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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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, solve subsequent parts in terms of unknown answer define clearly. All parts 10 pts., max=120. Problem 1 required, attempt 2 of remaining problems; circle which ones you want graded. a 0 Possibly helpful formulae and constants ˆL + ψ lm = h ll + 1) mm + 1)ψ lm+1 ˆL ψ lm = h ll + 1) mm 1)ψ lm 1 E n = me4 1 2 h 2 n 2 E n = hω n + 1 ) 2 ˆL 2 = ˆL + ˆL + ˆL 2 z hˆl z = ˆL ˆL+ + ˆL 2 z + hˆl z Hψ = Eψ Hψ = i h ψ t H = h2 2m 2 + V r) dx x n e x = n! sin 2 ax dx = a2 cosa 2 ) sina 2 ) 2a ) 2 1 ψ 0 x) = e 1 x 2 x 0 π 1/2 x 0 ψ 1 x) = ψ 2 x) = ψ x) = ) ) 2 1 2x e 1 x 2 x 0 2π 1/2 x 0 x 0 ) ) 1 x π 1/2 x x0 0 ) 1 x x π 1/2 x x0 0 x 0 e 1 2 ) ) 2 x x 0 e 1 2 ) 2 x x 0 1

2 1. Short answer. Must attempt only) 4 of 6. a) Explain what is meant by a 2p state of an atomic electron. The 2 refers to the principal quantum number n = 2, so state has energy E 2 = 1 Ryd/4, and p means total angular momentum quantum number l = 1. b) What is the degeneracy of the 1st excited state E = 5/2) hω) of the isotropic D simple harmonic oscillator? The D simple harmonic oscillator has energies E nx + E ny + E nz, where each E n is hωn + 1/2), so E tot = hωn x + n y + n z + /2). The first excited state has one quantum of excitation in either x, y, or z, so degeneracy =. c) Sketch the first eigenfunctions of the 1D infinite square well with V = 0 for a x a and V = otherwise. Label them according to their parity. Ground state ψ 1 and 2nd excited state ψ are both even functions of x even parity, in other words parity eigenvalue π = +1. 1st excited state ψ 2 is odd function odd parity, in other words parity eigenvalue π = 1. 2

3 d) State whether the following operators are self-adjoint, anti-self-adjoint, unitary, or none of the above and why. i. ˆLx L x = r p) x is a product of self-adjoint operators representing observables, and therefore also self-adjoint. It is also an observable itself, of course. ii. xˆp x xˆp x) = x ˆp x = xˆp x. Self-adjoint. iii. d dx Operator is anti-self-adjoint as discussed in class. Proof: definition of adjoint is O χ, ψ) = χ, Oψ) χ, ψ. For d/dx we have χ, d d x ψ) = = χ x) dψ dx dx d ) dx χ x) ψx)dx, where I ve performed an integration by parts and assumed that χ, ψ 0 at. From defintition, ) d = d dx dx iv. a + a a, a are raising & lowering operators for the 1D simple harmonic oscillator.) Since a) = a, this is self-adjoint. e) If a particle is in the state ψ, and n is the nth eigenvector of ˆQ corresponding to eigenvalue q n, what is the probability of measuring q? ψ 2.

4 f) Identify: i. Photoelectric effect Photons of wavelength ν are shone on a metal surface, and found to kick out an electron only if ν is greater than some threshold value, independent of intensity of light. Evidence Einstein) for quantum nature of light, E = hν. ii. Davisson-Germer effect Davisson-Germer experiment 1927) proved wave nature of electron hypothesized by de Broglie by diffracting electrons from a crystal. iii. Ehrenfest theorem says that expectation values of quantum mechanical observables obey classical equations of motion. For example, d p dt = ī h [H, p] = ī h i hdv dx = d V dx, where the right hand side is now just the classical force on a particle, i.e. this is Newton s law on the average. iv. Stefan-Boltzmann law Total radiation energy density emitted from a blackbody at temperature T is T Hydrogenic orbitals. An electron moving in the Coulomb field of a proton is in a state described by the wave function ignoring spin) Ψr, θ, φ) = 1 10 [ψ 100 r) + ψ 11 r, θ, φ)] 1) a) What is the expectation value of the energy? H = 1 10 E 1 + 9E ) = 1 Ryd /2 ) = 1 5 Ryd b) What is the expectation value of ˆL 2? L 2 = h2 ) = 1 5 h2 4

