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

Size: px
Start display at page:

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

Transcription

1 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 the Bohr energy levels of the Hydrogen atom are E n = m h n ( e 4πɛ ) () Compare the wavelengths of the p s transitions in i. hydrogen The energy which must be carried away by a photon in the transition is E E = hc/λ, so λ H = hc/(e E ) = 8 h3 3m ( 4πɛ e ) ii. deuterium (mass of nucleus that of H). The mass which appears in the Bohr formula is the electron mass m, which is replaced by the reduced mass µ = mm p /(m+m p ) if one allows for the proton motion. Changing m p m p doesn t change much, so to a good approximation the photon wavelength doesn t change, λ D λ H. iii. positronium. In this case the positron is not heavier than the electron, but in fact has the same mass. The reduced mass of identical particles of mass m is m/, so we obtain the correct answer from the Hydrogen expression by substituting m m/, or λ pos = λ H. (b) An electron is in the ground state of tritium, a form of heavy hydrogen where the nucleus has two neutrons in addition to a proton. A nuclear reaction now changes the nucleus instantaneously to 3 He, i.e. two protons

2 and a neutron. Calculate the probability that the electron remains in the ground state of 3 He. You may need ψ = (/ 4πa 3 ) exp r/a, with a = h /(mze ), where Z is the nuclear charge. Z, the nuclear charge or number of protons, is for tritium and for 3 He. Thus the Bohr radius for 3 He is twice as small as for tritium or hydrogen, a 3 He at. The electron in the 3 He ion is more tightly bound to the nucleus. We re told the system starts out in the tritium ground state, ψ t = e r/at 4π(a H ) 3 which can be expressed as a linear combination of any complete set of states in Hilbert space, for example the eigenstates of the 3 He Hamiltonian, ψ t = aψ 3 He + bψ 3 He + cψ 3 He +... meaning the electron in the tritium ground state will be found after a measurement in the 3 He ground state with probability amplitude a = ψ t ψ 3 He = 4 4π = (a t ) 3 (a 3 He ) 3 d 3 r e r 4 8 dr r e 3r/at (a t ) 3 = 4 8 and probability = a =.7. dy y e 3y = 6 7, ( a t + a 3 He =.838 (c) An electron moving in the Coulomb field of a proton is in a state described by the wave function Ψ = 6 [4ψ + 3ψ ψ + ψ ] () i. What is the expectation value of the energy? ) Ψ H Ψ = (6 H + 9 H + H 36 + H ) = 36 (6E + 9E + E + E ) = 36 (6E + E ). Note this is an example of general rule ψ O ψ = n o n c n, where the o n are the eigenvalues of the Hermitian operator O, and the c n the expansion coefficients of ψ in the basis of O eigenstates.

3 ii. What is the expectation value of ˆL? ψ L ψ = h l(l + ) c nlm = h h ( ) = nlm 36 9 iii. What is the expectation value of ˆL z? ψ L z ψ = hm c nlm = h nlm 36 ( ( )) = h 36 (d) How large would a constant magnetic field have to be to split two H-atom states which are degenerate in zero field by an amount so as to maximally absorb light of wavelength λ? H = µ B = e m S B = = e m S zb z = e m hm sb z, where the last step where the operator is replaced by its eigenvalues holds only when applied to S z eigenstates, and where I have used m s for the S z quantum number. The two S z states have a difference of m s =, so the energy of the photon produced must be e m hb z hc λ, or λ = (πmc/(eb z)). (e) For two particles a and b such that l a = and l b =, argue that it must be true that l =, m = = α m a =, m b = + β m a =, m b =, where l, m are the quantum numbers corresponding to total angular momentum L = L a + L b, and find the coefficients α and β. Hint: use L ± lm = l(l + ) m(m ± ) lm ±. a) states given are the only possible ones with m = m a + m b and m a l a, m b l b. b) Start at top of angular momentum ladder, where we know there is only one possible lm = l state, equal to the one possible m a = l a, m b = l b, then apply lowering operator as in HW, remembering square root factors to keep states normalized: L lm = (L a + L b ) mam b 3 = ( + ) 3

