UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet

Size: px
Start display at page:

Download "UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet"

Transcription

1 UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Solution for take home exam: FYS311, Oct. 7, The Hamiltonian of a charged particle in a weak magnetic field is Ĥ = P /m q mc P A where we have omitted the A term because the field is weak. The vector potential A = B ( y, x, ) gives a uniform field in the z-direction ( B = A). Inserting this into the P A -term we get: qb mc (xp y yp x ) = qb mc ˆL z. Writing out the Hamiltonian in polar coordinates we find Ĥ = h 1 m r r ( r ) L + r mr qb mc ˆL z The particle is constrained to move on a sphere, thus r is constant, and is therefore not a degree of freedom. This reduces the Hamiltonian to the quoted expression Ĥ = ᾱ h L + β ˆL z with α/ h = 1/(mr ) and β = (qb)/(mc). (q < for an electron). Eigenstates of Ĥ are the l, m with energy eigenvalues: E l,m = ᾱ h h l(l + 1) + β hm = hα (l(l + 1) + βα ) m where l {, 1,,...} and m { l, l + 1,..., l 1, l}. The L s represent a spatial degree of freedom(in contrast to spin), thus only integer l s are allowed E[h _ α β/α Figure 1: Energy levels. Shown are the 1 lowest (at β/α = ) levels. 1

2 1. The energy eigenstates are the l, m. Which of these have the lowest energy? Inspecting E l,m we see that the value of m giving the lowest energy for a given l is the smallest possible value of m, m = l. In this state l, l ˆL z l, l = hl. The value of l in the ground state is found by inspecting which l minimizes E l, l for a given value of β/α. From the plot in 1.1 it is clear that for small β/α, E, is the lowest energy. Then as β/α increases higher values of l takes over successively. The ground state will change from l to l + 1 discontinuously when E l+1, (l+1) = E l, l which is equivalent to hα [(l + 1)(l + ) βα (l + 1) = hα [l(l + 1) βα l = β α = l + Thus the ground state will have l = for β/α <. For < β/α < the ground state will have l = 1, etc... This can also be seen from the plot in 1.1. At β/α = {,, 6,...} 1 <L z >[h _ β/α Figure : Plot of ψ ˆL z ψ the ground state manifold is two-dimensional. It is spanned by the states l, l and l + 1, l 1. At these points the value of L z will depend on a further specification of the ground state. Since there are actually two states a further specification would be to specify the probabilities for each of these states. This can be achieved using the framework of density operators which is not part of this course. Without a further specification of the ground state at these points, all we can conclude is that the observed value must lie somewhere in between the values l h and (l + 1) h. In the following we consider the slightly more general state than in the problem set: ψ = c ( l = 1, m = 1 + b l = 1, m = l = 1, m = 1 )

3 To obtain the state in the exam, set c = 1 and b = i. 1.3 ψ(t) = c ( e ie1, 1t/ h 1, 1 + be ie1,t/ h 1, e ie1,1t/ h 1, 1 ) = ce ( iαt e iβt 1, 1 + b 1, e iβt 1, 1 ) = 1 (e e iαt iβt 1, 1 + i ) 1, e iβt 1, 1 The corresponding bra is ψ(t) = c e ( iαt e iβt 1, 1 + b 1, e iβt 1, 1 ) = 1 (e eiαt iβt 1, 1 i ) 1, e iβt 1, 1 The operator ˆL z acting on ψ(t) gives ˆL z ψ(t) = hce iαt ( e iβt 1, 1 e iβt 1, 1 ). Thus the expectation value of ˆL z becomes ψ(t) ˆL z ψ(t) = h c ( 1 + 1) = h ( 1 + 1) = 1. The state ψ(t) has total momentum l = 1. Thus the possible values of the z-axis component of the angular momentum are ± h,. The probabilities for the different values are gotten by projections onto the corresponding eigenfunctions: P ( h) = 1, 1 ψ(t) = ce iαt+iβt = c = 1 P () = 1, ψ(t) = ce iαt b = cb = 1 P (+ h) = 1, 1 ψ(t) = ce iαt iβt = c = 1 These probabilities give the expected value ˆL z = ( h) c + cb + h c = h c ( 1+1) = which agrees with problem 1.3. Note that it is not enough here to show that the most probable value of m is. The value of L z does not depend on how probable it is to get m =. Rather the crucial point here is that it is equally likely to get h as + h. 3