5 c) Is the wavefunction an eigenstate of parity? Yes or no? Explain either answer. The parity of the hydrogenic wavefunctions is given by 1) l. Therefore the wavefunction given is an admixture of an even parity and an odd parity wavefunction meaning it is not an eigenstate of the parity operator. d) What is the expectation value of the operator φ in this state? Easy way: recall L z = i h φ. The expectation value of L z is therefore φ = 9/10)i. L z = 1 10 ψ ψ 11, L z ψ ψ 11 )) = 9 h/10,. Electron in hydrogenic state. The electron in a hydrogen atom occupies a state ψ = R 21 r) 1 2 Y Y 1 1 2) where Y 0 1 = 1 1 R 21 r) = 4π cos θ, Y 1 1 = )/2 ) r e r/2a 0, ) 2a 0 a 0 8π sin θeiφ 4) a) What values) could a measurement of the z-component of the orbital angular momentum, ˆL z yield, and what is the probability of each? What is the expectation value of ˆL z in this state? m is either 0 or 1, so L z can be either 0 or h, with probability 1/ or 2/, respectively. L z = 1 = 2 h/, ψ ψ 211, L z ψ ψ 211 ) ) 5

6 b) Calculate the average distance of the electron from the nucleus in this state. r = 1 ψ ψ 211, rψ ψ 211 ) ) = 1 ψ 210, r ψ 210 ) + 2 ψ 211, r ψ 211 ) = )/2 ) ) r 2 dr r 1 1 r 0 2a 0 a 0 = a 0 dy y 5 exp y 24 0 = 5a 0 } {{ } e r/2a 0 c) What is the expectation value of ˆL x in this state? L x = 1 ψ ) L+ + L 2ψ 211, ψ ) 2ψ 211 ) 2 = h ψ ψ 211, 2ψ ψ ) 2ψ = h 2 + 2) = 2 h/ 6 d) If you measured the z-component of the angular momentum and the distance of the electron from the origin r simultaneously, what is the probability density for finding ˆL z with eigenvalue zero at a distance r? P m=0 r) = r 2 1 R2 21r) 6

7 4. Scattering potential. For a 1D potential as shown in the figure and E < V 0, a) write down the Schrödinger equation and its general solution in the regions I,II, and III assuming the particle is incident from the left. Hψ = Eψ, H = h2 2m 2. with ψ I = Ae ipx/ h + Be ipx/ h, ψ II = Ce qx/ h + De qx/ h ψ III = F e ipx/ h p = 2mE q = 2mV 0 E) b) Write down the matching conditions at the boundaries x = a, b. Require continuity of ψ and its derivatives at the interfaces: ψ I a) = ψ II a) ψ II b) = ψ III b) ψ Ia) = ψ IIa) ψ IIb) = ψ IIIb), in other words Ae ipa/ h + Be ipa/ h = Ce qa/ h + De qa/ h Ce qb/ h + De qb/ h = F e ipb/ h Ae ipa/ h Be ipa/ h )ip = Ce qa/ h De qa/ h )q qce qb/ h De qb/ h ) = F ipe ipb/ h 7

8 c) Do not solve for all coefficients, but reduce the problem to a single equation determining the eigenvalues. This is tedious, not a good exam question. 2iap/ h p + iq B/A = e p iq or F/A = 4e ipa/ h e ipb/ h e qa+b)/ h pq iq p)ipe 2qa/ h ipe 2qb/ h + qe 2qa/ h + qe 2qb/ h ) d) Sketch the probability of finding the electron in all three regions. 8