4 So divide by to get = ( + ).. Angular momentum. Consider an angular momentum system. (a) What are the possible eigenvalues of L and L z corresponding to the eigenvectors l =, m? L m = h L z m = hm (b) In the basis where the eigenvectors l =, m of the operator ˆL z are given by (,, ), (,, ) and (,, ), construct the matrix representation of the operator ˆL x (Hint: you will need to calculate the matrix elements m ˆL x m.) Write L x = (L + + L )/, find, e.g. L + + L = L + + L = h L + + L =... where I ve used the effect of L ± acting on m, and the orthonormality of the m. It is a little tedious to do them all, but eventually we find (c) Find the eigenvectors of ˆL x. L x = h Now we just have a matrix eigenvalue problem, and I assume you can find the eigenvectors, then normalize them. The results are = / / / = / / = / / / 4

5 (d) If a system is prepared in the state vector ψ = 6 4 3, (3) what is the probability that a measurement of ˆL x yields the value? P = ψ = 4/3 3. Pauli principle. Consider two electrons described by the Hamiltonian where H = ˆp m + ˆp m + V (x ) + V (x ) (4) x < a/ V (x) = a/ x a/ x > a/ Assume both electrons are in same spin state. (a) What is the lowest (ground state) energy? (b) What is the energy eigenfunction for this ground state? Parts a) and b): Denote the single-particle eigenfunctions of the ordinary infinite square well by ψ, ψ,... The two-particle ground state must be antisymmetric under exchange of all particle labels according to Pauli, but we are told that the spins are both the same, e.g. up. The wave function must be (5) Ψ (, ) = (ψ (x )ψ (x ) ψ (x )ψ (x ))χ From our theorem about additive Hamiltonians, the energy of this state is the sum of the energies of the single particle states and : E ground = E + E = h (π/a) m + h (π/a) m = 5 h (π/a) m (c) What is the energy and the wave function of the first excited state (still with equal-spin condition!)? Ψ = (ψ ψ ψ ψ )χ E exc, = E + E = h (π/a) m + h (3π/a) m = 5 h (π/a) m 5

### 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,

### 5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM

5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,

### Masses in Atomic Units

Nuclear Composition - the forces binding protons and neutrons in the nucleus are much stronger (binding energy of MeV) than the forces binding electrons to the atom (binding energy of ev) - the constituents

### Basic Nuclear Concepts

Section 7: In this section, we present a basic description of atomic nuclei, the stored energy contained within them, their occurrence and stability Basic Nuclear Concepts EARLY DISCOVERIES [see also Section

### Review of the isotope effect in the hydrogen spectrum

Review of the isotope effect in the hydrogen spectrum 1 Balmer and Rydberg Formulas By the middle of the 19th century it was well established that atoms emitted light at discrete wavelengths. This is in

### Photons. ConcepTest 27.1. 1) red light 2) yellow light 3) green light 4) blue light 5) all have the same energy. Which has more energy, a photon of:

ConcepTest 27.1 Photons Which has more energy, a photon of: 1) red light 2) yellow light 3) green light 4) blue light 5) all have the same energy 400 nm 500 nm 600 nm 700 nm ConcepTest 27.1 Photons Which

### 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.

### Basic Concepts in Nuclear Physics

Basic Concepts in Nuclear Physics Paolo Finelli Corso di Teoria delle Forze Nucleari 2011 Literature/Bibliography Some useful texts are available at the Library: Wong, Nuclear Physics Krane, Introductory

### Nuclear Physics. Nuclear Physics comprises the study of:

Nuclear Physics Nuclear Physics comprises the study of: The general properties of nuclei The particles contained in the nucleus The interaction between these particles Radioactivity and nuclear reactions

### Objectives. PAM1014 Introduction to Radiation Physics. Constituents of Atoms. Atoms. Atoms. Atoms. Basic Atomic Theory

PAM1014 Introduction to Radiation Physics Basic Atomic Theory Objectives Introduce and Molecules The periodic Table Electronic Energy Levels Atomic excitation & de-excitation Ionisation Molecules Constituents

