Lecture 10: The Schrödinger Equation. Lecture 10, p 2
|
|
- Marilynn Mills
- 6 years ago
- Views:
Transcription
1 Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct eperience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen. --Richard P. Feynman Lecture 10, p 1
2 Lecture 10: The Schrödinger Equation Lecture 10, p
3 This week and last week are critical for the course: Week 3, Lectures 7-9: Week 4, Lectures 10-1: Light as Particles Schrödinger Equation Particles as waves Particles in infinite wells, finite wells Probability Simple Harmonic Oscillator Uncertainty Principle Midterm Eam Monday, week 5 It will cover lectures 1-1 (ecept Simple Harmonic Oscillators) Practice eams: Old eams are linked from the course web page. Review Sunday before Midterm Office hours: Sunday and Monday Net week: Homework 4 covers material in lecture 10 due on Thur. after midterm. We strongly encourage you to look at the homework before the midterm! Discussion: Covers material in lectures There will be a quiz. Lab: Go to 57 Loomis (a computer room). You can save a lot of time by reading the lab ahead of time It s a tutorial on how to draw wave functions. Lecture 10, p 3
4 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels ψ() U= n=1 n=3 U= 0 L n= Nice descriptions in the tet Chapter 40 Good web site for animations Lecture 10, p 4
5 Matter Waves - Quantitative Having established that matter acts qualitatively like a wave, we want to be able to make precise quantitative predictions, under given conditions. Usually the conditions are specified by giving a potential energy U(,y,z) in which the particle is located. Eamples: Electron in the coulomb potential produced by the nucleus Electron in a molecule Electron in a solid crystal Electron in a nanostructure quantum dot Proton in the nuclear potential inside the nucleus U() For simplicity, consider a 1-dimensional potential energy function, U(). Classically, a particle in the lowest energy state would sit right at the bottom of the well. In QM this is not possible. (Why?) Lecture 10, p 5
6 Act 1: Classical probability distributions Start a classical (large) object moving in a potential well (two are shown here). At some random time later, what is the probability of finding it near position? U() Ball in a bo: U() Ball in a valley: E KE Total energy E = KE + U() E P() P() a b c a b c HINT: Think about speed vs position. Lecture 10, p 6
7 Solution Start a classical (large) object moving in a potential well (two are shown here). At some random time later, what is the probability of finding it near position? U() Ball in a bo: U() Ball in a valley: E KE Total energy E = KE + U() E KE P() P() a b c a b c Probability is equally distributed Lecture 10, p 7
8 Solution Start a classical (large) object moving in a potential well (two are shown here). At some random time later, what is the probability of finding it near position? U() Ball in a bo: U() Ball in a valley: E KE Total energy E = KE + U() E KE P() P() a b c a b c Probability is equally distributed More likely to spend time at the edges. To predict a quantum particle s behavior, we need an equation that tells us how the particle s wave function, Ψ(,y,z,t), changes in space and time. Lecture 10, p 8
9 The Schrödinger Equation (SEQ) In 196, Erwin Schrödinger proposed an equation that described the time- and space-dependence of the wave function for matter waves (i.e., electrons, protons,...) There are two important forms for the SEQ. First we will focus on a very important special case of the SEQ, the time-independent SEQ. Also simplify to 1-dimension: ψ(,y,z) ψ(). ħ d ψ ( ) + U( ) ψ ( ) = Eψ ( ) m d This special case applies when the particle has a definite total energy (E in the equation). We ll consider the more general case (E has a probability distribution), and also D and 3D motion, later. ħ = h π QM entities don t always have a definite energy. Time does not appear in the equation. Therefore, ψ(,y,z) is a standing wave, because the probability density, ψ(), is not a function of time. We call ψ(,y,z) a stationary state. Notation: Distinguish Ψ(,y,z,t) from ψ(,y,z). Lecture 10, p 9