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

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

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

CHAPTER 16: Quantum Mechanics and the Hydrogen Atom

CHAPTER 16: Quantum Mechanics and the Hydrogen Atom CHAPTER 16: Quantum Mechanics and the Hydrogen Atom Waves and Light Paradoxes in Classical Physics Planck, Einstein, and Bohr Waves, Particles, and the Schrödinger equation The Hydrogen Atom Questions

More information

λν = c λ ν Electromagnetic spectrum classification of light based on the values of λ and ν

λν = c λ ν Electromagnetic spectrum classification of light based on the values of λ and ν Quantum Theory and Atomic Structure Nuclear atom small, heavy, positive nucleus surrounded by a negative electron cloud Electronic structure arrangement of the electrons around the nucleus Classical mechanics

More information

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

Spectroscopic Notation

Spectroscopic Notation Spectroscopic Notation Most of the information we have about the universe comes to us via emission lines. However, before we can begin to understand how emission lines are formed and what they tell us,

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

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

Topic 1. Atomic Structure and Periodic Properties

Topic 1. Atomic Structure and Periodic Properties Topic 1 1-1 Atomic Structure and Periodic Properties Atomic Structure 1-2 History Rutherford s experiments Bohr model > Interpretation of hydrogen atom spectra Wave - particle duality Wave mechanics Heisenberg

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

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

Problem Set 1 Solutions

Problem Set 1 Solutions Chemistry 36 Dr. Jean M. Standard Problem Set Solutions. The first 4 lines in the visible region of atomic line spectrum of hydrogen atom occur at wavelengths of 656., 486., 434.0, and 40. nm (this is

More information

Lecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.)

Lecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.) Lecture 18: Quantum Mechanics Reading: Zumdahl 1.5, 1.6 Outline Basic concepts of quantum mechanics and molecular structure A model system: particle in a box. Demos how Q.M. actually obtains a wave function.

More information

Chemistry 417! 1! Fall Chapter 2 Notes

Chemistry 417! 1! Fall Chapter 2 Notes Chemistry 417! 1! Fall 2012 Chapter 2 Notes September 3, 2012! Chapter 2, up to shielding 1. Atomic Structure in broad terms a. nucleus and electron cloud b. nomenclature, so we may communicate c. Carbon-12

More information

CHAPTER 5 THE HARMONIC OSCILLATOR

CHAPTER 5 THE HARMONIC OSCILLATOR CHAPTER 5 THE HARMONIC OSCILLATOR The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment

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

Final Exam, Chem 311, 120 minutes, Dr. H. Guo, Dec. 17, You are allowed to bring a two page sheet containing equations and a calculator

Final Exam, Chem 311, 120 minutes, Dr. H. Guo, Dec. 17, You are allowed to bring a two page sheet containing equations and a calculator Final Exam, Chem 311, 120 minutes, Dr. H. Guo, Dec. 17, 2008 You are allowed to bring a two page sheet containing equations and a calculator I. Answer the following multiple choice questions (5 pts each),

More information

TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points

TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points 1. Check your examination for completeness prior to starting.

More information

AP Chemistry A. Allan Chapter 7 Notes - Atomic Structure and Periodicity

AP Chemistry A. Allan Chapter 7 Notes - Atomic Structure and Periodicity AP Chemistry A. Allan Chapter 7 Notes - Atomic Structure and Periodicity 7.1 Electromagnetic Radiation A. Types of EM Radiation (wavelengths in meters) 10-1 10-10 10-8 4 to 7x10-7 10-4 10-1 10 10 4 gamma

More information

Chemistry 431. NC State University. Lecture 3. The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics

Chemistry 431. NC State University. Lecture 3. The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics Chemistry 431 Lecture 3 The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics NC State University Derivation of the Schrödinger Equation The Schrödinger

More information

HW to be handed in: Extra (do not hand in):

HW to be handed in: Extra (do not hand in): CHEM344 HW#5 Due: Fri, Feb 28@2pm BEFORE CLASS! HW to be handed in: Atkins(9 th ed.) Chapter 7: Exercises: 7.6(b), 7.8(b), 7.10(b), 7.13(b) (moved to HW6), 7.15(b), 7.17(b), Problems: 7.2, 7.6, 7.10, 7.18,

More information

Solved Problems on Quantum Mechanics in One Dimension

Solved Problems on Quantum Mechanics in One Dimension Solved Problems on Quantum Mechanics in One Dimension Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada

More information

Chem 81 Fall, Exam 2 Solutions

Chem 81 Fall, Exam 2 Solutions Chem 81 Fall, 1 Exam Solutions 1. (15 points) Consider the boron atom. (a) What is the ground state electron configuration for B? (If you don t remember where B is in the periodic table, ask me. I ll tell

More information

COLLEGE PHYSICS. Chapter 29 INTRODUCTION TO QUANTUM PHYSICS

COLLEGE PHYSICS. Chapter 29 INTRODUCTION TO QUANTUM PHYSICS COLLEGE PHYSICS Chapter 29 INTRODUCTION TO QUANTUM PHYSICS Quantization: Planck s Hypothesis An ideal blackbody absorbs all incoming radiation and re-emits it in a spectrum that depends only on temperature.

More information

Chapter 11 Modern Atomic Theory

Chapter 11 Modern Atomic Theory Chapter 11 Modern Atomic Theory Rutherford s Atom The concept of a nuclear atom (charged electrons moving around the nucleus) resulted from Ernest Rutherford s experiments. Question left unanswered: how

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

The Electronic Structures of Atoms Electromagnetic Radiation

The Electronic Structures of Atoms Electromagnetic Radiation The Electronic Structures of Atoms Electromagnetic Radiation The wavelength of electromagnetic radiation has the symbol λ. Wavelength is the distance from the top (crest) of one wave to the top of the

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

EUF. Joint Entrance Examination for Postgraduate Courses in Physics

EUF. Joint Entrance Examination for Postgraduate Courses in Physics EUF Joint Entrance Examination for Postgraduate Courses in Physics For the second semester 204 23 April 204 Part Instructions Do not write your name on the test. It should be identified only by your candidate

More information

From Einstein to Klein-Gordon Quantum Mechanics and Relativity

From Einstein to Klein-Gordon Quantum Mechanics and Relativity From Einstein to Klein-Gordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic

More information

Hydrogen Atom. Dragica Vasileska Arizona State University

Hydrogen Atom. Dragica Vasileska Arizona State University Hydrogen Atom Dragica Vasileska Arizona State University Importance of Hydrogen Atom Hydrogen is the simplest atom The quantum numbers used to characterize the allowed states of hydrogen can also be used

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

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

- the total energy of the system is found by summing up (integrating) over all particles n(ε) at different energies ε

- the total energy of the system is found by summing up (integrating) over all particles n(ε) at different energies ε Average Particle Energy in an Ideal Gas - the total energy of the system is found by summing up (integrating) over all particles n(ε) at different energies ε - with the integral - we find - note: - the

More information

CHEM344 HW#7 Due: Fri, Mar BEFORE CLASS!