### Atomic Structure: Chapter Problems

Atomic Structure: Chapter Problems Bohr Model Class Work 1. Describe the nuclear model of the atom. 2. Explain the problems with the nuclear model of the atom. 3. According to Niels Bohr, what does n stand

### Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 7 : Symmetries and the Quark Model. Introduction/Aims

Particle Physics Michaelmas Term 2011 Prof Mark Thomson Handout 7 : Symmetries and the Quark Model Prof. M.A. Thomson Michaelmas 2011 206 Introduction/Aims Symmetries play a central role in particle physics;

### 1 Lecture 3: Operators in Quantum Mechanics

1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators

### 2, 8, 20, 28, 50, 82, 126.

Chapter 5 Nuclear Shell Model 5.1 Magic Numbers The binding energies predicted by the Liquid Drop Model underestimate the actual binding energies of magic nuclei for which either the number of neutrons

### CHEM 1411 Chapter 5 Homework Answers

1 CHEM 1411 Chapter 5 Homework Answers 1. Which statement regarding the gold foil experiment is false? (a) It was performed by Rutherford and his research group early in the 20 th century. (b) Most of

### Chapter 18: The Structure of the Atom

Chapter 18: The Structure of the Atom 1. For most elements, an atom has A. no neutrons in the nucleus. B. more protons than electrons. C. less neutrons than electrons. D. just as many electrons as protons.

### 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

### 13- What is the maximum number of electrons that can occupy the subshell 3d? a) 1 b) 3 c) 5 d) 2

Assignment 06 A 1- What is the energy in joules of an electron undergoing a transition from n = 3 to n = 5 in a Bohr hydrogen atom? a) -3.48 x 10-17 J b) 2.18 x 10-19 J c) 1.55 x 10-19 J d) -2.56 x 10-19

### 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.

### Atomic Calculations. 2.1 Composition of the Atom. number of protons + number of neutrons = mass number

2.1 Composition of the Atom Atomic Calculations number of protons + number of neutrons = mass number number of neutrons = mass number - number of protons number of protons = number of electrons IF positive

### Wave Function, ψ. Chapter 28 Atomic Physics. The Heisenberg Uncertainty Principle. Line Spectrum

Wave Function, ψ Chapter 28 Atomic Physics The Hydrogen Atom The Bohr Model Electron Waves in the Atom The value of Ψ 2 for a particular object at a certain place and time is proportional to the probability

### Lecture 5 Motion of a charged particle in a magnetic field

Lecture 5 Motion of a charged particle in a magnetic field Charged particle in a magnetic field: Outline 1 Canonical quantization: lessons from classical dynamics 2 Quantum mechanics of a particle in a

### AP* Atomic Structure & Periodicity Free Response Questions KEY page 1

AP* Atomic Structure & Periodicity ree Response Questions KEY page 1 1980 a) points 1s s p 6 3s 3p 6 4s 3d 10 4p 3 b) points for the two electrons in the 4s: 4, 0, 0, +1/ and 4, 0, 0, - 1/ for the three

### An Introduction to Hartree-Fock Molecular Orbital Theory

An Introduction to Hartree-Fock Molecular Orbital Theory C. David Sherrill School of Chemistry and Biochemistry Georgia Institute of Technology June 2000 1 Introduction Hartree-Fock theory is fundamental

### Quantum Mechanics: Postulates

Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, x-ray, etc.) through

### 5.61 Fall 2012 Lecture #19 page 1

5.6 Fall 0 Lecture #9 page HYDROGEN ATOM Consider an arbitrary potential U(r) that only depends on the distance between two particles from the origin. We can write the Hamiltonian simply ħ + Ur ( ) H =

### 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

### Homework #10 (749508)

Homework #10 (749508) Current Score: 0 out of 100 Description Homework on quantum physics and radioactivity Instructions Answer all the questions as best you can. 1. Hewitt10 32.E.001. [481697] 0/5 points

### Name Date Class ELECTRONS IN ATOMS. Standard Curriculum Core content Extension topics

13 ELECTRONS IN ATOMS Conceptual Curriculum Concrete concepts More abstract concepts or math/problem-solving Standard Curriculum Core content Extension topics Honors Curriculum Core honors content Options

