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hw_solutions_3

Course: PHY 4604, Spring 2009

School: UCF

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# 5.21 Expand the operators in the basis set of the eigenvectors of A: A= n A|n n| = n an |n n| Bnm |n m| mn B= mn |n n|B |m m| = where an are the eigenvalues of A and Bnm = n|B |m (the matrix elements ). Note that in the basis of its own eigenvectors, an operator such as A of B is diagonal, in that its o-diagonal (m = n) matrix elements are all 0. Then, noting that k |n = kn , AB = kmn ak Bnm |k k...

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Expand 5.21 the operators in the basis set of the eigenvectors of A: A= n A|n n| = n an |n n| Bnm |n m| mn B= mn |n n|B |m m| = where an are the eigenvalues of A and Bnm = n|B |m (the matrix elements ). Note that in the basis of its own eigenvectors, an operator such as A of B is diagonal, in that its o-diagonal (m = matrix n) elements are all 0. Then, noting that k |n = kn , AB = kmn ak Bnm |k k |n m| = mn an Bnm |n m| am Bnm |n m| mn BA = kmn ak Bnm |n m|k k | = [A, B ] = mn So (an am )Bnm |n m| This will only vanish if the matrix element (an am )Bnm is zero for all n and m. Now, we know that A and B share one eigenvector, say |

