# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

6 Pages

### 3220_PS5

Course: PHYSICS 3220, Fall 2008
School: Colorado
Rating:

Word Count: 1969

#### Document Preview

3220 Physics Quantum Mechanics 1 Fall 2008 Problem Set #5 Due 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 through them. a) Given the definitions 1 (-ip + mx) a+ 2 m h 1 a- (ip + mx) , 2 m h (1) ^ demonstrate that the simple harmonic oscillator Hamiltonian H =...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Colorado >> Colorado >> PHYSICS 3220

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
3220 Physics Quantum Mechanics 1 Fall 2008 Problem Set #5 Due 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 through them. a) Given the definitions 1 (-ip + mx) a+ 2 m h 1 a- (ip + mx) , 2 m h (1) ^ demonstrate that the simple harmonic oscillator Hamiltonian H = -( 2 /2m)( 2 /x2 ) + h 2 2 (1/2)m x can be written in two ways, 1 1 ^ H = h a+ a- + = h a- a+ - 2 2 . (2) b) Use the energy formula En = h(n + 1/2) along with part a) to demonstrate that a- a+ un = (n + 1)un , a+ a- un = nun , (3) where un (x) are the normalized stationary state wavefunctions. c) Use (??) and integration by parts to demonstrate that - f (a g) dx = - (a f ) g dx , (4) for any functions f (x), g(x) that go to zero at infinity, f () = g() = 0; for example, they could be normalizable wavefunctions. (As we will discuss more later, this implies that a+ and a- are adjoints or Hermitian conjugates of each other.) d) The raising and lowering operators must take one stationary state to the next, times an overall constant: a+ un = cn un+1 , a- un = dn un-1 , (5) where cn and dn are constants to be determined. Consider the expression - (a+ un ) (a+ un )dx. Evaluate it two ways, one of them using the results from parts b) and c), to solve for cn . Now consider - (a- un ) (a- un )dx and do something similar to solve for dn . You should find: a+ un = n + 1un+1 , a- un = nun-1 . (6) 1 By using the formulas demonstrated in this problem, you will be able to evaluate a lot of things about the SHO without ever having to "get your hands dirty" with the explicit forms of the wavefunctions. You will see this in the next couple problems. Problem 5.2: Expectation values in the SHO. (20 points) a) Find an expression for the operators x and p in terms of the raising and lowering operators ^ ^ a+ and a- as well as constants. b) Calculate x , p , x2 and p2 in the nth stationary state using the expressions from part a). Hint: You don't ever need to write out the functional form of the un if you use results from the previous problem. c) How must H be related to the expectation values you calculated in the previous part? Check that this relationship works given what you know H must be for a stationary state. How much do the kinetic and potential energies each contribute to the total expectation value H ? d) Calculate the product of uncertainties x p for the nth stationary state and verify the Heisenberg Uncertainty Principle x p h/2. For which values of n is the minimum possible uncertainty achieved? Besides learning some basic properties of the stationary states, here you get to practice with the algebraic method of using operators. These sorts of techniques will come up again when we study angular momentum in three dimensions. Problem 5.3: Coherent states. (20 points) In this problem, consider a wavefunction (x) in the simple harmonic oscillator defined as (x) = A n un (x) , n! n=0 (7) where is a complex number, and we assume A is chosen so as to normalize the wavefunction; don't worry about the value of A. This coherent state wavefunction has a lot of applications in optics and atomic physics, and we will explore some of its properties. a) Demonstrate that a coherent state is an eigenvector of the lowering operator with eigenvalue : a- (x) = (x) . (8) b) Calculate the expectation values x and p . As in the last problem, you should be able to do this without writing down the exact form of the wavefunction! Equation (??) will help. 2 c) Let (x, t) be the wavefunction at all times, so (x, t = 0) = (x). Demonstrate that (x, t) is also an eigenvector of a- , with eigenvalue (t); what is (t) in terms of and other quantities? d) In this part, for simplicity assume is real (but (t) might not be real). Take the results for x and p from part b) and put the value of (t) into them to find x (t) and p (t) for (x, t). How does the result compare to the classical motion of a particle in a simple harmonic oscillator? Lasers and Bose-Einstein condensates are examples of coherent states. Another property of the coherent state is that it minimzes the uncertainty, for all values of . This can be seen by calculating x2 and p2 -- if you want extra practice, take a look! Problem 5.4: Analytic solution of simple harmonic oscillator. (20 points) There are two ways to solve the harmonic oscillator. In class we pursued the method using raising and lowering operators, which is more powerful but difficult to guess if you didn't already know about it. In this problem we go through the method of solving the differential equation directly. Recall that the TISE for the simple harmonic oscillator is h2 d2 u 1 - + m 2 x2 u = Eu . 2 2m dx 2 (9) a) It is convenient to simplify the problem by introducing a dimensionless variable. To define x/ with dimensionless, we must construct x with units of length. What are the units x of the constants h, m and ? What is the unique combination (without any extra pure numbers) that gives a unit of length? Define this as x and switch variables to x/. x Show that the equation can be written d2 u = ( 2 - K)u , d 2 where K 2E/ contains the energy E. What are the units of h/2? h b) Consider the limit of the equation far from the center, |x| . Explain why 2 this limit, leading to the approximate equation d2 u 2u . 