4 1.5 The operator ˆL x = (ˆL + + ˆL )/, and ˆL ± l, m = h l(l + 1) m(m ± 1) l, m ± 1. In particular this gives ˆL x 1, 1 = h 1, ˆL x 1, = h ( 1, 1 + 1, 1 ) ˆL x 1, 1 = h 1, Using these relations we find Thus ˆL x ψ(t) = h ce iαt ( b 1, 1 + (e iβt e iβt ) 1, + b 1, 1 ) = h ( e iαt i 1, 1 + (e iβt e iβt ) 1, + i ) 1, 1 ψ(t) ˆL x ψ(t) = h c ( be iβt + b (e iβt e iβt ) be iβt) = h c ( e iβt (b b) e ibt (b b) ) = h c sin(βt)i(b b) = h sin(βt) Applying ˆL x once more gives which results in ˆL x ψ(t) = h ce iαt ( (e iβt e iβt ) 1, 1 + b 1, + (e iβt e iβt ) 1, 1 ) ψ(t) ˆL x ψ(t) = h c ( (1 e iβt ) + b (e iβt 1) ) = h c ( (1 e iβt ) + b (e iβt 1) ) = h c ( + b cos(βt) ) = h c ( + b cos(βt) ) = h = h (3 cos(βt)) ( 1 + sin (βt) )

5 1.6 The state ψ(t) has l = 1, thus the possible measurement values of ˆL x is ± h,. In order to find the probabilities of each we need the eigenstates of ˆL x. From the action of ˆL x l, m on the l = 1 states listed above (1.5) it is not difficult to see that ˆL x ( 1, 1 1, 1 ) = ˆL x ( 1, 1 + ) ( 1, + 1, 1 = h 1, 1 + ) 1, + 1, 1 ˆL x ( 1, 1 ) ( 1, + 1, 1 = h 1, 1 ) 1, + 1, 1 Normalizing these, the eigenstates of L x are And the probabilities 1 are Added together we find = 1 ( 1, 1 ) 1, + 1, 1 = 1 ( 1, 1 1, 1 ) + = 1 ( 1, 1 + ) 1, + 1, 1 P ( h) = ψ(t) = c eiβt b e iβt = 1 (1 sin(βt)) P () = ψ(t) = c eiβt + e iβt = 1 cos (βt) P (+ h) = + ψ(t) = c eiβt + b e iβt = 1 (1 + sin(βt)) P ( h) + P () + P (+ h) = 1 (1 sin(βt)) + 1 cos (βt) + 1 (1 + sin(βt)) = sin(βt) + 1 cos (βt) = = In the position representation the state ψ(t) is represented as ψ(θ, φ, t) = ce ( iαt e iβt Y1 1 (θ, φ) + by1 (θ, φ) + e iβt Y1 1 (θ, φ) ) = 1 ( e iαt e iβt Y1 1 (θ, φ) + i ) Y1 (θ, φ) + e iβt Y1 1 (θ, φ) 1 Alternative (equal) expressions are P (± h) = 3/8 ± sin(βt)/ cos(βt)/8 and P () = 1/ + cos(βt)/. 5

6 where Yl m (θ, φ) are the spherical harmonic functions. Specifically 3 3 Y 1 ±1 (θ, φ) = 8π sin θe±iφ, Y1 (θ, φ) = π cos θ Alternatively we can project ψ(t) on the position eigenkets θ, φ, ψ(θ, φ, t) = θ, φ ψ(t), and use θ, φ l, m = Yl m (θ, φ). The probability per area of finding the particle with angular coordinates θ and φ at time t is ψ(θ, φ, t) which is normalized such that π π dθ sin θ dφ ψ(θ, φ) = 1. We are interested in the most likely value of θ, specific values of φ do not matter, thus we integrate over φ in order to obtain a probability distribution over θ alone P (θ, t) = sin θ π dφ ψ(θ, φ, t) which is normalized as π dθp (θ, t) = 1 The extra factor sin θ comes from the measure of the integral in polar coordinates. One can see that it must be included in the probability density of finding specific θ values from the following argument. If the particle is equally likely to be anywhere on the sphere, the most probable value of θ will θ = π/ simply because there is more area on the sphere which has θ = π/ (the equator) than any other values of the polar angle. Compare this for instance to θ = the north pole. The sin θ factor expresses this fact. P (θ, t) = sin θ π dφ ψ(θ, φ, t) = sin θ c π ( Y1 1 + b Y1 + Y1 1 ) = sin θ c 3 ( sin (θ) + b cos (θ) ) = sin θ c 3 ( (1 b ) sin (θ) + b ) = sin θ 3 ( sin (θ) ) 8 Where we have used the fact that the integral over φ is zero for products of spherical harmonics that differ in their m-value. Therefore all cross-terms vanish and with it the 6

