CHAPTER 6 Quantum Mechanics II
|
|
- Elizabeth Long
- 5 years ago
- Views:
Transcription
1 CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well 6.6 Simple Harmonic Oscillator 6.7 Barriers and Tunneling I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, But how can it be like that? because you will get down the drain into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that. - Richard Feynman
2 Quantum Mechanics Origin of Quantum Mechanics is credited to Werner Heisenberg and Erwin Schrödinger. Heisenberg Matrix formalism (1925). Schrödinger Wave Mechanics (1926). similar to wave descriptions in classical physics. Both methods give identical results. QM complex subject and probabilistic nature is in contrast to cause and effect of classical physics. In this chapter will use QM for some simple cases.
3 Models of the Atom Thomson Plum Pudding Why? Known that negative charges can be removed from atom. Problem: just a random guess Rutherford Solar System + Why? Scattering showed hard core. Problem: electrons should spiral into nucleus in ~10-11 sec. Bohr fixed energy levels Why? Explains spectral lines. Problem: No reason for fixed energy levels + de Broglie electron standing waves Why? Explains fixed energy levels + Problem: still only works for Hydrogen. Schrödinger probability distribution
4 6.1: The Schrödinger Wave Equation Schrodinger equation development new approach to physics If particles exhibit wave properties (de Broglie) then, like any other wave phenomena, there should be a wave equation for a particle (a differential equation which describes the wave propagation). Like Newton s laws in classical physics for particles we need an equation to describe their wave motion. Schrödinger found an equation for the wave function using ideas from optics. the solution to the wave equation (the wave function) must be consistent with the notion of the uncertainty principle and the notion of probability. Like Newton s laws in classical physics there is no way to derive the Schrödinger wave equation from more basic principles. the ultimate test is to see if it describes experimental results.
5 Wave Equations differential equations which describe the propagation of waves EM Waves (light/photons) Amplitude = E or M field 2 tells you the probability of detecting a photon. Maxwell s Equations: x c t Solutions are sine/cosine waves: ( x, t) Asin( kxt) ( x, t) Acos( kxt) Matter Waves (electrons/etc) Amplitude = matter field 2 tells you the probability of detecting a particle. Schrödinger Equation: 2 2 i 2 2m x t Solutions are complex sine/cosine waves: ( x, t) Aexp i kxt A cos( kx t) i sin( kx t)
6 Schrodinger s starting point: Which aspects of a particle wave equation need to be similar and which different from classical wave equations? Not going to derive it, because there is no derivation Schrodinger started to think x c t Works for light, why not for an electron? E pc t x c 2 2 E : E ~ partial derivative with respect to t t p : p ~ partial derivative with respect to x x For electron, 2 p E 2m V So, one may guess that 2 1 ~ V 2 t 2m x
7 Schrodinger s starting point: E : E ~ first partial derivative with respect to t t p : p ~ first partial derivative with respect to x x The Planck Einstein and De Broglie relations:, Complex wave solution: E pk ( xt, ) Aexp i kx t Aexp i pxet / ( xt, ) i p ( x, t ) i p p i x x x ( xt, ) i E ( x, t ) i E E i t t t p ( x, t) ( x, t) E V i V( x, t) ( x, t) 2 2m t 2m x There is no way to derive the Schrödinger wave equation from more basic principles. the ultimate test is to see if it describes experimental results.
8 The Schrödinger Wave Equation The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is The extension into three dimensions is Schrödinger Eqn. is linear in the wave function Ψ (example 6.1). this required since principle of superposition is used to form wave packets
9 General Solution of the Schrödinger Wave Equation The general form of the solution of the Schrödinger wave equation is given by: which also describes a wave moving in the +x direction. In general the amplitude may also be complex. This is called the wave function of the particle. The wave function must not be restricted to being real. Notice that the sine term has an imaginary number. Only the physically measurable quantities however must be real. Can A exp(kx + wt) be also a solution to the equation?
10 Normalization and Probability The probability P(x) dx of a particle being between x and X + dx was given in the equation * here denotes the complex conjugate of The probability of the particle being between x 1 and x 2 is given by The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.