CHEM344 HW#7 Due: Fri, Mar BEFORE CLASS! CHEM344 HW#7 Due: Fri, Mar 14@2pm BEFORE CLASS! HW to be handed in: Atkins Chapter 8: Exercises: 8.11(b), 8.16(b), 8.19(b), Problems: 8.2, 8.4, 8.12, 8.34, Chapter 9: Exercises: 9.5(b), 9.7(b), Extra (do

More information

Module -1: Quantum Mechanics - 2

Module -1: Quantum Mechanics - 2 Quantum Mechanics - Assignment Question: Module -1 Quantum Mechanics Module -1: Quantum Mechanics - 01. (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization

More information

Quantum Mechanics I Physics 325. Importance of Hydrogen Atom

Quantum Mechanics I Physics 325. Importance of Hydrogen Atom Quantum Mechanics I Physics 35 Atomic spectra and Atom Models Importance of Hydrogen Atom Hydrogen is the simplest atom The quantum numbers used to characterize the allowed states of hydrogen can also

More information

Atomic Spectra and Energy Levels. Atomic Spectra

Atomic Spectra and Energy Levels. Atomic Spectra Atomic Spectra and Energy Levels Atomic Spectra Excited atoms emit light (neon signs, etc.) Emission from different elements is different colors. Emission of only certain wavelengths Spectral lines Existence

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

An Introduction to Quantum Cryptography

An Introduction to Quantum Cryptography An Introduction to Quantum Cryptography J Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu An Introduction to Quantum Cryptography p.1 Acknowledgments Quantum

More information

Quantum interference with slits

Quantum interference with slits 1 Quantum interference with slits Thomas V Marcella Department of Physics and Applied Physics, University of Massachusetts Lowell, Lowell, MA 01854-2881, USA E-mail: thomasmarcella@verizon.net In the experiments

More information

Chapter 38C - Atomic Physics. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 38C - Atomic Physics. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 38C - Atomic Physics A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 007 Objectives: After completing this module, you should be able to:

More information

Physics 53. Wave Motion 1

Physics 53. Wave Motion 1 Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can

More information

Chapter 35: Quantum Physics

Chapter 35: Quantum Physics Newton himself was better aware of the weakness inherent in his intellectual edifice than the generations which followed him. This fact has always aroused my admiration. Albert Einstein 35.1 The Particle

More information

Practice questions for Ch. 7

Practice questions for Ch. 7 Name: Class: Date: ID: A Practice questions for Ch. 7 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. When ignited, a uranium compound burns with a green

More information

Arrangement of Electrons in Atoms

Arrangement of Electrons in Atoms CHAPTER 4 PRE-TEST Arrangement of Electrons in Atoms In the space provided, write the letter of the term that best completes each sentence or best answers each question. 1. Which of the following orbital

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

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 31 Atomic Physics 31.1 Early Model of the Atom 31.2 The Spectrum of Atomic Hydrogen 31.3 Bohr s Model of the Hydrogen Atom 31.

Chapter 31 Atomic Physics 31.1 Early Model of the Atom 31.2 The Spectrum of Atomic Hydrogen 31.3 Bohr s Model of the Hydrogen Atom 31. Chapter 31 Atomic Physics 31.1 Early Model of the Atom 31.2 The Spectrum of Atomic Hydrogen 31.3 Bohr s Model of the Hydrogen Atom 31.4 de Broglie Waves and the Bohr Model 31.5 The Quantum Mechanical Hydrogen

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

Thomson and Rayleigh Scattering

Thomson and Rayleigh Scattering Thomson and Rayleigh Scattering In this and the next several lectures, we re going to explore in more detail some specific radiative processes. The simplest, and the first we ll do, involves scattering.

More information

THE NATURE OF THE ATOM

THE NATURE OF THE ATOM CHAPTER 30 THE NATURE OF THE ATOM CONCEPTUAL QUESTIONS 1. REASONING AND SOLUTION A tube is filled with atomic hydrogen at room temperature. Electromagnetic radiation with a continuous spectrum of wavelengths,

More information

Lecture Outlines Chapter 31. Physics, 3 rd Edition James S. Walker

Lecture Outlines Chapter 31. Physics, 3 rd Edition James S. Walker Lecture Outlines Chapter 31 Physics, 3 rd Edition James S. Walker 2007 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the use of instructors in