7 time-dependence. to. We get To find the most likely value of θ we differentiate p(θ) and set it equal ( 3(1 b ) sin (θ) + b ) cos(θ) = which has solutions cos(θ) = and sin (θ) = b 3( b 1) = 3 Thus the extrema of the probability distribution is at θ = π/ and at sin θ = /3. The θ = π/ is a minimum while the other corresponds to a maximum. This can be seen by differentiating once more or by plotting the probability density p(θ). Two angles fulfill the..15 p(θ) θ maximum condition: θ = sin 1 ( /3) =.957 rad = 5.7 θ = π.957 rad =.1863 rad = These corresponds to a latitude of roughly 36 north and south of the equator (Gibraltar/Cape Town). 1.8 We use ˆL z = x ˆP y y ˆP x which gives x = [x ˆP y y ˆP x, ˆP [ x = x, ˆP x ˆPy = i h ˆP y y = [x ˆP y y ˆP x, ˆP [ y = y, ˆP y ˆPx = i h ˆP x These commutators imply [ˆL z, [ˆL z, ˆP x = [ˆL z, i h ˆP y = i h( i) h ˆP x = h ˆPx Alternatively we could first have computed ψ(θ, φ, t) = 3 8π (sin θ cos (βt φ) + cos θ) and then performed the integral over φ. From this expression it is clear that when time changes by t the effect on the state is to rotate it an angle β t about the z-axis. Therefore the probabilities of getting specific values of φ are time-dependent, while probabilities for values of θ are not. 7

8 Using this we find h l, m ˆP x lm = l m [ˆL z, [ˆL z, ˆP x l, m = l m ˆL ˆP z x ˆL z ˆPx ˆLz + ˆP x ˆL z l, m = h ( (m ) m m + m ) l m ˆP x l, m = h (m m) l m ˆP x l, m Assembling everything on one side of the equation we get h [(m m) 1 l m ˆP x lm = Which implies that l m ˆP x lm = whenever the parenthesis [ (m m) 1 which is equvivalent to the condition m m ± 1. q.e.d. We might as well have used the commutation relations to establish the double commutator [ˆL z, [ˆL z, ˆP y = [ˆL z, i h ˆP x = i hi h ˆP y = h ˆPy. Thus the same relation holds for l m ˆP y lm. For ˆP z we have z =. Thus = l, m z lm = (m m) l, m ˆP z lm This implies that l, m ˆP z lm = for m m. So this differs from the other components. Note here that although z =, the commutator [L, ˆP z. So although ˆP z and ˆL z have common eigenstates the state l, m is not an eigenstate of ˆPz. By considering the commutator with L one can also show that l m ˆP i lm = for l l ± 1. The relations shown here are very general. In fact they hold for all vector operators, not just the momentum operator treated here. 1.9 Using the defining commutation relations for the angular momentum operators, among them [ˆL y, ˆL z = iˆl x it becomes clear that the extra term can be written H e = γ h i (ˆLy ˆLz ˆL z ˆLy ) = γ ˆL x Thus the Hamiltonian now reads Ĥ = ᾱ h L + (γ,, β) L = ᾱ h L + γ + β n L 8

9 where n is the unit vector (γ,, β)/ γ + β This problem is equivalent to our initial problem if we redefine our z-axis to lie along n and identify γ + β with β. Thus the energy eigenvalues are E l,m = hαl(l + 1) + γ + β hm The physical explanation for the same form of the energy levels is that the extra term corresponds to a change of the magnetic field, it is tilted towards the x-direction and changed in magnitude. As there is nothing special about which direction the magnetic field points in, the form of the energy levels must remain the same, regardless of the direction of the field, and we need only to correct for the changed magnitude. This is achieved by letting β β + γ. 9

For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N 1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

More information

1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 )

1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 ) CHEM 352: Examples for chapter 2. 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = 1 14 2ψ 1,, 3ψ 2,, ψ 3,2,2 ) where ψ n,l,m are eigenfunctions of the

More information

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006

Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 Rutgers - Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J

More information

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1) CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

More information

Lecture 2: Angular momentum and rotation

Lecture 2: Angular momentum and rotation Lecture : Angular momentum and rotation Angular momentum of a composite system Let J and J be two angular momentum operators. One might imagine them to be: the orbital angular momentum and spin of a particle;

More information

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004 PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,

More information

Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

More information

Part IB. Quantum Mechanics. Year

Part IB. Quantum Mechanics. Year Part IB Year 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2016 41 Paper 4, Section I 6B (a) Define the quantum orbital angular momentum operator ˆL = (ˆL 1, ˆL 2, ˆL

More information

Quantum Mechanics. December 17, 2007

Quantum Mechanics. December 17, 2007 Quantum Mechanics Based on lectures given by J.Billowes at the University of Manchester Sept-Dec 7 Please e-mail me with any comments/corrections: jap@watering.co.uk J.Pearson December 7, 7 Contents Review.