11 Example 6.4
12 Properties of Valid Wave Functions Boundary conditions 1) In order to avoid infinite probabilities, the wave function must be finite everywhere. 2) In order to avoid multiple values of the probability, the wave function must be single valued. 3) For finite potentials, the wave function and its derivative must be continuous. I( x) II( x) ( ) ( ) at boundary I x II x x x This is required because the second-order derivative term in the wave equation must be single valued. (There are exceptions to this rule when V is infinite.) 4) In order to normalize the wave functions, they must approach zero as x approaches positive and negative infinity. Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.
13 Time-Independent Schrödinger Wave Equation The potential in many cases will not depend explicitly on time. The dependence on time and position can then be separated in the Schrödinger wave equation. Let, which yields: Now divide by the wave function: The left side of this last equation depends only on time, and the right side depends only on spatial coordinates. Hence each side must be equal to a constant. The time dependent side is
14 Time-Independent Schrödinger Wave Equation (con t) We integrate both sides and find: where C is an integration constant that we may choose to be 0. Comparing with wavefunction for a free particle, A e i(kx t) : B E Plugging back into spatial equation yields: This is known as the time-independent Schrödinger wave equation, It is a fundamental equation in quantum mechanics. The wave function (with time dependence) can be written: i t ( xt, ) ( xe )
15 Stationary State Recalling the separation of variables: (x,t) (x) f (t) and with f(t) = the wave function can be written as: e it The probability density becomes: The probability distributions are constant in time. This is a standing wave phenomena that is called the stationary state.
16 Comparison of Classical and Quantum Mechanics Newton s second law and Schrödinger s wave equation are both differential equations. Both postulated as fundamental equations to explain observed behavior and verified by experiment. Newton s second law can be derived from the Schrödinger wave equation, so the latter is the more fundamental. Classical mechanics only appears to be more precise because it deals with precise values rather than probabilities. only appears precise because it deals with macroscopic phenomena. The underlying uncertainties in macroscopic measurements are just too small to be significant. Classical mechanics is accurate enough at large quantum numbers, but as far as we know, there is only one correct theory, QM.
CHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationModern Physics for Scientists and Engineers International Edition, 4th Edition
Modern Physics for Scientists and Engineers International Edition, 4th Edition http://optics.hanyang.ac.kr/~shsong Review: 1. THE BIRTH OF MODERN PHYSICS 2. SPECIAL THEORY OF RELATIVITY 3. THE EXPERIMENTAL
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationSemiconductor Physics and Devices
Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential
More informationWe also find the development of famous Schrodinger equation to describe the quantization of energy levels of atoms.
Lecture 4 TITLE: Quantization of radiation and matter: Wave-Particle duality Objectives In this lecture, we will discuss the development of quantization of matter and light. We will understand the need
More informationElectron in a Box. A wave packet in a square well (an electron in a box) changing with time.
Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5. De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle Many experimental
More informationEarly Quantum Theory & Models of the Atom (Ch 27) Discovery of electron. Blackbody Radiation. Blackbody Radiation. J. J. Thomson ( )
Early Quantum Theory & Models of the Atom (Ch 27) Discovery of electron Modern physics special relativity quantum theory J. J. Thomson (1856-1940) measured e/m directly set-up was similar to mass spectrometer
More informationQuantum Mechanics. The Schrödinger equation. Erwin Schrödinger
Quantum Mechanics The Schrödinger equation Erwin Schrödinger The Nobel Prize in Physics 1933 "for the discovery of new productive forms of atomic theory" The Schrödinger Equation in One Dimension Time-Independent
More informationRichard Feynman: Electron waves are probability waves in the ocean of uncertainty.
Richard Feynman: Electron waves are probability waves in the ocean of uncertainty. Last Time We Solved some of the Problems with Classical Physics Discrete Spectra? Bohr Model but not complete. Blackbody
More informationPHYS 571 Radiation Physics
PHYS 571 Radiation Physics Prof. Gocha Khelashvili http://blackboard.iit.edu login Bohr s Theory of Hydrogen Atom Bohr s Theory of Hydrogen Atom Bohr s Theory of Hydrogen Atom Electrons can move on certain
More informationPhysics-I. Dr. Anurag Srivastava. Web address: Visit me: Room-110, Block-E, IIITM Campus
Physics-I Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus Syllabus Electrodynamics: Maxwell s equations: differential
More informationCHAPTER 2: POSTULATES OF QUANTUM MECHANICS
CHAPTER 2: POSTULATES OF QUANTUM MECHANICS Basics of Quantum Mechanics - Why Quantum Physics? - Classical mechanics (Newton's mechanics) and Maxwell's equations (electromagnetics theory) can explain MACROSCOPIC
More informationThe Photoelectric Effect
Stellar Astrophysics: The Interaction of Light and Matter The Photoelectric Effect Methods of electron emission Thermionic emission: Application of heat allows electrons to gain enough energy to escape
More informationChapter 7. Bound Systems are perhaps the most interesting cases for us to consider. We see much of the interesting features of quantum mechanics.