10 Time-Independent SEQ What does the time-independent SEQ represent? It s actually not so puzzling it s just an epression of a familiar result: Kinetic Energy (KE) + Potential Energy (PE) = Total Energy (E) ħ d ψ ( ) + U( ) ψ ( ) = Eψ ( ) m d KE term PE term Total E term Can we understand the KE term? Consider a particle with a definite momentum. Its wave function is: ψ() cos(k), where p = h/λ = ħk. dψ d ψ p = k sin ( k) = k cos( k) = ψ ( ) d d ħ So, the first term in the SEQ is (p /m)ψ. Note that the KE of the particle depends on the curvature (d ψ/d ) of the wave function. This is sometimes useful when analyzing a problem. Lecture 10, p 10
11 Particle Wavefunctions: : Eamples What do the solutions to the SEQ look like for general U()? Eamples of ψ() for a particle in a given potential U() (but with different E): ψ() We call these wavefunctions eigenstates of the particle. ψ() These are special states: energy eigenstates. ψ() The corresponding probability distributions ψ() of these states are: ψ ψ ψ Key point: Particle cannot be associated with a specific location. -- like the uncertainty that a particle went through slit 1 or slit. Question: Which corresponds to the lowest/highest kinetic energy? The particle kinetic energy is proportional to the curvature of the wave function. Lecture 10, p 11
12 Probability distribution Difference between classical and quantum cases U() Classical (particle with same energy as in qunatum case) U() Quantum (lowest energy state state) E E P() In classical mechanics, the particle is most likely to be found where its speed is slowest P() = ψ In classical mechanics, the particle moves back and forth coming to rest at each turning point In quantum mechanics, the particle can be most likely to be found at the center. In quantum mechanics, the particle can also be found where it is forbidden in classical mechanics.
13 Solutions to the time-independent SEQ ħ d ψ ( ) + U( ) ψ ( ) = Eψ ( ) m d Notice that if U() = constant, this equation has the simple form: d ψ = Cψ ( ) d where C m = ( ) U E ħ is a constant that might be positive or negative. For positive C (i.e., U > E), what is the form of the solution? a) sin k b) cos k c) e a d) e -a For negative C (U < E) what is the form of the solution? a) sin k b) cos k c) e a d) e -a Most of the wave functions in P14 will be sinusoidal or eponential. Lecture 10, p 13
14 Solution ħ d ψ ( ) + U( ) ψ ( ) = Eψ ( ) m d Notice that if U() = constant, this equation has the simple form: d ψ = Cψ ( ) d where C m = ( ) U E ħ is a constant that might be positive or negative. For positive C (i.e., U > E), what is the form of the solution? a) sin k b) cos k c) e a d) e -a For negative C (U < E) what is the form of the solution? a) sin k b) cos k c) e a d) e -a Most of the wave functions in P14 will be sinusoidal or eponential. Lecture 10, p 14
15 Eample: Particle in a Bo As a specific important eample, consider a quantum particle confined to a region, 0 < < L, by infinite potential walls. We call this a one-dimensional (1D) bo. U = 0 for 0 < < L U = everywhere else U() We already know the form of ψ when U = 0: sin(k) or cos(k). However, we can constrain ψ more than this. 0 L The waves have eactly the same form as standing waves on a string, sound waves in a pipe, etc. The wavelength is determined by the condition that it fits in the bo. On a string the wave is a displacement y() and the square is the intensity, etc. The discrete set of allowed wavelengths results in a discrete set of tones that the string can produce. In a quantum bo, the wave is the probability amplitude ψ() ond the square ψ() is the probability of finding the electron near point. The discrete set of allowed wavelengths results in a discrete set of allowed energies that the particle can have. Lecture 10, p 15
16 Boundary conditions Standing waves A standing wave is the solution for a wave confined to a region Boundary condition: Constraints on a wave where the potential changes Displacement = 0 for wave on string conductor E = 0 at surface of a E = 0 If both ends are constrained (e.g., for a cavity of length L), then only certain wavelengths λ are possible: n λ f 1 L v/l L v/l 3 L/3 3v/L nλ = L n = 1,, 3 mode inde L 4 L/ v/l n L/n nv/l
17 The Energy is Quantized Due to Confinement by the Potential The discrete E n are known as energy eigenvalues : E n n nλ = L n electron p h ev nm = = = m mλ λ n n h 8mL 1 where E1 E = En n λ (= v/f) E 4 L/ 16E 1 3 L/3 9E 1 L 4E 1 1 L E 1 Important features: Discrete energy levels. E 1 0 Standing wave (±p for a given E) n = 0 is not allowed. (why?) an eample of the uncertainty principle U = E n U = n=3 n= n=1 0 L Lecture 10, p 17