More information

3-D Dynamics of Rigid Bodies

3-D Dynamics of Rigid Bodies 3-D Dynamics of Rigid Bodies Introduction of third dimension :: Third component of vectors representing force, linear velocity, linear acceleration, and linear momentum :: Two additional components for

More information

A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>.

A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>. Quantum Mechanical Operators and Commutation C I. Bra-Ket Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates

More information

MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions

MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in one-dimension. Its position at time t is x(t. As a function of

More information

We can represent the eigenstates for angular momentum of a spin-1/2 particle along each of the three spatial axes with column vectors: 1 +y =

We can represent the eigenstates for angular momentum of a spin-1/2 particle along each of the three spatial axes with column vectors: 1 +y = Chapter 0 Pauli Spin Matrices We can represent the eigenstates for angular momentum of a spin-/ particle along each of the three spatial axes with column vectors: +z z [ ] 0 [ ] 0 +y y [ ] / i/ [ ] i/

More information

The Schrödinger Equation

The Schrödinger Equation The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development

More information

How is a vector rotated?

How is a vector rotated? How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

More information

Ground State of the He Atom 1s State

Ground State of the He Atom 1s State Ground State of the He Atom s State First order perturbation theory Neglecting nuclear motion H m m 4 r 4 r r Ze Ze e o o o o 4 o kinetic energies attraction of electrons to nucleus electron electron repulsion

More information

Mixed states and pure states

Mixed states and pure states Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

More information

Chapter 3 The Earth's dipole field

Chapter 3 The Earth's dipole field Chapter 3 The Earth's dipole field 1 Previously The Earth is associated with the geomagnetic field that has an S-pole of a magnet near the geographic north pole and an N-pole of a magnet near the geographic

More information

The force equation of quantum mechanics.

The force equation of quantum mechanics. The force equation of quantum mechanics. by M. W. Evans, Civil List and Guild of Graduates, University of Wales, (www.webarchive.org.uk, www.aias.us,, www.atomicprecision.com, www.upitec.org, www.et3m.net)

More information

How Map Projections Transform Velocity Vectors

How Map Projections Transform Velocity Vectors September 9, 2008 9:0 pm MDT Page of 8 How Map Projections Transform Velocity Vectors. Bullock Let P be a point moving on the surface of the earth, which we ll take to be a sphere of radius. If we re using

More information

Chapter 4. Rotations

Chapter 4. Rotations Chapter 4 Rotations 1 CHAPTER 4. ROTATIONS 2 4.1 Geometrical rotations Before discussing rotation operators acting the state space E, we want to review some basic properties of geometrical rotations. 4.1.1

More information

PHY411. PROBLEM SET 3

PHY411. PROBLEM SET 3 PHY411. PROBLEM SET 3 1. Conserved Quantities; the Runge-Lenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the

More information

Mathematical Formulation of the Superposition Principle

Mathematical Formulation of the Superposition Principle Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.

More information

Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics

Lecture 22 Relevant sections in text: 3.1, 3.2. Rotations in quantum mechanics Lecture Relevant sections in text: 3.1, 3. Rotations in quantum mechanics Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in

More information

Physics 70007, Fall 2009 Solutions to HW #4

Physics 70007, Fall 2009 Solutions to HW #4 Physics 77, Fall 9 Solutions to HW #4 November 9. Sakurai. Consider a particle subject to a one-dimensional simple harmonic oscillator potential. Suppose at t the state vector is given by ipa exp where

More information

Three Body Problem for Constant Angular Velocity

Three Body Problem for Constant Angular Velocity Three Body Problem for Constant Angular Velocity P. Coulton Department of Mathematics Eastern Illinois University Charleston, Il 61920 Corresponding author, E-mail: cfprc@eiu.edu 1 Abstract We give the

More information

Average Angular Velocity

Average Angular Velocity Average Angular Velocity Hanno Essén Department of Mechanics Royal Institute of Technology S-100 44 Stockholm, Sweden 199, December Abstract This paper addresses the problem of the separation of rotational

More information

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

More information

Figure 1: Volume between z = f(x, y) and the region R.

