oct 7, 2PM at start of class 1. Momentum operator (Total: 10 pts) Since the momentum p is an observable, its expectation value < p > should be a real value. However, the comple
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Spring 2009 Problem Set #8Due Wednesday, March 11 at 9am Problem 8.1: Qualitative methods for stationary states. (20 points)a) The potential energy for a particle is given by V0 , V (x) = 0 , V0 /2 , V0 , x < 0, 0 < x <
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #8 Due Wednesday, Oct 22 at 2pm Problem 8. 1 Review with some realistic numbers [15 pts] A CO (carbon monoxide) molecule can be modeled as a vibrating spring. When you have two objects ("C" and "O" he
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#8 due Wed, Oct 15, 2PM at start of class 1. Minimum energy (Griffiths, Problem 2.2, Total: 10 pts) Show that E must exceed the minimum value of V (x), for every normalizable solution to t
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #9 Due Wednesday, Oct 29 at 2pm 9. 1 Commutators [15 pts] Prove the following commutator identities: A) [A+B,C] = [A,C] + [B,C] B) [AB, C] = A[B,C] + [A,C]B C) [f (x), p] = ih df , where x and p are t
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#9 due Wed, Oct 22, 2PM at start of class 1. Potential barrier (Total: 20 pts) For the potential with a barrier of height V0 0, V (x) = V0 0,x<0 0<x<a a<xthe transmission coefficient for
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #10 Due Wednesday, Nov 5 at 2pm Problem 10. 1 Generalized Uncertainty [15 pts] A) If operators and B are Hermitian, what must be true about the constant in order to ensure that A, B is also a Hermitia
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#10 due Wed, Oct 28, 2PM at start of class 1. Symmetric potentials (Griffiths, Problem 2.1(c), Total: 10 pts) Prove the following statement: If V (x) is an even function (that is, V (-x) =
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Spring 2009 Problem Set #11Due Wednesday, April 15 at 9am Problem 11.1: Modeling molecules: the Quantum Rigid Rotor. (20 points) Simple diatomic molecules can be modeled as two particles of mass m (representing the atoms)
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #11 Due Wednesday, Nov 19 at 2pm Problem 11. 1 Modeling molecules: a quantum rigid rotator [15 pts] Simple molecules can be modeled as two particles of mass m, attached to the ends of a massless rod o
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#11 due Wed, Nov 4, 2PM at start of class 1. Analytic solution of the harmonic oscillator (Total: 20 pts) In this problem we go through the analytic solution of the time-independent Schrdi
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#12 due Wed, Nov 18, 2PM at start of class 1. Vectors (Total: 20 pts) ^ a) (Griffiths, Problem A.1) Consider the ordinary vectors in 3D (ax^ + ay ^ + az k), with i j complex components. Fo
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #12Due Wednesday, December 3 at 2pm Problem 12.1: Analytic solution of radial equation for hydrogen. (20 points) Stationary states for the hydrogen atom that are also eigenstates of L2 and Lz were fo
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#13 due Wed, Dec 2, 2PM at start of class 1. Matrix representation of the eigenvalue problem (Total: 20 pts) ^ Suppose there are two observables A and B with corresponding Hermitian operat
Colorado - PHYSICS - 3220
Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #13Due Wednesday, December 10 at 2pm Problem 13.1: Surveys! (20 points) Please take the following surveys. You will not be graded for accuracy for these surveys, you get credit just for participating
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, HW#14 due Wed, Dec 9, 2PM at start of class 1. Survey (Total: 20 pts) Please take the following survey under http:/www.colorado.edu/sei/surveys/Fall09/Clicker Phys3220 fa09-post.html Note: C
Colorado - PHYSICS - 3220
Brief guide to instructors TUTORIALS IN UPPER DIVISIONSteven Pollock, Spring 2009 Please feel free to add comments and improve this document. Week by week materials available in the "Tutorials" folder at http:/www.colorado.edu/sei/departments/physics_331
Colorado - PHYSICS - 3220
Midterm 1 Review PretestName: _ CU ID: _The first two questions refer to the x ( 0, normalized wave function, ,t= ) which is shown at the right. Q1: a) what, if anything, can we say about <x>?A. It must be zero B. It must be positive C. It depends on t
Colorado - PHYSICS - 3220
Quantum Mechanics I, Review Problems for Midterm 2 I: Scattering A. Half-Infinite Square Well Consider the potential shown at the right where the potential is infinite for x < 0, zero in the region, 0 x < L and V0 in the region x L. 1. Are there any bound
Colorado - PHYSICS - 3220
Midterm2 Review PretestName: _ CU ID: _ Consider a quantum mechanical system with the potential shown at the right. The system is prepared so that at t=0, it is in a state described by a wave function identical to that of the ground state of the infinite
Colorado - PHYSICS - 3220
F-23 Notice that in 1D problems, like the 1D infinite well or the 1D SHO, we only needed one number (n) to uniquely specify an eigenstate. This state label is called a quantum number or q-number and it is always in a 1-to-1 correspondence with the eigenva
Colorado - PHYSICS - 3220
1 of 1Quantum Mechanics is fundamentally a probabilistic theory. This indeterminacy was deeply disturbing to some of the founders of quantum mechanics. Einstein and Schodinger were never happy with this postulate. Einstein was particularly unhappy and ne
Colorado - PHYSICS - 3220
Course Scale Learning GoalsLearning Goals Phys 3220 Quantum IPhys 3220 introduces students to the formal theory of quantum mechanics. Most of the class focuses on problems in one dimension although the class also covers problems such as the hydrogen ato
Colorado - PHYSICS - 3220