18 Quantum Wire Eample An electron is trapped in a quantum wire that is L = 4 nm long. Assume that the potential seen by the electron is approimately that of an infinite square well. 1: Calculate the ground (lowest) state energy of the electron. : What photon energy is required to ecite the trapped electron to the net available energy level (i.e., n = )? U= E n U= n=3 n= n=1 The idea here is that the photon is absorbed by the electron, which gains all of the photon s energy (similar to the photoelectric effect). 0 L Lecture 10, p 18
19 Solution An electron is trapped in a quantum wire that is L = 4 nm long. Assume that the potential seen by the electron is approimately that of an infinite square well. 1: Calculate the ground (lowest) state energy of the electron. h ev nm En = En = = 8mL 4L E 1 with E ev nm = = ev 4(4 nm) 1 Using: h E = = mλ where λ =L ev nm λ : What photon energy is required to ecite the trapped electron to the net available energy level (i.e., n = )? U= E n U= n=3 n= n=1 0 L En = n E1 So, the energy difference between the n = and n = 1 levels is: E = ( - 1 )E 1 = 3E 1 = ev Lecture 10, p 19
20 Boundary conditions We can solve the SEQ wherever we know U(). However, in many problems, including the 1D bo, U() has different functional forms in different regions. In our bo problem, there are three regions: 1: < 0 : 0 < < L 3: > L ψ() will have different functional forms in the different regions. We must make sure that ψ() satisfies the constraints (e.g., continuity) at the boundaries between these regions. The etra conditions that ψ must satisfy are called boundary conditions. They appear in many problems. Lecture 10, p 0
21 Particle in a Bo (1) Regions 1 and 3 are identical, so we really only need to deal with two distinct regions, (I) outside, and (II) inside the well Region I: When U =, what is ψ()? U() ψ ( ) + ( ) ( ) = 0 d ħ d m E U ψ I ψ Ι II I ψ Ι For U =, the SEQ can only be satisfied if: ψ I () = 0 0 L U = 0 for 0 < < L U = everywhere else Otherwise, the energy would have to be infinite, to cancel U. Note: The infinite well is an idealization. There are no infinitely high and sharp barriers. Lecture 10, p 1
22 Region II: When U = 0, what is ψ()? Particle in a Bo () d ψ m E U ψ ( ) + ( ) ( ) = 0 d ħ d ψ ( ) me = ψ ( ) d ħ U() II ψ 0 L The general solution is a superposition of sin and cos: ψ ( ) = B sink + B cosk 1 where, k = π λ Remember that k and E are related: E p ħ k h = = = m m mλ because U = 0 B 1 and B are coefficients to be determined by the boundary conditions. Lecture 10, p
23 Particle in a Bo (3) U() Now, let s worry about the boundary conditions. Match ψ at the left boundary ( = 0). I II I Region I: ψ I( ) = 0 ψ Ι ψ ΙΙ ψ Ι Region II: II( ) B1 sink B cosk ψ = + 0 L Recall: The wave function ψ() must be continuous at all boundaries. Therefore, at = 0: ψ (0) = ψ (0) I II ( ) B ( ) 0 = B sin 0 + cos = B because cos(0) = 1 and sin(0) = 0 This boundary condition requires that there be no cos(k) term! Lecture 10, p 3
24 Particle in a Bo (4) Now, match ψ at the right boundary ( = L). At = L: This constraint requires k to have special values: k n ψ ( L) = ψ ( L) I nπ π = n = 1,,... Using k, w e fin : nλn L λ = d = L This is the same condition we found for confined waves, e.g., waves on a string, EM waves in a laser cavity, etc.: n II 1 λ (= v/f) 4 L/ 3 L/3 L 1 L ( ) 0 = B sin kl U() For matter waves, the wavelength is related to the particle energy: E = h /mλ I ψ Ι II ψ ΙΙ 0 L h En = En where E 8mL 1 1 I ψ Ι Lecture 10, p 4
25 Net Lectures Normalizing the wavefunction General properties of bound-state wavefunctions Finite-depth square well potential (more realistic) Lecture 10, p 5
Lecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
More informationLecture 10: The Schrödinger Equation Lecture 10, p 1
Lecture 10: The Schrödinger Equation Lecture 10, p 1 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels y() U= n=1
More informationLecture 12: Particle in 1D boxes, Simple Harmonic Oscillators
Lecture 1: Particle in 1D boes, Simple Harmonic Oscillators U U() ψ() U n= n=0 n=1 n=3 Lecture 1, p 1 This week and last week are critical for the course: Week 3, Lectures 7-9: Week 4, Lectures 10-1: Light
More informationLecture 8: Wave-Particle Duality. Lecture 8, p 2
We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.