### MASTER OF SCIENCE IN PHYSICS MASTER OF SCIENCES IN PHYSICS (MS PHYS) (LIST OF COURSES BY SEMESTER, THESIS OPTION)

MASTER OF SCIENCE IN PHYSICS Admission Requirements 1. Possession of a BS degree from a reputable institution or, for non-physics majors, a GPA of 2.5 or better in at least 15 units in the following advanced

### Till now, almost all attention has been focussed on discussing the state of a quantum system.

Chapter 13 Observables and Measurements in Quantum Mechanics Till now, almost all attention has been focussed on discussing the state of a quantum system. As we have seen, this is most succinctly done

### NMR - Basic principles

NMR - Basic principles Subatomic particles like electrons, protons and neutrons are associated with spin - a fundamental property like charge or mass. In the case of nuclei with even number of protons

### Level 3 Achievement Scale

Unit 1: Atoms Level 3 Achievement Scale Can state the key results of the experiments associated with Dalton, Rutherford, Thomson, Chadwick, and Bohr and what this lead each to conclude. Can explain that

### 0.1 Phase Estimation Technique

Phase Estimation In this lecture we will describe Kitaev s phase estimation algorithm, and use it to obtain an alternate derivation of a quantum factoring algorithm We will also use this technique to design

### Free Electron Fermi Gas (Kittel Ch. 6)

Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)

### CHAPTER 13 MOLECULAR SPECTROSCOPY

CHAPTER 13 MOLECULAR SPECTROSCOPY Our most detailed knowledge of atomic and molecular structure has been obtained from spectroscopy study of the emission, absorption and scattering of electromagnetic radiation

### Main properties of atoms and nucleus

Main properties of atoms and nucleus. Atom Structure.... Structure of Nuclei... 3. Definition of Isotopes... 4. Energy Characteristics of Nuclei... 5. Laws of Radioactive Nuclei Transformation... 3. Atom

### Chemistry 1000 Lecture 2: Nuclear reactions and radiation. Marc R. Roussel

Chemistry 1000 Lecture 2: Nuclear reactions and radiation Marc R. Roussel Nuclear reactions Ordinary chemical reactions do not involve the nuclei, so we can balance these reactions by making sure that

### Operator methods in quantum mechanics

Chapter 3 Operator methods in quantum mechanics While the wave mechanical formulation has proved successful in describing the quantum mechanics of bound and unbound particles, some properties can not be

### [1] Diagonal factorization

8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

### Time dependence in quantum mechanics Notes on Quantum Mechanics

Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:37-05:00 Copyright 2003 Dan

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### Lecture 12 Atomic structure

Lecture 12 Atomic structure Atomic structure: background Our studies of hydrogen-like atoms revealed that the spectrum of the Hamiltonian, Ĥ 0 = ˆp2 2m 1 Ze 2 4πɛ 0 r is characterized by large n 2 -fold

### Fundamental Quantum Mechanics for Engineers

Fundamental Quantum Mechanics for Engineers Leon van Dommelen 5/5/07 Version 3.1 beta 3. ii Dedication To my parents iii iv Preface Why Another Book on Quantum Mechanics? This document was written because

### Chem 1A Exam 2 Review Problems

Chem 1A Exam 2 Review Problems 1. At 0.967 atm, the height of mercury in a barometer is 0.735 m. If the mercury were replaced with water, what height of water (in meters) would be supported at this pressure?