Figure 1: Volume between z = f(x, y) and the region R. 3. Double Integrals 3.. Volume of an enclosed region Consider the diagram in Figure. It shows a curve in two variables z f(x, y) that lies above some region on the xy-plane. How can we calculate the volume

More information

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

Class 26: Rutherford-Bohr atom

Class 26: Rutherford-Bohr atom Class 6: Rutherford-Bohr atom Atomic structure After the discovery of the electron in 1897 by J. J. Thomson, Thomson and others developed models for the structure of atoms. It was known that atoms contained

More information

Estimating Dynamics for (DC-motor)+(1st Link) of the Furuta Pendulum

Estimating Dynamics for (DC-motor)+(1st Link) of the Furuta Pendulum Estimating Dynamics for (DC-motor)+(1st Link) of the Furuta Pendulum 1 Anton and Pedro Abstract Here the steps done for identification of dynamics for (DC-motor)+(1st Link) of the Furuta Pendulum are described.

More information

The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

More information

Assignment 2: Transformation and Viewing

Assignment 2: Transformation and Viewing Assignment : Transformation and Viewing 5-46 Graphics I Spring Frank Pfenning Sample Solution Based on the homework by Kevin Milans kgm@andrew.cmu.edu Three-Dimensional Homogeneous Coordinates (5 pts)

More information

Symbols, conversions, and atomic units

Symbols, conversions, and atomic units Appendix C Symbols, conversions, and atomic units This appendix provides a tabulation of all symbols used throughout this thesis, as well as conversion units that may useful for reference while comparing

More information

1 Variational calculation of a 1D bound state

1 Variational calculation of a 1D bound state TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN- ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,

More information

8.04 Spring 2013 April 17, 2013 Problem 1. (15 points) Superposition State of a Free Particle in 3D

8.04 Spring 2013 April 17, 2013 Problem 1. (15 points) Superposition State of a Free Particle in 3D Problem Set 8 Solutions 8.04 Spring 013 April 17, 013 Problem 1. (15 points) Superposition State of a Free Particle in 3D (a) (4 points) Recall from lecture that the energy eigenstates of a free particle

More information

Chapter 9 Rotation of Rigid Bodies

Chapter 9 Rotation of Rigid Bodies Chapter 9 Rotation of Rigid Bodies 1 Angular Velocity and Acceleration θ = s r (angular displacement) The natural units of θ is radians. Angular Velocity 1 rad = 360o 2π = 57.3o Usually we pick the z-axis

More information

Quantum Chemistry Exam 2 Solutions (Take-home portion)

Quantum Chemistry Exam 2 Solutions (Take-home portion) Chemistry 46 Spring 5 Name KEY Quantum Chemistry Exam Solutions Take-home portion 5. 5 points In this problem, the nonlinear variation method will be used to determine approximations to the ground state

More information

Chapter 24 Physical Pendulum

Chapter 24 Physical Pendulum Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

More information

07. Quantum Mechanics Basics.

07. Quantum Mechanics Basics. 07. Quantum Mechanics Basics. Stern-Gerlach Experiment (Stern & Gerlach 1922) magnets splotches sliver atoms detection screen Suggests: Electrons possess 2-valued "spin" properties. (Goudsmit & Uhlenbeck

More information

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at

More information

Laboratory 2 Application of Trigonometry in Engineering

Laboratory 2 Application of Trigonometry in Engineering Name: Grade: /26 Section Number: Laboratory 2 Application of Trigonometry in Engineering 2.1 Laboratory Objective The objective of this laboratory is to learn basic trigonometric functions, conversion

More information

The data were explained by making the following assumptions.

