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### ComplexNbrReview

Course: PHYSICS 3220, Fall 2008
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PHYS 3220Concept Tests Fall 2009In classical mechanics, given the state (i.e. position and velocity) of a particle at a certain time instant, the state of the particle at a later time . A) cannot be determined B) is known more or less C) is uniquely det
Quantum I (PHYS 3220)concept questionsClicker IntroDo you have an iClicker? (Set your frequency to CB and vote.)A) Yes B) No2Have you looked at the web lecture notes for this class, before now?A) Yes B) No3Intro to Quantum MechanicsIn Classical
Quantum I (PHYS 3220)concept questionsSchrdinger EquationConsider the eigenvalue equationd [ f ( x)] = C f ( x) 2 dxHow many of the following give an eigenfunction and corresponding eigenvalue? I. f(x) = sin(kx), C = +k2 II. f(x) = exp(-x), C = +1 II
Quantum I (PHYS 3220)concept questionsPhys3220, Michael Dubson U.Colorado at BoulderOperators,A wavefunction (x) has been expressed as a sum of energy eigenfunctions (un(x)'s): )= c u(x (x n ) nnCompared to the original (x), the set of numbers cfw_
Quantum I (PHYS 3220)concept questionsPhys3220, U.Colorado at3-DConsider a particle in 3D. Is there a state where the result of position in the y-direction and momentum in the z-direction can both be predicted with 100% accuracy?A) Yes, every state B
Fall 2008 3220 Tutorial Schedule 1 - Aug. 29: &quot;Classical Probability&quot; (U. Wash) 2 - Sep. 05: &quot;Wave Functions and Probability&quot; (Goldhaber &amp; Pollock) 3 - Sep. 12: &quot;Relating Classical and Quantum Mechanics&quot; (U. Wash) 4 - Sep. 19: &quot;Time Dependence in Quantum
Texas Tech - NRM - 2307
1 Summary Biodiversity is the variability Levels of biodiversity Description We stillamong living organisms at all levels of organization. Genetic, species, ecosystemof diversity in space Alpha (a), beta (b), and gamma (g)do not know Earth's biodive
PHYS 3221 Spring 2009 Tutorial Schedule1/15/2009 Tutorial #1 &quot;Classical Probability&quot; (Wash U) 1/22/2009 Tutorial #2 &quot;Wave Functions and Probability&quot; (S2) 1/29/2009 Tutorial #3 &quot;Relating Classical and Quantum Mechanics&quot; (Wash U) 2/05/2009 Tutorial #4 &quot;Tim
Faculty Disagreement about the Teaching of Quantum MechanicsMichael Dubson1, Steve Goldhaber1,2, Steven Pollock1, and Katherine Perkins1,21Department of Physics, UCB 390, University of Colorado at Boulder, Boulder CO 80309 2 Science Education Initiativ
Energy and the Art of Sketching Wave Functions I: Sketching wave functions A. Review: The figure to the right shows an infinite square well potential (V = 0 from -L/2 to L/2 and is infinite everywhere else). 1. Write down the formula for the energies of t
PHYS 3320Week 6Tutorial Energy and the art of sketching wave functionsGoals this week: 1. Developing intuition about the curvature and general behaviours of wave functions for bound states. (LG: Math/phys connection, sketching, checking) 2. Classical l
University of Colorado, Department of Physics PHYS3220, Fall 09, Some final review problems 1. At time t=0, a particle is represented by the wave function: A x , if 0 x a a b-x (x, t = 0) = A b-a , if a x b 0, else where a and b are constants. At which x
Formula Sheet for Exam 1 (These formulas will be given.)(I will not give the de Broglie relations, or the definitions of the wave number k and the angular frequency . I expect you to memorize those.)2The classical wave equation:f (x, t) 1 = 2 2 x v2
Formula Sheet for Exam 1 (These formulas will be given.)(I will not give the de Broglie relations, or the definitions of the wave number k and the angular frequency . I expect you to memorize those.) The classical wave equation: 2 f (x, t ) 1 2 f ( x, t
Formula Sheet for Exam 2 (These formulas will be given.)2The classical wave equation:f (x, t) 1 = 2 2 x v2f (x, t) t2The time-dependent Schrdinger Equation:ih h2 = - t 2m2 x2+ V(x) The standard deviation = ^ Momentum operator: p x =( x )2=(
Formula Sheet for Exam 2 (These formulas will be given.) 2 f (x, t) 1 2 f (x, t) = 2 The classical wave equation: x2 v t2The time-dependent Schrdinger Equation:i2 2 = - t 2 m x2+ V(x) The standard deviation =^ Momentum operator: p x =( x ) i x2