More informationLecture 12: Particle in 1D boxes & Simple Harmonic Oscillator
Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound
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 informationToday: Particle-in-a-Box wavefunctions
Today: Particle-in-a-Bo wavefunctions 1.Potential wells.solving the S.E. 3.Understanding the results. HWK11 due Wed. 5PM. Reading for Fri.: TZ&D Chap. 7.1 PE for electrons with most PE. On top work function
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 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 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 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 informationThe Schrödinger Equation in One Dimension
The Schrödinger Equation in One Dimension Introduction We have defined a comple wave function Ψ(, t) for a particle and interpreted it such that Ψ ( r, t d gives the probability that the particle is at
More informationChapter 4 (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence
V, E, Chapter (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence Potential Total Energies and Probability density
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 informationLecture 16: 3D Potentials and the Hydrogen Atom. 1 = π. r = a 0. P(r) ( ) h E. E n. Lecture 16, p 2
It was almost as incredible as if ou fired a 15-inch shell at a piece of tissue paper, and it came back to hit ou! --E. Rutherford (on the discover of the nucleus) ecture 16, p 1 ecture 16, p ecture 16:
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 informationThere is light at the end of the tunnel. -- proverb. The light at the end of the tunnel is just the light of an oncoming train. --R.
A vast time bubble has been projected into the future to the precise moment of the end of the universe. This is, of course, impossible. --D. Adams, The Hitchhiker s Guide to the Galaxy There is light at
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 informationFrom Last time. Exam 3 results. Probability. The wavefunction. Example wavefunction. Discrete vs continuous. De Broglie wavelength
From ast time Eam 3 results De Broglie wavelength Uncertainty principle Eam average ~ 70% Scores posted on learn@uw D C BC B AB A Wavefunction of a particle Course evaluations: Tuesday, Dec. 9 Tue. Dec.
More informationLecture 13: Barrier Penetration and Tunneling
Lecture 13: Barrier Penetration and Tunneling nucleus x U(x) U(x) U 0 E A B C B A 0 L x 0 x Lecture 13, p 1 Today Tunneling of quantum particles Scanning Tunneling Microscope (STM) Nuclear Decay Solar
More informationLecture-XXVI. Time-Independent Schrodinger Equation
Lecture-XXVI Time-Independent Schrodinger Equation Time Independent Schrodinger Equation: The time-dependent Schrodinger equation: Assume that V is independent of time t. In that case the Schrodinger equation
More informationLecture 15: Time-Dependent QM & Tunneling Review and Examples, Ammonia Maser
ecture 15: Time-Dependent QM & Tunneling Review and Examples, Ammonia Maser ψ(x,t=0) 2 U(x) 0 x ψ(x,t 0 ) 2 x U 0 0 E x 0 x ecture 15, p.1 Special (Optional) ecture Quantum Information One of the most
More informationThe Simple Harmonic Oscillator
The Simple Harmonic Oscillator Michael Fowler, University of Virginia Einstein s Solution of the Specific Heat Puzzle The simple harmonic oscillator, a nonrelativistic particle in a potential ½C, is a
More informationDavid J. Starling Penn State Hazleton PHYS 214
Not all chemicals are bad. Without chemicals such as hydrogen and oxygen, for example, there would be no way to make water, a vital ingredient in beer. -Dave Barry David J. Starling Penn State Hazleton
More informationPhysics 342 Lecture 17. Midterm I Recap. Lecture 17. Physics 342 Quantum Mechanics I
Physics 342 Lecture 17 Midterm I Recap Lecture 17 Physics 342 Quantum Mechanics I Monday, March 1th, 28 17.1 Introduction In the context of the first midterm, there are a few points I d like to make about
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 informationProblem Set 5: Solutions
University of Alabama Department of Physics and Astronomy PH 53 / eclair Spring 1 Problem Set 5: Solutions 1. Solve one of the exam problems that you did not choose.. The Thompson model of the atom. Show
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 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 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 informationToday: Finite box wavefunctions