Chapter 7 In chapter 6 we learned about a set of rules for quantum mechanics. Now we want to apply them to various cases and see what they predict for the behavior of quanta under different conditions.
More informationQuantum mechanics (QM) deals with systems on atomic scale level, whose behaviours cannot be described by classical mechanics.
A 10-MINUTE RATHER QUICK INTRODUCTION TO QUANTUM MECHANICS 1. What is quantum mechanics (as opposed to classical mechanics)? Quantum mechanics (QM) deals with systems on atomic scale level, whose behaviours
More informationLECTURE 6 QUANTUM PHYSICS II. Instructor: Shih-Chieh Hsu
LECTURE 6 QUANTUM PHYSICS II Instructor: Shih-Chieh Hsu Development of Quantum Mechanics 2 In 1862, Kirchhoff coined black body radiation or known as cavity radiation The experiments raised the question
More informationWave nature of particles
Wave nature of particles We have thus far developed a model of atomic structure based on the particle nature of matter: Atoms have a dense nucleus of positive charge with electrons orbiting the nucleus
More informationPHYS 3313 Section 001 Lecture #20
PHYS 3313 Section 001 ecture #0 Monday, April 10, 017 Dr. Amir Farbin Infinite Square-well Potential Finite Square Well Potential Penetration Depth Degeneracy Simple Harmonic Oscillator 1 Announcements
More informationComplementi di Fisica Lectures 10-11
Complementi di Fisica - Lectures 1-11 15/16-1-1 Complementi di Fisica Lectures 1-11 Livio Lanceri Università di Trieste Trieste, 15/16-1-1 Course Outline - Reminder Quantum Mechanics: an introduction Reminder
More informationThe Nature of Energy
The Nature of Energy For atoms and molecules, one does not observe a continuous spectrum, as one gets from a white light source.? Only a line spectrum of discrete wavelengths is observed. 2012 Pearson
More informationPhysics. Light Quanta
Physics Light Quanta Quantum Theory Is light a WAVE or a PARTICLE? Particle tiny object like a bullet, has mass and travels in straight lines unless a force acts upon it Waves phenomena that extend in
More informationNotes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates.
Notes on wavefunctions IV: the Schrödinger equation in a potential and energy eigenstates. We have now seen that the wavefunction for a free electron changes with time according to the Schrödinger Equation
More informationMidterm Examination 1
CHEM 332 Physical Chemistry Spring 2014 Name: Answer Key Midterm Examination 1 1. Match the individual with their accomplishment: A. Millikan G Proposed a probabilistic interpretation for *. B. Planck
More informationCHEM-UA 127: Advanced General Chemistry I
1 CHEM-UA 127: Advanced General Chemistry I I. RATIONALIZATION OF THE ELECTRON DIFFRACTION EXPERIMENT We will consider two different rationalizations of the electron double-slit experiment. A. Particle-wave
More informationPhysics 1C. Lecture 28D
Physics 1C Lecture 28D "I ask you to look both ways. For the road to a knowledge of the stars leads through the atom; and important knowledge of the atom has been reached through the stars." --Sir Arthur
More informationAtoms, nuclei, particles
Atoms, nuclei, particles Nikolaos Kidonakis Physics for Georgia Academic Decathlon September 2016 Age-old questions What are the fundamental particles of matter? What are the fundamental forces of nature?