City College of San Francisco - CE - ce162
Load Dead Load Concrete Weight: (Density taken from.) (Im not so sure about this one, Lam calculated it but it actually equals 312.5 lbs, not 3750 lbs) Side Beam: (Density taken from.) Wall: (Density taken from.) Tiles: (Load taken from.) Mechanical Duct:
1 of 6Formula Sheet for Final Exam (These formulas will be given.)2The classical wave equation:f (x, t) 1 = 2 2 x v2f (x, t) t2The time-dependent Schrdinger Equation:ih h2 = - t 2m2 x2+ V(x) The standard deviation = ^ Momentum operator: p x =
1 of 5Formula Sheet for Final Exam (These formulas will be given.) 2 f (x, t) 1 2 f (x, t) The classical wave equation: = 2 x2 v t2 The time-dependent Schrdinger Equation:2 2 i = - t 2 m x2+ V(x) The standard deviation =^ Momentum operator: p x =(
FREE PARTICLES I: One-dimensional wave functions A. Free particle with momentum p:11. Write down the de Broglie relations relating momentum (p) to wavelength () and energy to frequency. Momentum is a vector but wavelength is not. How do you reconcile th
Free Particle PretestName: _ CU ID: _ For each of the questions below, you are given a plot of a wave function and are asked if this is a physically possible wave function for a particle confined to a one-dimensional infinite square as well as shown at r
OutlineAckQM IssuesTransLGFacQMATDevResultsSP1SignupSP2ReferencesWhat are they learning in quantum mechanics? A conceptual post test for Quantum ISteve Goldhaber , Steven Pollock , Mike Dubson , Paul Beale and Katherine Perkins Physics Scie
Transforming Upper-Division Quantum Mechanics Learning Goals and AssessmentSteve, GoldhaberSteven, PollockMike, DubsonPauland BealeKatherine PerkinsPhysics Dept., University of Colorado, Boulder, CO (per.colorado.edu),The Science Education Ini
Transforming Upper-Division Quantum Mechanics: Learning Goals and AssessmentSteve Goldhaber, Steven Pollock, Mike Dubson, Paul Beale and Katherine PerkinsDepartment of Physics, University of Colorado, Boulder, Colorado 80309, USAAbstract. In order to h
PHYS 3220 Fall 2008 Observations from Homework Help SessionsThese notes are completely raw and unprocessed. It is unlikely they will be of much use, unless you are working on modifying or developing problems and want some sense of where our students stru
PHYS 3220, Fall 09, Homework #1due 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 &quot;correctness of final
3220 HW#1Due start of second class (yikes, that's fast!) Wed Aug 271On all homeworks 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 &quot;correctness of final answer&quot;! This f
PHYS 3220, Fall 09, Homework #1due 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 &quot;correctness of final
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
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
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
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
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 &lt; px &gt; V = - dt x (1)where the potential V is a real quantity. (This theorem te
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
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
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
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 &quot;wave packets&quot;, meaning we start with an initial Gaussian 2 wave function, (x,t =
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(-
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 &lt; 0 for 0 &lt; x &lt; a for a &lt; x &lt; 2a for 2a &lt; xA) The potential energy V(x) for a particle is giv
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 &lt; p &gt; should be a real value. However, the comple
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 &lt; 0, 0 &lt; x &lt;
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 (&quot;C&quot; and &quot;O&quot; he
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
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
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&lt;0 0&lt;x&lt;a a&lt;xthe transmission coefficient for
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
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) =
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)