### Teoretisk Fysik KTH. Advanced QM (SI2380), test questions 1

Teoretisk Fysik KTH Advanced QM (SI238), test questions NOTE THAT I TYPED THIS IN A HURRY AND TYPOS ARE POSSIBLE: PLEASE LET ME KNOW BY EMAIL IF YOU FIND ANY (I will try to correct typos asap - if you

### Weak decays of H like 140 Pr 58+ and He like 140 Pr 57+ ions

Weak decays of H like 40 Pr 58+ and He like 40 Pr 57+ ions A. N. Ivanov a M. Faber a R. Reda c P. Kienle bc arxiv:07.384v [nucl-th] 0 Feb 008 December 03 a Atominstitut der Österreichischen Universitäten

### Atomic structure. Chapter 9

Chapter 9 Atomic structure Previously, we have seen that the quantum mechanics of atomic hydrogen, and hydrogen-like atoms is characterized by a large degeneracy with eigenvalues separating into multiplets

### Atoms and Elements. Outline Atoms Orbitals and Energy Levels Periodic Properties Homework

Atoms and the Periodic Table The very hot early universe was a plasma with cationic nuclei separated from negatively charged electrons. Plasmas exist today where the energy of the particles is very high,

### Lecture 3: Optical Properties of Bulk and Nano. 5 nm

Lecture 3: Optical Properties of Bulk and Nano 5 nm First H/W#1 is due Sept. 10 Course Info The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model)

α α λ α = = λ λ α ψ = = α α α λ λ ψ α = + β = > θ θ β > β β θ θ θ β θ β γ θ β = γ θ > β > γ θ β γ = θ β = θ β = θ β = β θ = β β θ = = = β β θ = + α α α α α = = λ λ λ λ λ λ λ = λ λ α α α α λ ψ + α =

### Proton Nuclear Magnetic Resonance Spectroscopy

Proton Nuclear Magnetic Resonance Spectroscopy Introduction: The NMR Spectrum serves as a great resource in determining the structure of an organic compound by revealing the hydrogen and carbon skeleton.

### Concepts in Theoretical Physics

Concepts in Theoretical Physics Lecture 6: Particle Physics David Tong e 2 The Structure of Things 4πc 1 137 e d ν u Four fundamental particles Repeated twice! va, 9608085, 9902033 Four fundamental forces

### Theory of electrons and positrons

P AUL A. M. DIRAC Theory of electrons and positrons Nobel Lecture, December 12, 1933 Matter has been found by experimental physicists to be made up of small particles of various kinds, the particles of

### Chemistry 2 Chapter 13: Electrons in Atoms Please do not write on the test Use an answer sheet! 1 point/problem 45 points total

Chemistry 2 Chapter 13: Electrons in Atoms Please do not write on the test Use an answer sheet! 1 point/problem 45 points total 1. Calculate the energy in joules of a photon of red light that has a frequency

### EQUATION OF STATE. e (E µ)/kt ± 1 h 3 dp,

EQUATION OF STATE Consider elementary cell in a phase space with a volume x y z p x p y p z = h 3, (st.1) where h = 6.63 1 7 erg s is the Planck constant, x y z is volume in ordinary space measured in

### Chapter 9 Unitary Groups and SU(N)

Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three

### Lecture 3: Optical Properties of Bulk and Nano. 5 nm

Lecture 3: Optical Properties of Bulk and Nano 5 nm The Previous Lecture Origin frequency dependence of χ in real materials Lorentz model (harmonic oscillator model) 0 e - n( ) n' n '' n ' = 1 + Nucleus

### Quantum Mechanics and Representation Theory

Quantum Mechanics and Representation Theory Peter Woit Columbia University Texas Tech, November 21 2013 Peter Woit (Columbia University) Quantum Mechanics and Representation Theory November 2013 1 / 30

### Department of Physics and Geology The Elements and the Periodic Table

Department of Physics and Geology The Elements and the Periodic Table Physical Science 1422 Equipment Needed Qty Periodic Table 1 Part 1: Background In 1869 a Russian chemistry professor named Dmitri Mendeleev

### CHAPTER 9 ATOMIC STRUCTURE AND THE PERIODIC LAW

CHAPTER 9 ATOMIC STRUCTURE AND THE PERIODIC LAW Quantum mechanics can account for the periodic structure of the elements, by any measure a major conceptual accomplishment for any theory. Although accurate

### Instructors Guide: Atoms and Their Isotopes

Instructors Guide: Atoms and Their Isotopes Standards Connections Connections to NSTA Standards for Science Teacher Preparation C.3.a.1 Fundamental structures of atoms and molecules. C.3.b.27 Applications