More informationChapter 38 Quantum Mechanics
Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the Double-Slit Experiment 38-3 The Heisenberg Uncertainty Principle
More informationChapter. 5 Bound States: Simple Case
Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ Textbook PHYS-2402 Lecture 12 HW3 (due 3/2) 13, 15, 20, 31, 36, 41, 48, 53, 63, 66 ***** Exam: 3/12 Ch.2, 3, 4, 5 Feb. 26, 2015 Physics
More informationQuantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :
Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in
More informationQUANTUM MECHANICS Intro to Basic Features
PCES 4.21 QUANTUM MECHANICS Intro to Basic Features 1. QUANTUM INTERFERENCE & QUANTUM PATHS Rather than explain the rules of quantum mechanics as they were devised, we first look at a more modern formulation
More informationWave properties of matter & Quantum mechanics I. Chapter 5
Wave properties of matter & Quantum mechanics I Chapter 5 X-ray diffraction Max von Laue suggested that if x-rays were a form of electromagnetic radiation, interference effects should be observed. Crystals
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationQuantum Mechanics. Semester /2015. (Introduction)
EMT 295/3 Quantum Mechanics Semester 1 2014/2015 (Introduction) EMT 295 Course Outcomes (COs): CO1: Ability to explain the concept and principles of modern physics, quantization and postulates of quantum
More informationChemistry 3502/4502. Exam I Key. September 19, ) This is a multiple choice exam. Circle the correct answer.
D Chemistry 350/450 Exam I Key September 19, 003 1) This is a multiple choice exam. Circle the correct answer. ) There is one correct answer to every problem. There is no partial credit. 3) A table of
More informationThe Bohr Model of Hydrogen, a Summary, Review
The Bohr Model of Hydrogen, a Summary, Review Allowed electron orbital radii and speeds: Allowed electron energy levels: Problems with the Bohr Model Bohr s model for the atom was a huge success in that
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationIntroduction to Quantum Physics. Early Atomic Physics
Introduction to Quantum Physics Early Atomic Physics What is Quantum Physics Quantum Physics is a collection of laws which explain observations of the tiny building blocks of all matter. The world of the
More informationConstants & Atomic Data. The birth of atomic physics and quantum mechanics. debroglie s Wave Equations. Energy Calculations. λ = f = h E.
Constants & Atomic Data The birth of atomic physics and quantum mechanics Honors Physics Don Rhine Look inside back cover of book! Speed of Light (): c = 3.00 x 10 8 m/s Elementary Charge: e - = p + =
More informationPHYS 262. George Mason University. Professor Paul So
PHYS 6 George Mason University Professor Paul So Chapter 40/41: Quantum Mechanics Wave Functions & 1D Schrodinger Eq Particle in a Box Wave function Energy levels Potential Wells/Barriers & Tunneling The
More informationTheoretical Biophysics. Quantum Theory and Molecular Dynamics. Pawel Romanczuk WS 2017/18
Theoretical Biophysics Quantum Theory and Molecular Dynamics Pawel Romanczuk WS 2017/18 http://lab.romanczuk.de/teaching/ 1 Introduction Two pillars of classical theoretical physics at the begin of 20th
More informationThe birth of atomic physics and quantum mechanics. Honors Physics Don Rhine
The birth of atomic physics and quantum mechanics Honors Physics Don Rhine Constants & Atomic Data Look inside back cover of book! Speed of Light (vacuum): c = 3.00 x 10 8 m/s Elementary Charge: e - =
More information20th Century Atomic Theory- Hydrogen Atom
Background for (mostly) Chapter 12 of EDR 20th Century Atomic Theory- Hydrogen Atom EDR Section 12.7 Rutherford's scattering experiments (Raff 11.2.3) in 1910 lead to a "planetary" model of the atom where
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationPhysics 227 Exam 2. Rutherford said that if you really understand something you should be able to explain it to your grandmother.
Physics 227 Exam 2 Rutherford said that if you really understand something you should be able to explain it to your grandmother. For each of the topics on the next two pages, write clear, concise, physical
More informationLecture 14: Superposition & Time-Dependent Quantum States
Lecture 14: Superposition & Time-Dependent Quantum States U= y(x,t=0) 2 U= U= y(x,t 0 ) 2 U= x 0 x L 0 x L Lecture 14, p 1 Last Week Time-independent Schrodinger s Equation (SEQ): 2 2m d y ( x) U ( x)
More informationThe Schrodinger Equation and Postulates Common operators in QM: Potential Energy. Often depends on position operator: Kinetic Energy 1-D case:
The Schrodinger Equation and Postulates Common operators in QM: Potential Energy Often depends on position operator: Kinetic Energy 1-D case: 3-D case Time Total energy = Hamiltonian To find out about
More information4/14/2015. Models of the Atom. Quantum Physics versus Classical Physics The Thirty-Year War ( ) Classical Model of Atom
Quantum Physics versus Classical Physics The Thirty-Year War (1900-1930) Models of the Atom Interactions between Matter and Radiation Models of the Atom Bohr s Model of the Atom Planck s Blackbody Radiation
More informationPhysics 280 Quantum Mechanics Lecture
Spring 2015 1 1 Department of Physics Drexel University August 3, 2016 Objectives Review Early Quantum Mechanics Objectives Review Early Quantum Mechanics Schrödinger s Wave Equation Objectives Review
More informationLecture 21 Matter acts like waves!