The data were explained by making the following assumptions. Chapter Rutherford Scattering Let usstart fromtheoneofthefirst steps which was donetowards understanding thedeepest structure of matter. In 1911, Rutherford discovered the nucleus by analysing the data

More information

We consider a hydrogen atom in the ground state in the uniform electric field

We consider a hydrogen atom in the ground state in the uniform electric field Lecture 13 Page 1 Lectures 13-14 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and

More information

ORIENTATIONS OF LINES AND PLANES IN SPACE

ORIENTATIONS OF LINES AND PLANES IN SPACE GG303 Lab 1 9/10/03 1 ORIENTATIONS OF LINES AND PLANES IN SPACE I Main Topics A Definitions of points, lines, and planes B Geologic methods for describing lines and planes C Attitude symbols for geologic

More information

Electron Scattering: Form Factors and Nuclear Shapes

Electron Scattering: Form Factors and Nuclear Shapes Electron Scattering: Form Factors and Nuclear Shapes Kyle Foster November 16, 011 1 1 Brief History of Early Electron Scattering Experiments 1951 Early electron scattering experiments are performed at

More information

4. The Infinite Square Well

4. The Infinite Square Well 4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the

More information

Section 10.6 Vectors in Space

Section 10.6 Vectors in Space 36 Section 10.6 Vectors in Space Up to now, we have discussed vectors in two-dimensional plane. We now want to epand our ideas into three-dimensional space. Thus, each point in three-dimensional space

More information

Qualification Exam: Quantum Mechanics

Qualification Exam: Quantum Mechanics Qualification Exam: Quantum Mechanics Name:, QEID#26080663: March, 2014 Qualification Exam QEID#26080663 2 1 Undergraduate level Problem 1. 1983-Fall-QM-U-1 ID:QM-U-2 Consider two spin 1/2 particles interacting

More information

Strategy. Theorem Fermat s Last Theorem: If n > 2, then there are no nontrivial integer solutions to x n + y n = z n.

Strategy. Theorem Fermat s Last Theorem: If n > 2, then there are no nontrivial integer solutions to x n + y n = z n. 1. Rewrite equation as α n + β n + γ 3 = 0. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β, γ is divisible by 3. 1. Rewrite equation as α n + β n + γ 3 = 0. 2. Exactly one of α, β,

More information

E-Appendix: Bell s Theorem

E-Appendix: Bell s Theorem E-Appendix: Bell s Theorem Christopher T. Hill and Leon M. Lederman, 01 Let s recapitulate Bell s thought experiment as described in the text. In a tropical aquarium we notice (1) that every fish comes

More information

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004 PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall

More information

Fermi s golden rule. 1 Main results

Fermi s golden rule. 1 Main results Fermi s golden rule Andreas Wacker 1 Mathematical Physics, Lund University October 1, 216 Fermi s golden rule 2 is a simple expression for the transition probabilities between states of a quantum system,

More information

University of Illinois at Chicago Department of Physics. Electricity and Magnetism PhD Qualifying Examination

University of Illinois at Chicago Department of Physics. Electricity and Magnetism PhD Qualifying Examination University of Illinois at Chicago Department of Physics Electricity and Magnetism PhD Qualifying Examination January 6, 2015 9.00 am 12:00 pm Full credit can be achieved from completely correct answers

More information

Theory of Angular Momentum and Spin

Theory of Angular Momentum and Spin Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO3 of the associated rotation matrices and the corresponding transformation matrices of spin states forming

More information

Engineering Mathematics 233 Solutions: Double and triple integrals

Engineering Mathematics 233 Solutions: Double and triple integrals Engineering Mathematics s: Double and triple integrals Double Integrals. Sketch the region in the -plane bounded b the curves and, and find its area. The region is bounded b the parabola and the straight

More information

Harmonic Oscillator Physics

Harmonic Oscillator Physics Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the time-independent Schrödinger equation: d ψx

More information

Equilibrium. To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium

Equilibrium. To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium Equilibrium Object To determine the mass of unknown objects by utilizing the known force requirements of an equilibrium situation. 2 Apparatus orce table, masses, mass pans, metal loop, pulleys, strings,

More information

5.1 Angles and Their Measure. Objectives

5.1 Angles and Their Measure. Objectives Objectives 1. Convert between decimal degrees and degrees, minutes, seconds measures of angles. 2. Find the length of an arc of a circle. 3. Convert from degrees to radians and from radians to degrees.

More information

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers

Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Complex Numbers Basic Concepts of Complex Numbers Complex Solutions of Equations Operations on Complex Numbers Identify the number as real, complex, or pure imaginary. 2i The complex numbers are an extension

More information

Concepts for specific heat

Concepts for specific heat Concepts for specific heat Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 8, 1 1 Introduction In this notes I want to briefly eplain general results for the internal

More information

Math Review: Circular Motion 8.01

Math Review: Circular Motion 8.01 Math Review: Circular Motion 8.01 Position and Displacement r ( t) : position vector of an object moving in a circular orbit of radius R Δr ( t) : change in position between time t and time t+δt Position