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UCF - PHY - 4604
Homework Solutions # 4 (Libo Chapter 6) 6.7 Note that (x, 0) is an eigenstate of H . (a) (x, t) = (x, 0)eit where = E/h = hk0 /2m. 2 (b) 1 ik0 x e eik0 x eit 2i in terms of p eigenfunctions. Therefore, hk0 can be observed, with 1 equal probability ( 2 ).
UCF - PHY - 4604
Homework Solutions # 5 (Libo Chapter 7)7.5 Using the denition of a in terms of x and p, [, a ] = a m0 2i 1 [, p] x [x, x] [, p] + 2 2 [, p] = x p =1 2 h m0 m 0 ih 7.9 Say a |n = Cn |n + 1 . Then the norm of this state is n|aa |n = n|( a + 1)|n = n|(N
UCF - PHY - 4604
Homework Solutions # 6 (Libo Ch. 7)7.37 Use the 1D form, /t + J/x = 0, and go through the derivationin equations 7.102 to 7.105. You will get ih 2 2 2 t 2m x x2 i + (V V ) = 0 h Since V = V , the last term does not vanish, and the continuity equation d
UCF - PHY - 4604
Homework Solutions # 7 (Libo Ch. 8)8.7 The eigenfunctions of the semi-innite potential well are just those forthe nite one, with the extra boundary condition that (0) = 0. This is satised by the x &gt; 0 halves of the odd-parity solutions to the nite well,
UCF - PHY - 4604
Physics 139BSolutions to Homework Set 1Fall 20091. Libo, problem 11.27 on page 498. (a) Let A be an hermitian operator. We rst demonstrate that (eiA ) = eiA . To prove this, we use the series expansion that denes the exponential,eiA =n=0(iA)n . n!T
UCF - PHY - 4604
Physics 139BSolutions to Homework Set 2Fall 20091. Libo, problem 9.32 on page 395. (a) The key equation is given in Table 9.4 on p. 379 of Libo: L | , m = [( m)( m + 1)]1/2 | , m 1 . We will also need to use: L2 | , m =2(1)( + 1) | , m .(2)When co
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 1 due 9/1/2005, in class.1. Recapping angular momentum. Libo Problem 1.5 Libo Problem 1.7 Libo Problem 1.8 2. Cyclic coordinates for the simple harmonic oscillator. Libo Problem 1.11 3. Poisson brackets Libo Pro
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 1 Solutions.1. Recapping angular momentum. Liboff Problem 1.5 In cartesian coordinates, the z-component of angular momentum is given by L z = xp y yp x where p x = mx and p y = my. To express it in spherical coo
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 2 due 9/8/2005, in class.1. Recapping complex variables. Liboff Problem 1.21, without parts (b), (d), (f), (j). 2. Black-body radiation and the photoelectric effect. Liboff Problem 2.1 Liboff Problem 2.14 3. The
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 2 Solutions. First of all, here is a general comment. Liboff uses Gaussian units throughout the book - so, for example, Coulombs law for the force between two charges, q 1 and q 2 separated by a distance r, would
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 3 due 9/15/2005, in class.1. Getting more familiar with functions. Libo Problem 3.6 Remember to regard these equalities in the following sense. To show blah = BLAH , you need to demonstrate that blah f (y ) dy =
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 3 Solutions.1. Getting more familiar with functions. Liboff Problem 3.6 Remember to regard these equalities in the following sense. To show blah = BLAH, you need to demonstrate that blah f(y )dy = BLAH f(y )dy
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 4 due 9/22/2005, in class.1. Becoming more familiar with operators. Liboff Problem 3.3 Liboff Problem 3.16 2. and in particular, the displacement operator: Liboff Problem 3.4 Liboff Problem 3.17 3. Why is the Sc
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 4 Solutions.1. Becoming more familiar with operators. Liboff Problem 3.3 Nonlinear operators are: P, - why? P[a 1 (x) + b 2 (x)] = [a 1 (x) + b 2 (x)] 3 3[a 1 (x) + b 2 (x)] 2 4 which is not equal to : aP 1 (x)
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 5 due 10/6/2005, in class.1. One-dimensional problems. Liboff Problem 3.23 Liboff Problem 4.1 2. Dirac notation and operators Liboff Problem 4.4 Liboff Problem 4.12 3. Extra credit: The time evolution operator f
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PHYSICS 4455 QUANTUM MECHANICS Problem Set 5 Solutions.1. One-dimensional problems. Liboff Problem 3.23 Since the particle is free, V(x) = 0 for x 0. So, the stationary Schrdinger equation for this problem is 2 2 d 2 (x) = E(x) (x) + 2m E(x) = 0 2m dx 2
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 6 due 10/13/2005, in class.1. More practice with one-dimensional problems. Liboff Problem 4.33 Liboff Problem 4.35 2. Math: basic linear algebra Liboff Problems 4.19, 20, 21, 22 3. Extra credit: Liboff Problem 5
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 6 due 10/13/2005, in class.1. More practice with one-dimensional problems. Liboff Problem 4.33 (a) A particle in a one-dimensional box on the interval (0, a) is in the superposition state (x) = b 1 e i 1 1 (x) +
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 7 due 10/20/2005, in class.The finite potential well in one dimension. Consider a structureless, non-relativistic particle of mass m in one dimension, moving in the presence of a finite square well potential: V
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 7 Solutions.The finite potential well in one dimension. Consider a structureless, non-relativistic particle of mass m in one dimension, moving in the presence of a finite square well potential: V (x ) = V 0 |x |
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 8 due 10/27/2005, in class.No room to wiggle - exercises with function potentials 1. Starting with the solution for the bound states in HW7, set V 0 = /L and take the limit L 0. Notice that, although the depth o