2 d Show that in this limit, an approximate solution is u Ae- 2 /2 (10) K in (11) + Be 3 2 /2 . (12) Be sure to explain any terms you neglect. We have to contstrain one of A and B to make sure that u can be normalized; what is this constraint? Explain why. (Don't actually try to normalize function.) the c) The original equation turns out to simplify if we "extract" the asymptotic (large |x|) behavior of u and solve for what's left. Accordingly, define h() by means of u() h()e- 2 /2 . (13) Substitute this back into the full TISE (not just the asymptotic version) and show that it is equivalent to h () - 2h () + (K - 1)h() = 0 . d) To solve this differential equation, postulate a series solution for h: (14) h() = j=0 aj j , (15) where the aj are constants. Show that the result of part c) implies the recursion relation for the constants: aj+2 = (2j + 1 - K) aj . (j + 1)(j + 2) (16) If we had tried to do this for u directly, we would have obtained a more complicated recursion relation. e) Normalizability of the wavefunction implies that the series (??) can't go on forever. Thus to ensure we can normalize our wavefunction, the series must stop at some point. Assume that there exists a value n such that an = 0 but an+2 = 0, and find the resulting constraint on the energy En associated with this wavefunction. The SHO wavefunctions are thus of the form un = (nth order polynomial in ) e- = (nth order polynomial in x) e 2 /2 , , (17) (18) -mx2 /2 h where the polynomials are determined by the recursion relation (??) and are called (up to an overall constant) Hermite polynomials. This agrees with the results dervied using raising and lowering operators. These sorts of methods are common in solving differential equations, and also show up in solving the radial part of the hydrogen wavefunction. 4 Problem 5.5: Finding quantized energies numerically. (20 points) Only certain values of the energy E lead to normalizable solutions of the TISE for the simple harmonic oscillator. If a different E is picked, solutions exist, but they cannot be normalized they blow up at large |x|. They can blow up either to positive or negative infinity; the allowed values of E "thread the needle" in between blowing up to plus infinity or to minus infinity by going to zero instead. One can thus numerically determine the allowed energies for the simple harmonic oscillator (or a different potential) by guessing values of E and trying to find solutions that don't blow up at large |x|, tweaking your guess of E each time to get closer. Even if you don't guess the exact value of E, if you can find the cross-over region where the wavefunction switches from blowing up to positive infinity to blowing up to negative infinity, the allowed value of E must be in between. a) Consider the version of the simple harmonic oscillator TISE derived in the previous problem, 2u = ( 2 - K)u , 2 (19) where K = 2E/ contains the energy and x/ is the rescaled position variable. h x Numerically solve and graph this equation for various values of K to find the ground state energy and plot the results. Use the boundary conditions: u(0) = 1, u (0) = 0. What property of the ground state are we assuming in order to set the first derivative equal to zero at the origin? (The choice u(0) = 1 is for convenience; we won't worry about actually normalizing the answer.) If you use Mathematica, some useful code might be K = 1.1; a = 0; b = 10; c = -10; d= 10; Plot[ Evaluate[u[z] /. NDSolve[{u''[z] - (z2 - K)*u[z] == 0, u[0] == 1, u'[0] == 0}, u[z], {z, .000000001, 10}, MaxSteps -> 10000]], {z, a, b}, PlotRange -> {c, d}, PlotStyle -> Thick] Here we used z instead of for convenience. (a, b) and (c, d) are the ranges on the two axes of the plot; feel free to tweak these to get a better view. We already know the answer: K = 1. Try values like K = 1.1 and K = 0.9 and see how the asymptotic (large-|x|) behavior flips between these values; by tuning the guesses for K one can get closer and closer to the correct value. In your write-up, give plots for the result for one choice just above K = 1 and one choice just below K = 1. (Note that by numerically solving, you are only working to a finite precision, and so even K = 1 may seem to blow up; this is because not enough significant digits are being kept. This is okay because we can still see that the asymptotic behavior flips right around K = 1, and hence the allowed energy is there to within the precision of the solution.) b) Now consider the second-lowest energy state of the SHO, the so-called "first-excited state". How must you change the boundary conditions in the numerical evaluation to find this state? 5 What value of K corresponds with this state? Check that you've chosen the right boundary conditions by doing the numerical solution and seeing that the asymptotic behavior flips at this value of K. (You don't need to include the plots in your write-up.) c) Now let's do something new: consider a different system, with potential energy 1 V (x) = x4 . 2 (20) What are the units of ? We would like to write the TISE for this potential with a dimensionless variable; show that the unique combination of , m and h that gives a unit of length is of the form x= h2 m , (21) and determine the power . Now define x/ as before and rewrite the TISE as x 2u = ( 4 - K)u , 2 where K will be different from the SHO case, of the form K = E/E; what is E? d) Now find to three significant digits the values of K and the corresponding energies of both the ground state and first excited state of this system using the numerical method, assuming that = 1 in the appropriate MKS units. Don't forget to think about boundary conditions as in the SHO case. This method helps make explicit the mechanics of why only certain energies are allowed in bound state problems in quantum systems. The method gets very useful for cases where the potential is too complicated to solve analytically, as V = x4 /2 begins to show. (22) 6
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