More information

Conceptual Approaches to the Principles of Least Action

Conceptual Approaches to the Principles of Least Action Conceptual Approaches to the Principles of Least Action Karlstad University Analytical Mechanics FYGB08 January 3, 015 Author: Isabella Danielsson Supervisors: Jürgen Fuchs Igor Buchberger Abstract We

More information

PHYS 311 HW 1 Solution Manual

PHYS 311 HW 1 Solution Manual PHYS 3 HW Solution Manual Shuting Zhang January 22, 205. (.30) - 8 points Object, moving at velocity, v collides with object 2. There are no external forces. They stick together. After the collision, what

More information

5. Spherically symmetric potentials

5. Spherically symmetric potentials TFY4215/FY1006 Tillegg 5 1 TILLEGG 5 5. Spherically symmetric potentials Chapter 5 of FY1006/TFY4215 Spherically symmetric potentials is covered by sections 5.1 and 5.4 5.7 in Hemmer s bok, together with

More information

0.1 Linear Transformations

0.1 Linear Transformations .1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

More information

Near horizon black holes in diverse dimensions and integrable models

Near horizon black holes in diverse dimensions and integrable models Near horizon black holes in diverse dimensions and integrable models Anton Galajinsky Tomsk Polytechnic University LPI, 2012 A. Galajinsky (TPU) Near horizon black holes and integrable models LPI, 2012

More information

Rotational Motion. Description of the motion. is the relation between ω and the speed at which the body travels along the circular path.

Rotational Motion. Description of the motion. is the relation between ω and the speed at which the body travels along the circular path. Rotational Motion We are now going to study one of the most common types of motion, that of a body constrained to move on a circular path. Description of the motion Let s first consider only the description

More information

Lecture-XXIV. Quantum Mechanics Expectation values and uncertainty

Lecture-XXIV. Quantum Mechanics Expectation values and uncertainty Lecture-XXIV Quantum Mechanics Expectation values and uncertainty Expectation values We are looking for expectation values of position and momentum knowing the state of the particle, i,e., the wave function

More information

DYNAMICS OF GALAXIES

DYNAMICS OF GALAXIES DYNAMICS OF GALAXIES 2. and stellar orbits Piet van der Kruit Kapteyn Astronomical Institute University of Groningen the Netherlands Winter 2008/9 and stellar orbits Contents Range of timescales Two-body

More information

CHAPTER 6 THE HYDROGEN ATOM OUTLINE. 3. The HydrogenAtom Wavefunctions (Complex and Real)

CHAPTER 6 THE HYDROGEN ATOM OUTLINE. 3. The HydrogenAtom Wavefunctions (Complex and Real) CHAPTER 6 THE HYDROGEN ATOM OUTLINE Homework Questions Attached SECT TOPIC 1. The Hydrogen Atom Schrödinger Equation. The Radial Equation (Wavefunctions and Energies) 3. The HydrogenAtom Wavefunctions

More information

PHYS 110A - HW #5 Solutions by David Pace Any referenced equations are from Griffiths

PHYS 110A - HW #5 Solutions by David Pace Any referenced equations are from Griffiths PHYS 110A - HW #5 Solutions by David Pace Any referenced equations are from Griffiths [1. Problem 3.8 from Griffiths Following Example 3.9 of Griffiths (page 14) consider a spherical shell of radius R

More information

OpenStax-CNX module: m Angular velocity. Sunil Kumar Singh. 1 General interpretation of angular quantities

OpenStax-CNX module: m Angular velocity. Sunil Kumar Singh. 1 General interpretation of angular quantities OpenStax-CNX module: m14314 1 Angular velocity Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Abstract Angular quantities are not

More information

We have seen in the previous Chapter that there is a sense in which the state of a quantum

We have seen in the previous Chapter that there is a sense in which the state of a quantum Chapter 8 Vector Spaces in Quantum Mechanics We have seen in the previous Chapter that there is a sense in which the state of a quantum system can be thought of as being made up of other possible states.

More information

Introduces the bra and ket notation and gives some examples of its use.