More information

Notes on wavefunctions III: time dependence and the Schrödinger equation

Notes on wavefunctions III: time dependence and the Schrödinger equation Notes on wavefunctions III: time dependence and the Schrödinger equation We now understand that the wavefunctions for traveling particles with momentum p look like wavepackets with wavelength λ = h/p,

More information

Introduction to quantum mechanics

Introduction to quantum mechanics Introduction to quantum mechanics Lecture 3 MTX9100 Nanomaterjalid OUTLINE -What is electron particle or wave? - How large is a potential well? -What happens at nanoscale? What is inside? Matter Molecule

More information

Exam 2 Solutions Chem 6, 9 Section, Spring 2002

Exam 2 Solutions Chem 6, 9 Section, Spring 2002 1. Dartmouth s FM radio station, WDCR, broadcasts by emitting from its antenna photons of frequency 99.3 MHz (99.3 10 6 Hz). (a) What is the energy of a single WDCR photon? The photon energy is simply

More information

The Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933

The Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933 The Schrödinger Equation Erwin Schrödinger 1887-1961 Nobel Prize in Physics 1933 The Schrödinger Wave Equation The Schrödinger wave equation in its time-dependent form for a particle of energy E moving

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

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

ABC Math Student Copy

ABC Math Student Copy Page 1 of 8 Line Spectra Physics Week 15(Sem. 2) Name The Atom Chapter Summary From the last section, we know that all objects emit electromagnetic waves. For a solid object, such as the filament of a

More information

Thomson and Rayleigh Scattering

Thomson and Rayleigh Scattering Thomson and Rayleigh Scattering Initial questions: What produces the shapes of emission and absorption lines? What information can we get from them regarding the environment or other conditions? In this

More information

ψ 2 ψ 3 h 2 d 2 2m dx 0 x > a V (x) = V0 x a x < a = B sin kx or B'cos kx a x a = Ce κx x > a k =, κ = h Since the solution has to be odd, ψ 2

ψ 2 ψ 3 h 2 d 2 2m dx 0 x > a V (x) = V0 x a x < a = B sin kx or B'cos kx a x a = Ce κx x > a k =, κ = h Since the solution has to be odd, ψ 2 .0 Proble Set 3 Solution.Solution The energy eigenvalue proble for the given syste is: Ĥψ = Eψ h d Ĥ = + V (x) dx 0 x > a V (x) = V0 x a For bounded particle, the solution is: ψ = Ae κx x < a ψ = B sin

More information

Atomic Theory and the Periodic Table

Atomic Theory and the Periodic Table Atomic Theory and the Periodic Table Petrucci, Harwood and Herring: Chapters 9 and 10 Aims: To examine the Quantum Theory, to understand the electronic structure of elements, To explain the periodic table

More information

Chapter 27 Early Quantum Physics and the Photon

Chapter 27 Early Quantum Physics and the Photon Chapter 27 Early Quantum Physics and the Photon 1. A problem with the classical theory for radiation from a blackbody was that the theory predicted too much radiation in the wavelengths. A. ultraviolet

More information

UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet

UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet UNIVERSITETET I OSLO Det matematisk-naturvitenskapelige fakultet Solution for take home exam: FYS311, Oct. 7, 11. 1.1 The Hamiltonian of a charged particle in a weak magnetic field is Ĥ = P /m q mc P A

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

Optical Spectroscopy and Atomic Structure. PHYS 0212 Optical Spectroscopy and Atomic Structure 1

Optical Spectroscopy and Atomic Structure. PHYS 0212 Optical Spectroscopy and Atomic Structure 1 Optical Spectroscopy and Atomic Structure PHYS 0212 Optical Spectroscopy and Atomic Structure 1 Optical Spectroscopy and Atomic Structure This experiment has four parts: 1. Spectroscope Setup - Your lab

More information

QUANTUM-MECHANICAL MODEL OF THE ATOM. Quantum Mechanics?