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UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 9 due 11/17/2005, in class.A few more items about harmonic oscillators. 1. Many of you wondered in class how the raising and lowering operator formalism connects back to the explicit wave functions in coordinate
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 10 due 12/2/2005, 5pm in D)r. Schmittmanns office.1. Angular momentum and its commutators part of every physicists rate of passage! Confirm Liboff Eq. (9.8) and (9.13). Do Liboff Problems 9.4 (takes just a few m
UCF - PHY - 4604
PHYSICS 4455 QUANTUM MECHANICS Problem Set 10 Solutions.1. Angular momentum and its commutators part of every physicists rite of passage! Confirm Liboff Eq. (9.8) and (9.13). We worked out Eq. (9.8) in class, using the Levi-Civita tensor. As for (9.13),
UCF - PHY - 4604
UCF - PHY - 4604
1I. SYLLABUS AND INTRODUCTIONLet us start with a brief overview of the items that will (hopefully) be covered in this course and to give a guideline of what we are trying to learn. Quantum mechanics 560 and 561 are advanced quantum mechanics courses des
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 1 Due: September 04, 2009You check out all problems in Chapter 1 of Libo. The following are to be handed in for grading: 1. Libo, 1.1 2. Libo, 1.4 3. Libo, 1.6 4. Libo,
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 2 Due: September 18, 2009From Chap. 2 of Libo: Check out all problems; the following are to be handed in for grading: 1. Libo, 2.12 2. Libo, 2.16 3. Libo, 2.19 4. Libo,
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 3 Due: October 2, 2009There are still some problems from Chapter 3. The book gives the answer to the most interesting problems from Chap. 4. Study them carefully. The li
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 4 Due: October 16, 2009There are still a few problems from Chap. 4 you should try to solve. Chapter 5 is full of interesting and useful problems. Notice that there are m
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 5 Due: October 30, 2009Please, read carefully secs. 52. to 5.4 in Libo. Check out all problems in Chap. 5. Hand in the following problems for grading: 1. Libo, 5.18 (aft
UCF - PHY - 4604
University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 6 Due: November 16, 20091. Libo, 6.33 2. Libo, 7.3 3. Libo, 7.8 4. Libo, 7.16 5. Using the algebraic method, compute the matrix elements in the energy basis of the posit
UCF - PHY - 3513
PHYS 3513 ThermodynamicsSolutions to Homework set 11. Classify the following system as open, closed, or isolated: a) A mass of biomolecules enclosed by a membrane that is permeable to positive ions. b) A mass of liquid in a container with impermeable, a
UCF - PHY - 3513
UCF - PHY - 3513
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PHYS 3513 ThermodynamicsHomework Set 1 (Due 09/08/09 before 5pm)1. Classify the following system as open, closed or isolated:a) A mass of biomolecules enclosed by a membrane that is permeable to positive ions. b) A mass of liquid in a container with im
UCF - PHY - 3513
PHYS 3513 ThermodynamicsHomework Set 2 (Due 09/17/09 before 5pm)1. 2. 3. 4. 5. 6.A.H. Carter 4-2 A.H. Carter 4-8 A.H. Carter 4-12 A.H. Carter 5-5 A.H. Carter 5-10 A.H. Carter 5-14
UCF - PHY - 3513
PHYS 3513 ThermodynamicsHomework Set 3 (Due 09/29/09 before 5pm)1. 2. 3. 4. 5. 6.A.H. Carter 6-1 A.H. Carter 6-11 A.H. Carter 6-13 A.H. Carter 7-5 A.H. Carter 7-6 A.H. Carter 7-13
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PHYS 3513 ThermodynamicsHomework Set 4 (Due 10/08/09 before 5pm)1. 2. 3. 4. 5.A.H. Carter 7-17 A.H. Carter 8-1 A.H. Carter 8-2 A.H. Carter 8-6 A.H. Carter 8-8
UCF - PHY - 3513
PHYS 3513 ThermodynamicsHomework Set 5 (Due 10/27/09 before 5pm)1. A.H. Carter 9-10 2. A.H. Carter 10-5 3. A.H. Carter 10-6
UCF - PHY - 3513
PHYS 3513 ThermodynamicsHomework Set 6 (Due 11/05/09 before 5pm)1. A.H. Carter 11-1 2. A.H. Carter 11-7 3. A.H. Carter 11-9
UCF - PHY - 3513
PHYS 3513 ThermodynamicsHomework Set 7 (Due 11/12/09 before 5pm)1. A.H. Carter 12-1 2. A.H. Carter 12-4 3. A.H. Carter 12-5
UCF - PHY - 3513
PHYSICS 101-102 - Conceptual PhysicsExam Formula Sheet Units (SI): Length: m = meter Time: s = second Mass: kg = kilogram; atomic mass unit u = 1.661.10-27 kg = m(12C)/12 Velocity: m/s Acceleration: m/s2 Momentum: kg m/s Force: N = Newton = kg m/s2 Energ
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UCF - AST - 4142
Physical Properties of NEOs by Binzel et al.OUTLINE1. 2. 3. Introduction Tabulation of NEO Physical Properties Analysis1. 2. 3. 4. 5. Taxonomy of NEOs Relationship of NEOs to Comets Relationship of NEOs to Ordinary Chondrites Shapes and Rotations Optic
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Assignment due Thurs. Oct 1. Hard-copy Chapter by Gaffey et al. In Asteroids II (NOT III) book.Abstract and Sections I, II, (not III) IV,V, VI, (not VII) and VIIIComparing the Terrestrial PlanetsAsteroids III Ch. 1 An Overview of the Asteroids: The Ast
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Review for Exam 2 October 20, 2009Asteroids III Ch. 1 An Overview of the Asteroids: The Asteroids III PerspectiveOutline1. 2. 3. 4. 5. Introduction Brief History of the Primordial Asteroid Belt Present State of Main Belt The Near-Earth-Object Populatio
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HW due Thurs. Nov. 5th Sections IV, V, VI VII and VIII from Chapter on Physics and Chemistry of Comets (pdf in website)INTRODUCTION TO COMETSOUTLINEI. II. Introduction: History, Nucleus, Coma, Tails, Sources Gas and Ice Composition Parent and Daughter
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