Introduces the bra and ket notation and gives some examples of its use. Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to

More information

Angular Momentum, Hydrogen Atom, and Helium Atom

Angular Momentum, Hydrogen Atom, and Helium Atom Chapter Angular Momentum, Hydrogen Atom, and Helium Atom Contents.1 Angular momenta and their addition..................4. Hydrogenlike atoms...................................38.3 Pauli principle, Hund

More information

221A Lecture Notes on Spin

221A Lecture Notes on Spin 22A Lecture Notes on Spin This lecture note is just for the curious. It is not a part of the standard quantum mechanics course. True Origin of Spin When we introduce spin in non-relativistic quantum mechanics,

More information

Physics 211 Week 12. Simple Harmonic Motion: Equation of Motion

Physics 211 Week 12. Simple Harmonic Motion: Equation of Motion Physics 11 Week 1 Simple Harmonic Motion: Equation of Motion A mass M rests on a frictionless table and is connected to a spring of spring constant k. The other end of the spring is fixed to a vertical

More information

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:

We can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation: Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region

More information

Electromagnetism - Lecture 4. Dipole Fields

Electromagnetism - Lecture 4. Dipole Fields Electromagnetism - Lecture 4 Dipole Fields Electric Dipoles Magnetic Dipoles Dipoles in External Fields Method of Images Examples of Method of Images 1 Electric Dipoles An electric dipole is a +Q and a

More information

Section V.4: Cross Product

Section V.4: Cross Product Section V.4: Cross Product Definition The cross product of vectors A and B is written as A B. The result of the cross product A B is a third vector which is perpendicular to both A and B. (Because the

More information

Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34 Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

More information

Lecture 20. PLANAR KINEMATIC-PROBLEM EXAMPLES

Lecture 20. PLANAR KINEMATIC-PROBLEM EXAMPLES Lecture 20. PLANAR KINEMATIC-PROBLEM EXAMPLES Figure 4.15 Slider-crank mechanism. TASK: For a given constant rotation rate, find the velocity and acceleration terms of the piston for one cycle of θ. Geometric

More information

Notes: Tensor Operators

Notes: Tensor Operators Notes: Tensor Operators Ben Baragiola I. VECTORS We are already familiar with the concept of a vector, but let s review vector properties to refresh ourselves. A -component Cartesian vector is v = v x

More information

Module17:Coherence Lecture 17: Coherence

Module17:Coherence Lecture 17: Coherence Module7:Coherence Lecture 7: Coherence We shall separately discuss spatial coherence and temporal coherence. 7. Spatial Coherence The Young s double slit experiment (Figure 7.) essentially measures the

More information

Bead moving along a thin, rigid, wire.

Bead moving along a thin, rigid, wire. Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing

More information

Solution Derivations for Capa #10

Solution Derivations for Capa #10 Solution Derivations for Capa #10 1) The flywheel of a steam engine runs with a constant angular speed of 172 rev/min. When steam is shut off, the friction of the bearings and the air brings the wheel

More information

Modern Geometry Homework.

Modern Geometry Homework. Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

More information

MATH 118, LECTURES 14 & 15: POLAR AREAS

MATH 118, LECTURES 14 & 15: POLAR AREAS MATH 118, LECTURES 1 & 15: POLAR AREAS 1 Polar Areas We recall from Cartesian coordinates that we could calculate the area under the curve b taking Riemann sums. We divided the region into subregions,

More information

Electromagnetic Waves

Electromagnetic Waves May 4, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 7 Maxwell Equations A basic feature of Maxwell equations for the EM field is the existence of travelling wave solutions which

More information

Notes on wavefunctions

Notes on wavefunctions Notes on wavefunctions The double slit experiment In the double slit experiment, a beam of light is send through a pair of slits, and then observed on a screen behind the slits. At first, we might expect

More information

Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

More information

Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

More information

Basic Quantum Mechanics

Basic Quantum Mechanics Basic Quantum Mechanics Postulates of QM - The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn) - To every observable (physical magnitude) there

More information

FALL 2005 EXAM C SOLUTIONS

FALL 2005 EXAM C SOLUTIONS FALL 005 EXAM C SOLUTIONS Question #1 Key: D S ˆ(300) = 3/10 (there are three observations greater than 300) H ˆ (300) = ln[ S ˆ (300)] = ln(0.3) = 1.0. Question # EX ( λ) = VarX ( λ) = λ µ = v = E( λ)

More information

REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.

REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example. REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described

More information

Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.

Write your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used. UNIVERSITY OF LONDON BSc/MSci EXAMINATION June 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship

More information

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing.

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. The amplitude of the sun (at sunrise, say) is the direction you look to see the sun come up. If it s rising exactly

More information

Appendix E - Elements of Quantum Mechanics

Appendix E - Elements of Quantum Mechanics 1 Appendix E - Elements of Quantum Mechanics Quantum mechanics provides a correct description of phenomena on the atomic or sub- atomic scale, where the ideas of classical mechanics are not generally applicable.

More information