Particles Act Like Waves! De Broglie s Matter Waves λ = h / p Schrodinger s Equation Announcements Schedule: Today: de Broglie and matter waves, Schrodinger s Equation March Ch. 16, Lightman Ch. 4 Net
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 1-1B: THE INTERACTION OF MATTER WITH RADIATION Introductory Video Quantum Mechanics Essential Idea: The microscopic quantum world offers
More informationdt r r r V(x,t) = F(x,t)dx
Quantum Mechanics and Atomic Physics Lecture 3: Schroedinger s Equation: Part I http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement First homework due on Wednesday Sept 14 at the beginning
More informationProbability and Normalization
Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ψ ( x) dx = 1 (normalization condition) L
More informationApplied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation
22.101 Applied Nuclear Physics (Fall 2006) Lecture 2 (9/11/06) Schrödinger Wave Equation References -- R. M. Eisberg, Fundamentals of Modern Physics (Wiley & Sons, New York, 1961). R. L. Liboff, Introductory
More informationFundamental of Spectroscopy for Optical Remote Sensing Xinzhao Chu I 10 3.4. Principle of Uncertainty Indeterminacy 0. Expression of Heisenberg s Principle of Uncertainty It is worth to point out that
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
More informationChemistry 3502/4502. Exam I. September 19, ) This is a multiple choice exam. Circle the correct answer.
D Chemistry 350/450 Exam I September 9, 003 ) This is a multiple choice exam. Circle the correct answer. ) There is one correct answer to every problem. There is no partial credit. 3) A table of useful
More informationComplementi di Fisica Lectures 5, 6
Complementi di Fisica - Lectures 5, 6 9/3-9-15 Complementi di Fisica Lectures 5, 6 Livio Lanceri Università di Trieste Trieste, 9/3-9-15 Course Outline - Reminder Quantum Mechanics: an introduction Reminder
More informationTime dependent Schrodinger equation
Lesson: Time dependent Schrodinger equation Lesson Developer: Dr. Monika Goyal, College/Department: Shyam Lal College (Day), University of Delhi Table of contents 1.1 Introduction 1. Dynamical evolution
More informationQuantum Mechanics. Physics April 2002 Lecture 9. Planck Bohr Schroedinger Heisenberg
Quantum Mechanics Physics 102 18 April 2002 Lecture 9 Planck Bohr Schroedinger Heisenberg From: http://www.th.physik.uni-frankfurt.de/~jr/portraits.html 18 Apr 2002 Physics 102 Lecture 9 1 Blackbody radiation
More information* = 2 = Probability distribution function. probability of finding a particle near a given point x,y,z at a time t
Quantum Mechanics Wave functions and the Schrodinger equation Particles behave like waves, so they can be described with a wave function (x,y,z,t) A stationary state has a definite energy, and can be written
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationPHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101
PHY 114 A General Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lecture 3 (Chapter 40-4): Some topics in Quantum Theory 1. Particle behaviors of electromagnetic waves. Wave behaviors of particles 3. Quantized
More informationThe Schrödinger Wave Equation Formulation of Quantum Mechanics
Chapter 5. The Schrödinger Wave Equation Formulation of Quantum Mechanics Notes: Most of the material in this chapter is taken from Thornton and Rex, Chapter 6. 5.1 The Schrödinger Wave Equation There
More informationProfessor Jasper Halekas Van Allen 70 MWF 12:30-1:20 Lecture
Professor Jasper Halekas Van Allen 70 MWF 1:30-1:0 Lecture Back on regular schedule for the next two weeks There *will* be labs and homeworks due this week and next EM Waves (light/photons) Amplitude E
More informationThe Particle in a Box
Page 324 Lecture 17: Relation of Particle in a Box Eigenstates to Position and Momentum Eigenstates General Considerations on Bound States and Quantization Continuity Equation for Probability Date Given:
More informationChapter 29 Atomic Physics. Looking Ahead. Slide 29-1