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
BYU - ID - CIT - CIT140
Bent CyclingAuthor Date Purpose Kory Johnson 3/20/2008 To analyze different investment and loan scenariosBent CyclingSavings Proposals02/27/2012 Prepared: 11/7/2009Bent CyclingInvestment Analysis Initial Investment (PV) (\$400,000) (\$400,000) (\$400,0
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
BYU - ID - CIT - CIT140
Bent CyclingAuthor Date Purpose Kory Johnson 3/21/2008 To project Bent Cycling's income statements for the next five fiscal yearsBent CyclingProjected Five-Year Income Statement02/27/2012 Prepared: 11/7/2009Bent CyclingProjected Five-Year Income Sta
Colorado - PHYSICS - 3220
Angular Momentum and Spin I: Hydrogen atoms, angular momenta, and probabilities Ignoring spin (for now), an electron is known to be in a hydrogen atom state given by (t = 0) = 1 R10 Y00 + 6 1 R21 Y11 + cR32 Y21 61A. Pick a value of c which normalizes th
Colorado - PHYSICS - 3220
PHYS 3220Week 11Tutorial - Spin and Angular MomentumGoals this week: 1. Developing intuition about measurement, bra-ket notation, and finite dimensional systems. (LG Math/physics connection, interpretation, building on earlier work/coherence of the cla
BYU - ID - CIT - CIT140
Bent CyclingAuthor Date Purpose To estimate the return of an investment in current dollarsBent CyclingDepreciation of Assets02/27/2012Bent CyclingReturn on the Building ProjectPrepared: 11/7/2009Yearly ReturnInitial Investment (\$1,200,000) Cumula
Colorado - PHYSICS - 3220
Angular Momentum and Spin PretestName: _ CU ID: _ In the following questions, we will use quantum states made up of the hydrogen energy eigenstates:(, , ) Rr 1(, ) ( m n r l m =n ) 1 YThe energy of one of these states is:E = n E 1 n2where E1 is the e
BYU - ID - CIT - CIT140
Global Travel Author Date Kory Johnson 2/28/2008Purpose New Mexico theme park ticket salesNew Mexico Fiscal Year- 2010Global Travel Theme Park Ticket Sales-Total# Tickets Sold Sales (\$) Adults Children Adults Children Animal World #REF! #REF! #REF! #R
Colorado - PHYSICS - 3220
Modern Physics-1A Brief History of Modern Physics and the development of the Schrdinger Equation&quot;Modern&quot; physics means physics discovered after 1900; i.e. twentieth-century physics. 1900: Max Planck (German) tried to explain blackbody radiation using Ma
BYU - ID - CIT - CIT140
LaFouch Museum Created By Date Purpose Track art objects in LaFouch MuseumSum - Appraised Value Location East PavilionCourtyardGardenSouth PavilionWest PavilionTotal ResultCondition Excellent Good Fair Total Result # \$75,769 \$44,350 Painting # \$41,
Colorado - PHYSICS - 3220
Modern Physics-1A Brief History of Modern Physics and the development of the Schrdinger Equation&quot;Modern&quot; physics means physics discovered after 1900; i.e. twentieth-century physics. 1900: Max Planck (German) tried to explain blackbody radiation using Ma
BYU - ID - CIT - CIT140
Bent CyclingAuthor Date Purpose To analyze different investment and loan scenariosBent CyclingSavings Proposals02/27/2012 Prepared: 11/7/2009Bent CyclingInvestment Analysis Initial Investment (PV) Investment Goal (FV) Annual Rate Months per Year Rat
Colorado - PHYSICS - 3220
Ch. 1 notes, part11 of 41Quantum MechanicsIntroductory Remarks: Q.M. is a new (and absolutely necessary) way of predicting the behavior of microscopic objects. It is based on several radical, and generally also counter-intuitive, ideas: 1) Many aspects
BYU - ID - CIT - CIT140