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### = N 2 = 3π2 n = k 3 F. The kinetic energy of the uniform system is given by: 4πk 2 dk h2 k 2 2m. (2π) 3 0

Chapter 1 Thomas-Fermi Theory The Thomas-Fermi theory provides a functional form for the kinetic energy of a non-interacting electron gas in some known external potential V (r) (usually due to impurities)

### 1. Degenerate Pressure

. Degenerate Pressure We next consider a Fermion gas in quite a different context: the interior of a white dwarf star. Like other stars, white dwarfs have fully ionized plasma interiors. The positively

### MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).

MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0

### Sample Exercise 6.1 Concepts of Wavelength and Frequency

Sample Exercise 6.1 Concepts of Wavelength and Frequency Two electromagnetic waves are represented in the margin. (a) Which wave has the higher frequency? (b) If one wave represents visible light and the

WAVES AND ELECTROMAGNETIC RADIATION All waves are characterized by their wavelength, frequency and speed. Wavelength (lambda, ): the distance between any 2 successive crests or troughs. Frequency (nu,):

### Review of Statistical Mechanics

Review of Statistical Mechanics 3. Microcanonical, Canonical, Grand Canonical Ensembles In statistical mechanics, we deal with a situation in which even the quantum state of the system is unknown. The

### 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.

### Lecture 3 September 14, 2009 Atomic Models: Rutherford & Bohr

Welcome to 3.091 Lecture 3 September 14, 2009 Atomic Models: Rutherford & Bohr 1 Periodic Table Quiz 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### The Quantum Harmonic Oscillator Stephen Webb

The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems

### Three Pictures of Quantum Mechanics. Thomas R. Shafer April 17, 2009

Three Pictures of Quantum Mechanics Thomas R. Shafer April 17, 2009 Outline of the Talk Brief review of (or introduction to) quantum mechanics. 3 different viewpoints on calculation. Schrödinger, Heisenberg,

### Auger width of metastable states in antiprotonic helium II

«Избранные вопросы теоретической физики и астрофизики». Дубна: ОИЯИ, 2003. С. 153 158. Auger width of metastable states in antiprotonic helium II J. Révai a and A. T. Kruppa b a Research Institute for

### Basics of Nuclear Physics and Fission

Basics of Nuclear Physics and Fission A basic background in nuclear physics for those who want to start at the beginning. Some of the terms used in this factsheet can be found in IEER s on-line glossary.

### The rate of change of velocity with respect to time. The average rate of change of distance/displacement with respect to time.

H2 PHYSICS DEFINITIONS LIST Scalar Vector Term Displacement, s Speed Velocity, v Acceleration, a Average speed/velocity Instantaneous Velocity Newton s First Law Newton s Second Law Newton s Third Law

### Electric Dipole moments as probes of physics beyond the Standard Model

Electric Dipole moments as probes of physics beyond the Standard Model K. V. P. Latha Non-Accelerator Particle Physics Group Indian Institute of Astrophysics Plan of the Talk Parity (P) and Time-reversal

### The quantum mechanics of particles in a periodic potential: Bloch s theorem

Handout 2 The quantum mechanics of particles in a periodic potential: Bloch s theorem 2.1 Introduction and health warning We are going to set up the formalism for dealing with a periodic potential; this

### Quantum Physics II (8.05) Fall 2013 Assignment 4

Quantum Physics II (8.05) Fall 2013 Assignment 4 Massachusetts Institute of Technology Physics Department Due October 4, 2013 September 27, 2013 3:00 pm Problem Set 4 1. Identitites for commutators (Based

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Unit 2: Chemical Bonding and Organic Chemistry

Chemistry AP Unit : Chemical Bonding and Organic Chemistry Unit : Chemical Bonding and Organic Chemistry Chapter 7: Atomic Structure and Periodicity 7.1: Electromagnetic Radiation Electromagnetic (EM)

### Solar Energy. Outline. Solar radiation. What is light?-- Electromagnetic Radiation. Light - Electromagnetic wave spectrum. Electromagnetic Radiation