Chapter 29 Atomic Physics Looking Ahead Slide 29-1 Atomic Spectra and the Bohr Model In the mid 1800s it became apparent that the spectra of atomic gases is comprised of individual emission lines. Slide
More informationA New Proposal Combining Quantum Mechanics and the Special Theory of Relativity
Apeiron, Vol. 9, No., April 00 0 A New Proposal Combining Quantum Mechanics and the Special Theory of Relativity Rajan Dogra H. no.391, Sector 7-D, Chandigarh, India e-mail:rajandogra_000@yahoo.com The
More informationSession 1: Solid State Physics. Classical vs. Quantum Mechanics
Session 1: Solid State Physics Classical vs. Quantum Mechanics 1 Outline Introduction History Thomson s atomic model Rutherford s atomic model Birth of QM Black body radiation Photoelectric effect Bohr
More informationThe Atom. Result for Hydrogen. For example: the emission spectrum of Hydrogen: Screen. light. Hydrogen gas. Diffraction grating (or prism)
The Atom What was know about the atom in 1900? First, the existence of atoms was not universally accepted at this time, but for those who did think atoms existed, they knew: 1. Atoms are small, but they
More informationProfessor Jasper Halekas Van Allen 70 MWF 12:30-1:20 Lecture
Professor Jasper Halekas Van Allen 70 MWF 1:30-1:0 Lecture Back on regular schedule for the next two weeks until Spring Break! There will be labs and homework due this week and next Labs this week and
More informationThe Birth of Quantum Mechanics. New Wave Rock Stars
The Birth of Quantum Mechanics Louis de Broglie 1892-1987 Erwin Schrödinger 1887-1961 Paul Dirac 1902-1984 Werner Heisenberg 1901-1976 New Wave Rock Stars Blackbody radiation: Light energy is quantized.
More informationLecture 2: simple QM problems
Reminder: http://www.star.le.ac.uk/nrt3/qm/ Lecture : simple QM problems Quantum mechanics describes physical particles as waves of probability. We shall see how this works in some simple applications,
More informationCHM 532 Notes on Wavefunctions and the Schrödinger Equation
CHM 532 Notes on Wavefunctions and the Schrödinger Equation In class we have discussed a thought experiment 1 that contrasts the behavior of classical particles, classical waves and quantum particles.
More informationPHYS 3313 Section 001 Lecture #16
PHYS 3313 Section 001 Lecture #16 Monday, Mar. 24, 2014 De Broglie Waves Bohr s Quantization Conditions Electron Scattering Wave Packets and Packet Envelops Superposition of Waves Electron Double Slit
More informationQuantum Mechanics of Atoms
Quantum Mechanics of Atoms Your theory is crazy, but it's not crazy enough to be true N. Bohr to W. Pauli Quantum Mechanics of Atoms 2 Limitations of the Bohr Model The model was a great break-through,
More informationChapter (5) Matter Waves
Chapter (5) Matter Waves De Broglie wavelength Wave groups Consider a one- dimensional wave propagating in the positive x- direction with a phase speed v p. Where v p is the speed of a point of constant
More information12/04/2012. Models of the Atom. Quantum Physics versus Classical Physics The Thirty-Year War ( )
Quantum Physics versus Classical Physics The Thirty-Year War (1900-1930) Interactions between Matter and Radiation Models of the Atom Bohr s Model of the Atom Planck s Blackbody Radiation Models of the
More informationAlan Mortimer PhD. Ideas of Modern Physics
Alan Mortimer PhD Ideas of Modern Physics Electromagnetic Waves Last Week Special Relativity General Relativity The Quantum World Index Planck s Law Atomic Structure and emission lines Matter waves Uncertainty
More informationScience One Physics Lecture 8. The Schrodinger Equation slides with commentary
Science One Physics Lecture 8 The Schrodinger Equation slides with commentary : Outline The Schrödinger equation Measurement in quantum mechanics: from the Stern-Gerlach experiment to the Born rule Entanglement
More informationClass 21. Early Quantum Mechanics and the Wave Nature of Matter. Physics 106. Winter Press CTRL-L to view as a slide show. Class 21.