Eugene Community TheatreAuthor: Created By: Purpose: Prepare invoices for theatre patronsInvoice Data Subscriber Michael Keller Address 1234 Main Street City State Zip Eugene, OR 70777 Phone (806)555-1111 Ticket Quantity 3 Series B Location Orchestra No
Colorado - PHYSICS - 3220
SJP QM 3220 Ch. 2, part 1Once again, the Schrdinger equation:Page 1 ( x, t ) 2 2 ( x, t ) i =- + V ( x, t ) t 2m 2 x (which can also be written (x,t) if you like.) And once again, assume V = V(x) (no t in there!)We can start to solve the PDE by SEPAR
BYU - ID - CIT - CIT140
New Mexico Fiscal Year 2011Global Travel Theme Park Ticket Sales-Total# Tickets Sold Sales (\$) Adults Children Adults Children Animal World Err:522 Err:522 Err:522 Err:522 Global Workplaces Florida Err:522 Err:522 Err:522 Err:522 Great Adventure Err:522
Colorado - PHYSICS - 3220
SJP QM 3220 Ch. 2, part 1Once again, the Schrdinger equation:Page 1(which can also be written (x,t) if you like.) And once again, assume V = V(x) (no t in there!)We can start to solve the PDE by SEPARATION OF VARIABLES. Assume (hope? wonder if?) we mi
BYU - ID - CIT - CIT140
Global Travel Author Date Purpose Kory Johnson 3/2/2008Utah theme park ticket salesUtah Fiscal Year - 2010Global Travel Theme Park Ticket Sales-Total Sales (\$) Adults Children #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF! #REF
Colorado - PHYSICS - 3220
BYU - ID - CIT - CIT140
New Century FundAuthor Date PurposeKory Johnson 2/25/2008 To report on the performance and financial details of the New Century mutual fundThe New Century FundNew Century Fund ReportThe New Century FundSummary Report (as of 12-31-09)50,000kGrowt h
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
SJP QM 3220 Formalism 1The Formalism of Quantum Mechanics: Our story so far . State of physical system: normalizable ( x, t ) h ,H Observables: operators x , p = i x Y = HY Dynamics of : TDSE ih t To solve, 1st solve TISE: = Ey Hy Solutions are stationa
Colorado - PHYSICS - 3220
SJP QM 3220 Formalism 1The Formalism of Quantum Mechanics: Our story so far . State of physical system: normalizable ( x, t ) ^ ^ Observables: operators x , p = ^ ,H i xDynamics of : TDSE i ^ = H t^ To solve, 1st solve TISE: H = E Solutions are st
Colorado - PHYSICS - 3220
Lecture notes (these are from ny earlier version of the course we may follow these at a slightly different order, but they should still be relevant!) Physics 3220, Steve Pollock. Basic Principles of Quantum MechanicsThe first part of Griffith's Ch 3 is
Colorado - PHYSICS - 3220
Lecture notes (these are from ny earlier version of the course - we may follow these at aslightly different order, but they should still be relevant!) Physics 3220, Steve Pollock.Basic Principles of Quantum MechanicsThe first part of Griffith's Ch 3 is
Colorado - PHYSICS - 3220
SJP QM 3220 3D 1Angular Momentum (warm-up for H-atom) Classically, angular momentum defined as (for a 1-particle system) y m Lrp ^ ^ ^ x y z p = mv r = x y z x px p y pz O Note: L defined w.r.t. an origin of coords. ^ ^ L = x ( yp z - zp y ) + y ( zp x -
Colorado - PHYSICS - 3220
SJP QM 3220 3D 1AngularMomentum(warmupforHatom) Classically,angularmomentumdefinedas(fora1particlesystem) y m x O Note: definedw.r.t.anoriginofcoords. (InQM,theoperatorcorrespondingtoLxis accordingtoprescriptionofPostulate2,part3.) Classically,torquedefi
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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_
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
Colorado - PHYSICS - 3220
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
Colorado - PHYSICS - 3220
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