QUANTUM-MECHANICAL MODEL OF THE ATOM. Quantum Mechanics? QUANTUM-MECHANICAL MODEL OF THE ATOM GENERAL CHEMISTRY by Dr. Istadi 1 Quantum Mechanics? Dual nature of matter and energy The uncertainty principle The wave nature of objects on the atomic scale Quantum

More information

CHAPTER 4 Structure of the Atom

CHAPTER 4 Structure of the Atom CHAPTER 4 Structure of the Atom 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydrogen Atom 4.5 Successes and Failures

More information

2. Wavefunctions. An example wavefunction

2. Wavefunctions. An example wavefunction 2. Wavefunctions Copyright c 2015 2016, Daniel V. Schroeder To create a precise theory of the wave properties of particles and of measurement probabilities, we introduce the concept of a wavefunction:

More information

5.111 Principles of Chemical Science

5.111 Principles of Chemical Science MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.111 Lecture Summary

More information

DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS

DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS DO PHYSICS ONLINE FROM QUANTA TO QUARKS QUANTUM (WAVE) MECHANICS Quantum Mechanics or wave mechanics is the best mathematical theory used today to describe and predict the behaviour of particles and waves.

More information

2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion.

2. The graph shows how the displacement varies with time for an object undergoing simple harmonic motion. Practice Test: 29 marks (37 minutes) Additional Problem: 31 marks (45 minutes) 1. A transverse wave travels from left to right. The diagram on the right shows how, at a particular instant of time, the

More information

Quantum Theory of the Hydrogen Atom

Quantum Theory of the Hydrogen Atom Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of

More information

PHY 481/581 Intro Nano-Materials Science and engineering: Some Basics of Quantum Mechanics

PHY 481/581 Intro Nano-Materials Science and engineering: Some Basics of Quantum Mechanics PHY 481/581 Intro Nano-Materials Science and engineering: Some Basics of Quantum Mechanics http://creativecommons.org/licenses/by-nc-sa/2.5/ 1 Basics of Quantum Mechanics - Why Quantum Physics? - Experimental

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

The Bohr atom and the Uncertainty Principle

The Bohr atom and the Uncertainty Principle The Bohr atom and the Uncertainty Principle Previous Lecture: Matter waves and De Broglie wavelength The Bohr atom This Lecture: More on the Bohr Atom The H atom emission and absorption spectra Uncertainty

More information

Chapter 1: Introduction to Quantum Physics

Chapter 1: Introduction to Quantum Physics Chapter 1: Introduction to Quantum Physics Luis M. Molina Departamento de Física Teórica, Atómica y Óptica Quantum Physics Luis M. Molina (FTAO) Chapter 1: Introduction to Quantum Physics Quantum Physics

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

Chapter 29: Atomic Structure. What will we learn in this chapter?

Chapter 29: Atomic Structure. What will we learn in this chapter? Chapter 29: Atomic Structure What will we learn in this chapter? Contents: Electrons in atoms Wave functions Electron spin Pauli exclusion principle Atomic structure Periodic table W. Pauli & N. Bohr Both

More information

Lecture 18 Time-dependent perturbation theory

Lecture 18 Time-dependent perturbation theory Lecture 18 Time-dependent perturbation theory Time-dependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are time-independent. In such cases,

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

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5

8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology. Problem Set 5 8.04: Quantum Mechanics Professor Allan Adams Massachusetts Institute of Technology Tuesday March 5 Problem Set 5 Due Tuesday March 12 at 11.00AM Assigned Reading: E&R 6 9, App-I Li. 7 1 4 Ga. 4 7, 6 1,2

More information

C10J ATOMIC STRUCTURE (6 lectures) Introduction The Atomic Structure course is considered as an important part of the core course for Introductory Chemistry as concepts which are learnt here will be employed