1 of 6Quantum MechanicsIntroductory Remarks Humans have divided physics into a few artificial categories, called theories, such as classical mechanics (non-relativistic and relativistic) electricity & magnetism (classical version) quantum mechanics (non
Colorado - PHYSICS - 3220
1 of 3 Statistics and the Wavefunction Let's review some elementary statistics about random variables that can assume discrete values. Suppose we make many repeated measurements of a random discrete variable called x. An example of x is the mass, rounded
Colorado - PHYSICS - 3220
1 of 3 Statistics and the Wavefunction Let's review some elementary statistics about random variables that can assume discrete values. Suppose we make many repeated measurements of a random discrete variable called x. An example of x is the mass, rounded
Colorado - PHYSICS - 3220
PHYS 3220Week 9Tutorial - Quantum Operator MethodsGoals this week: 1. Developing intuition about measurement in a finite dimensional systems. (LG Math/physics connection, interpretation, building on earlier work/coherence of the class) 2. Practice with
Colorado - PHYSICS - 3220
Quantum operator methods QM (quantum mouse) versionp. 1SETUP: Consider a quantum object (a "quantum lab mouse") and some new properties we can measure. E.g., suppose "quantum weight", W, is a hermitian operator. The corresponding physical measurement is
Colorado - PHYSICS - 3220
Quantum operator methodsp. 1SETUP: Consider a quantum particle with some new properties that we can measure. We won't talk much about the physics of these measurements yet, but the formalism of quantum mechanics will teach us a great deal, just from ope
Colorado - PHYSICS - 3220
Quantum operator methodsp. 1SETUP: Consider a quantum particle with some new properties that we can measure. We won't talk much about the physics of these measurements yet, but the formalism of quantum mechanics will teach us a great deal, just from ope
Colorado - PHYSICS - 3220
Quantum Operator Methods PretestName: _ CU ID: _ For the questions below, we will consider measurements on two observables, A and B, in ^ ^ a two-state system. Observable A has an associated operator A . A has two eigenvalues 1 and 2 and eigenfunctions a
Colorado - PHYSICS - 3220
(Name optional: _) Physics 3220 "QUIZ" (Not for credit, just to learn from!) Start it on your own. Consider quantum states with definite angular momentum quantum number l=1. There are just 3 basis states: Y1 , Y1 , and Y1 , or in Dirac notation, call them
Colorado - PHYSICS - 3220
(Name not required) Physics 3220 "QUIZ" (Not for credit, just to learn from!) To start, this will be CLOSED BOOK, CLOSED NOTES, CLOSED MOUTHS. (This will change in a few minutes! But try it first on your own.) A) In the space below, write a brief summary
Colorado - PHYSICS - 3220
In-class ActivitiesThese activities that we used in class to support relevant quantum concepts. The "quiz" files were done as activities at the start of class - sort of like a whiteboard, but they first worked about 5 minutes on their own, then another 5
Colorado - PHYSICS - 3220
Lecture Notes for PHYS 3220Quantum Mechanics lecture notes, made available to the students. Notes are by Steven Pollock (SJP) or Michael Dubson (MD). There are older versions (of scanned, handwritten) notes in a subfolder There are probably many typos, p
Colorado - PHYSICS - 3220
Concept tests, organized by Griffiths Chapter numbers.The concept test collections in this folder were largely constructed by Michael Dubson in Spring 2008, and then modified and added to by Steven Pollock and Oliver DeWolfe in Fall 2008, with some techn
Colorado - PHYSICS - 3220
Quantum I Homework AssignmentsThe "Homework assignments" folder has homework sets built by Mike Dubson, Steve Pollock, and Oliver DeWolfe. Some of them are basically Griffiths' problems, slightly disguised, or tweaked. I tried to add elements of explanat
Colorado - PHYSICS - 3220
Learning Goals for Quantum IThese learning goals were created by a working group of faculty both those in physics education research and those with other areas of research. This list represents what we want students to be able to do at the end of the cou
Colorado - PHYSICS - 3220
PublicationsPapers and Posters: Steve Goldhaber, Steven Pollock, Mike Dubson, Paul Beale and Katherine Perkins, "Transforming Upper-Division Quantum Mechanics: Learning Goals and Assessment", Physics Education Research Conference Proceedings (2009), AIP,
Colorado - PHYSICS - 3220
Student Learning DifficultiesThis directory contains documented student difficulties. For more information, contact: steven.goldhaber@colorado.edu PHYS 3220 Student Learning Difficulties (MS Word) Back to Introduction
Colorado - PHYSICS - 3220
Quantum I TutorialsOver the last four semesters, we have used several different tutorials from the University of Washington. Please contact Peter Shaffer (shaffer@phys.washington.edu) if you are interested in those materials. They are still under develop
Colorado - PHYSICS - 3220
University of Colorado, Department of Physics PHYS3220, Fall 09, Some review problems - will not be graded 1. The state of a particle in an infinite square well with V (x) = 0 for 0 < x < a and V (x) = elsewhere is given by the wave function (x, t = 0) =
Colorado - PHYSICS - 3220
Reflecting on Transmission I: An asymmetric potential well The figure at the right shows a graph of the potential energy for a one-dimensional system. V (x) = 0 for x < 0 (region I), V (x) = -V0 for 0 x < a (region II), and V (x) = +V1 for x a (region III
Colorado - PHYSICS - 3220
Scattering PretestName: _ CU ID: _For the two questions below, consider the graph of the potential energy for a one-dimensional system as shown at the right. V(x) = 0 for x<0 and V0 for x>0.Q1: Describe in words the boundary conditions for this system
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
Sketching Wave Functions PretestName: _ CU ID: _ In the two sketches below, we show the ground state wave functions of electrons trapped in two different wells. (The wells extend beyond points xa and xb, but we only show X(x) in this limited region.) The
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220