Early and the Wave Nature of Matter Winter 2018 Press CTRL-L to view as a slide show. Last Time Last time we discussed: Optical systems Midterm 2 Today we will discuss: Quick of X-ray diffraction Compton
More informationBasic Quantum Mechanics
Frederick Lanni 10feb'12 Basic Quantum Mechanics Part I. Where Schrodinger's equation comes from. A. Planck's quantum hypothesis, formulated in 1900, was that exchange of energy between an electromagnetic
More informationPSI AP Physics How was it determined that cathode rays possessed a negative charge?
PSI AP Physics 2 Name Chapter Questions 1. How was it determined that cathode rays possessed a negative charge? 2. J. J. Thomson found that cathode rays were really particles, which were subsequently named
More informationQuantum Physics and Atomic Models Chapter Questions. 1. How was it determined that cathode rays possessed a negative charge?
Quantum Physics and Atomic Models Chapter Questions 1. How was it determined that cathode rays possessed a negative charge? 2. J. J. Thomson found that cathode rays were really particles, which were subsequently
More informationProblems and Multiple Choice Questions
Problems and Multiple Choice Questions 1. A momentum operator in one dimension is 2. A position operator in 3 dimensions is 3. A kinetic energy operator in 1 dimension is 4. If two operator commute, a)
More informationECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline:
ECE 487 Lecture 6 : Time-Dependent Quantum Mechanics I Class Outline: Time-Dependent Schrödinger Equation Solutions to thetime-dependent Schrödinger Equation Expansion of Energy Eigenstates Things you
More informationCHAPTER NUMBER 7: Quantum Theory: Introduction and Principles
CHAPTER NUMBER 7: Quantum Theory: Introduction and Principles Art PowerPoints Peter Atkins & Julio De Paula 2010 1 mm 1000 m 100 m 10 m 1000 nm 100 nm 10 nm 1 nm 10 Å 1 Å Quantum phenomena 7.1 Energy quantization
More informationChapter 7 Atomic Structure -1 Quantum Model of Atom. Dr. Sapna Gupta
Chapter 7 Atomic Structure -1 Quantum Model of Atom Dr. Sapna Gupta The Electromagnetic Spectrum The electromagnetic spectrum includes many different types of radiation which travel in waves. Visible light
More informationElectronic Structure of Atoms. Chapter 6
Electronic Structure of Atoms Chapter 6 Electronic Structure of Atoms 1. The Wave Nature of Light All waves have: a) characteristic wavelength, λ b) amplitude, A Electronic Structure of Atoms 1. The Wave
More informationUNIT 4 Electrons in Atoms. Advanced Chemistry 235 Lanphier High School Mr. David Peeler
UNIT 4 Electrons in Atoms Advanced Chemistry 235 Lanphier High School Mr. David Peeler Section 4.1 Models of the Atom OBJECTIVES: Identify the inadequacies in the Rutherford atomic model. Section 4.1 Models
More informationReview Models of the Atom
Review Models of the Atom Copyright 2007 Pearson Benjamin Cummings. All rights reserved. Dalton proposes the indivisible unit of an element is the atom. Thomson discovers electrons, believed to reside
More information8 Wavefunctions - Schrödinger s Equation
8 Wavefunctions - Schrödinger s Equation So far we have considered only free particles - i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under
More informationHistorical Background of Quantum Mechanics
Historical Background of Quantum Mechanics The Nature of Light The Structure of Matter Dr. Sabry El-Taher 1 The Nature of Light Dr. Sabry El-Taher 2 In 1801 Thomas Young: gave experimental evidence for
More informationModels of the Atom. Spencer Clelland & Katelyn Mason
Models of the Atom Spencer Clelland & Katelyn Mason First Things First Electrons were accepted to be part of the atom structure by scientists in the1900 s. The first model of the atom was visualized as
More informationNotes for Special Relativity, Quantum Mechanics, and Nuclear Physics
Notes for Special Relativity, Quantum Mechanics, and Nuclear Physics 1. More on special relativity Normally, when two objects are moving with velocity v and u with respect to the stationary observer, the
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 8, February 3, 2006 & L "
Chem 352/452 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 26 Christopher J. Cramer Lecture 8, February 3, 26 Solved Homework (Homework for grading is also due today) Evaluate
More information