More information

Atomic Structure Ron Robertson

Atomic Structure Ron Robertson Atomic Structure Ron Robertson r2 n:\files\courses\1110-20\2010 possible slides for web\atomicstructuretrans.doc I. What is Light? Debate in 1600's: Since waves or particles can transfer energy, what is

More information

PRACTICE EXAM IV P202 SPRING 2004

PRACTICE EXAM IV P202 SPRING 2004 PRACTICE EXAM IV P202 SPRING 2004 1. In two separate double slit experiments, an interference pattern is observed on a screen. In the first experiment, violet light (λ = 754 nm) is used and a second-order

More information

Models of an atom and old quantum theory

Models of an atom and old quantum theory Models of an atom and old quantum theory Classical models of atoms Thompson's model Chemical elements dier by the number Z of electrons in their atoms. Atoms are electrically neutral, so that the charge

More information

Physics Formulae for GREs. 1 Mechanics. Gravitational potential energy. V = GMm. On Earth s surface, Kinematics (constant acceleration) V = mgh

Physics Formulae for GREs. 1 Mechanics. Gravitational potential energy. V = GMm. On Earth s surface, Kinematics (constant acceleration) V = mgh Physics Formulae for GREs Mechanics Kinematics constant acceleration) s = ut + at v = u + at v = u + as Newton s Second Law F = m a Gravitational potential energy On Earth s surface, Kinetic energy V =

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

Figure 1. Visible spectrum of atomic hydrogen.

Figure 1. Visible spectrum of atomic hydrogen. Chapter 7 THE HYDROGEN ATOM; ATOMIC ORBITALS Atomic Spectra When gaseous hydrogen in a glass tube is excited by a 5-volt electrical discharge, four lines are observed in the visible part of the emission

More information

hypothesis of Louis de Broglie (1924): particles may have wave-like properties

hypothesis of Louis de Broglie (1924): particles may have wave-like properties Wave properties of particles hypothesis of Louis de Broglie (1924): particles may have wave-like properties note: it took almost 20 years after noting that waves have particle like properties that particles

More information

Lecture 1: Microscopic Theory of Radiation

Lecture 1: Microscopic Theory of Radiation 253a: QFT Fall 2009 Matthew Schwartz Lecture : Microscopic Theory of Radiation Blackbody Radiation Quantum Mechanics began on October 9, 900 with Max Planck s explanation of the blackbody radiation spectrum.

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

1. Thegroundstatewavefunctionforahydrogenatomisψ 0 (r) =

1. Thegroundstatewavefunctionforahydrogenatomisψ 0 (r) = CHEM 5: Examples for chapter 1. 1. Thegroundstatewavefunctionforahydrogenatomisψ (r = 1 e r/a. πa (a What is the probability for finding the electron within radius of a from the nucleus? (b Two excited

More information

Chapter 9 Statistical Mechanics

Chapter 9 Statistical Mechanics Chapter 9 Statistical Mechanics 9. Statistical Distributions This first section is just an overview. Statistical mechanics deals with the behavior of systems of a large number of particles. It does this

More information

The Phenomenon of Photoelectric Emission:

The Phenomenon of Photoelectric Emission: The Photoelectric Effect. The Wave particle duality of light Light, like any other E.M.R (electromagnetic radiation) has got a dual nature. That is there are experiments that prove that it is made up of

More information

Lecture 12 Quantum Mechanics and Atomic Orbitals

Lecture 12 Quantum Mechanics and Atomic Orbitals Lecture 12 Quantum Mechanics and Atomic Orbitals Bohr and Einstein demonstrated the particle nature of light.e = hν. De Broglie demonstrated the wavelike properties of particles. λ = h/mv. However, these

More information

Multi-electron atoms

Multi-electron atoms Multi-electron atoms Today: Using hydrogen as a model. The Periodic Table HWK 13 available online. Please fill out the online participation survey. Worth 10points on HWK 13. Final Exam is Monday, Dec.

More information