Outline MAE 493R/593V- Renewable Energy Devices Solar Energy Electromagnetic wave Solar spectrum Solar global radiation Solar thermal energy Solar thermal collectors Solar thermal power plants Photovoltaics

### Chemistry - Elements Electron Configurations The Periodic Table. Ron Robertson

Chemistry - Elements Electron Configurations The Periodic Table Ron Robertson History of Chemistry Before 16 th Century Alchemy Attempts (scientific or otherwise) to change cheap metals into gold no real

### CHEM6085: Density Functional Theory Lecture 2. Hamiltonian operators for molecules

CHEM6085: Density Functional Theory Lecture 2 Hamiltonian operators for molecules C.-K. Skylaris 1 The (time-independent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction

### Chapter 13 Spectroscopy NMR, IR, MS, UV-Vis

Chapter 13 Spectroscopy NMR, IR, MS, UV-Vis Main points of the chapter 1. Hydrogen Nuclear Magnetic Resonance a. Splitting or coupling (what s next to what) b. Chemical shifts (what type is it) c. Integration

Nuclear Physics and Radioactivity 1. The number of electrons in an atom of atomic number Z and mass number A is 1) A 2) Z 3) A+Z 4) A-Z 2. The repulsive force between the positively charged protons does

### Solutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 7

Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid April 21, 2002 Chapter 7 Problem 7.2 Obtain the Lorentz transformation in which the velocity is at an infinitesimal angle

### Light as a Wave. The Nature of Light. EM Radiation Spectrum. EM Radiation Spectrum. Electromagnetic Radiation

The Nature of Light Light and other forms of radiation carry information to us from distance astronomical objects Visible light is a subset of a huge spectrum of electromagnetic radiation Maxwell pioneered

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### Precession of spin and Precession of a top

6. Classical Precession of the Angular Momentum Vector A classical bar magnet (Figure 11) may lie motionless at a certain orientation in a magnetic field. However, if the bar magnet possesses angular momentum,

### Chapters 21-29. Magnetic Force. for a moving charge. F=BQvsinΘ. F=BIlsinΘ. for a current

Chapters 21-29 Chapter 21:45,63 Chapter 22:25,49 Chapter 23:35,38,53,55,58,59 Chapter 24:17,18,20,42,43,44,50,52,53.59,63 Chapter 26:27,33,34,39,54 Chapter 27:17,18,34,43,50,51,53,56 Chapter 28: 10,11,28,47,52

### Electrons in Atoms & Periodic Table Chapter 13 & 14 Assignment & Problem Set

Electrons in Atoms & Periodic Table Name Warm-Ups (Show your work for credit) Date 1. Date 2. Date 3. Date 4. Date 5. Date 6. Date 7. Date 8. Electrons in Atoms & Periodic Table 2 Study Guide: Things You

### Objectives 404 CHAPTER 9 RADIATION

Objectives Explain the difference between isotopes of the same element. Describe the force that holds nucleons together. Explain the relationship between mass and energy according to Einstein s theory

### Elements in the periodic table are indicated by SYMBOLS. To the left of the symbol we find the atomic mass (A) at the upper corner, and the atomic num

. ATOMIC STRUCTURE FUNDAMENTALS LEARNING OBJECTIVES To review the basics concepts of atomic structure that have direct relevance to the fundamental concepts of organic chemistry. This material is essential

### Solar Energy Production

Solar Energy Production We re now ready to address the very important question: What makes the Sun shine? Why is this such an important topic in astronomy? As humans, we see in the visible part of the

### PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS

PHOTOELECTRIC EFFECT AND DUAL NATURE OF MATTER AND RADIATIONS 1. Photons 2. Photoelectric Effect 3. Experimental Set-up to study Photoelectric Effect 4. Effect of Intensity, Frequency, Potential on P.E.

### Section 5 Molecular Electronic Spectroscopy (lecture 9 ish)

Section 5 Molecular Electronic Spectroscopy (lecture 9 ish) Previously: Quantum theory of atoms / molecules Quantum Mechanics Vl Valence Molecular Electronic Spectroscopy Classification of electronic states