Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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=(

Colorado - PHYSICS - 3220

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:

Colorado - PHYSICS - 3220

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 =

Colorado - PHYSICS - 3220

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 =(

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

Colorado - PHYSICS - 3220

Colorado - PHYSICS - 3220

Colorado - PHYSICS - 3220

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

Colorado - PHYSICS - 3220

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 "correctness of final

Colorado - PHYSICS - 3220

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 "correctness of final answer"! This f

Colorado - PHYSICS - 3220

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 "correctness of final

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