Toay: Finite bo wavefunctions 1.Think outsie the bo!.solving the S.E. 3.Unerstaning the results. HWK1 ue We. 1AM. Reaing for Monay.: TZ&D Chap. 8 Reaing Quiz Classically forbien regions are where A. a
More informationThe Quantum Theory of Atoms and Molecules
The Quantum Theory of Atoms and Molecules The postulates of quantum mechanics Dr Grant Ritchie The postulates.. 1. Associated with any particle moving in a conservative field of force is a wave function,
More informationLecture 4 (19/10/2012)
4B5: Nanotechnology & Quantum Phenomena Michaelmas term 2012 Dr C Durkan cd229@eng.cam.ac.uk www.eng.cam.ac.uk/~cd229/ Lecture 4 (19/10/2012) Boundary-value problems in Quantum Mechanics - 2 Bound states
More informationModern Physics. Unit 3: Operators, Tunneling and Wave Packets Lecture 3.3: The Momentum Operator
Modern Physics Unit 3: Operators, Tunneling and Wave Packets Lecture 3.3: The Momentum Operator Ron Reifenberger Professor of Physics Purdue University 1 There are many operators in QM H Ψ= EΨ, or ˆop
More informationLecture 9: Introduction to QM: Review and Examples
Lecture 9: Introduction to QM: Review and Examples S 1 S 2 Lecture 9, p 1 Photoelectric Effect V stop (v) KE e V hf F max stop Binding energy F The work function: F is the minimum energy needed to strip
More informationThe Schrödinger Equation
Chapter 13 The Schrödinger Equation 13.1 Where we are so far We have focused primarily on electron spin so far because it s a simple quantum system (there are only two basis states!), and yet it still
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 informationExplanations of quantum animations Sohrab Ismail-Beigi April 22, 2009
Explanations of quantum animations Sohrab Ismail-Beigi April 22, 2009 I ve produced a set of animations showing the time evolution of various wave functions in various potentials according to the Schrödinger
More informationChemistry 125: Instructions for Erwin Meets Goldilocks
Chemistry 125: Instructions for Erwin Meets Goldilocks [Note the 5 problems for Monday s problem set are found at the end of this document. Many of the details on operating the program relate to an earlier
More informationECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline:
ECE 487 Lecture 5 : Foundations of Quantum Mechanics IV Class Outline: Linearly Varying Potential Triangular Potential Well Time-Dependent Schrödinger Equation Things you should know when you leave Key
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 informationLecture 19: Building Atoms and Molecules
Lecture 19: Building Atoms and Molecules +e r n = 3 n = 2 n = 1 +e +e r ψ even Lecture 19, p 1 Today Nuclear Magnetic Resonance Using RF photons to drive transitions between nuclear spin orientations in
More informationC/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11
C/CS/Phys C191 Particle-in-a-box, Spin 10/0/08 Fall 008 Lecture 11 Last time we saw that the time dependent Schr. eqn. can be decomposed into two equations, one in time (t) and one in space (x): space
More informationIntro to Quantum Physics
Physics 256: Lecture Q5 Intro to Quantum Physics Agenda for Today De Broglie Waves Electron Diffraction Wave-Particle Duality Complex Numbers Physics 201: Lecture 1, Pg 1 Are photons Waves or Particles?
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 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 informationAnyone who can contemplate quantum mechanics without getting dizzy hasn t understood it. --Niels Bohr. Lecture 17, p 1
Anone who can contemplate quantum mechanics without getting di hasn t understood it. --Niels Bohr Lecture 7, p Phsics Colloquium TODAY! Quantum Optomechanics Prof. Markus Aspelmeer, U. Vienna Massive mechanical
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 informationLecture 19: Building Atoms and Molecules
Lecture 19: Building Atoms and Molecules +e r n = 3 n = 2 n = 1 +e +e r y even Lecture 19, p 1 Today Nuclear Magnetic Resonance Using RF photons to drive transitions between nuclear spin orientations in
More informationFinal Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
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 informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Final Exam SOLUTIONS December 17, 2004, 7:30 a.m.- 9:30 a.m.
PHY464 Introduction to Quantum Mechanics Fall 4 Final Eam SOLUTIONS December 7, 4, 7:3 a.m.- 9:3 a.m. No other materials allowed. If you can t do one part of a problem, solve subsequent parts in terms
More informationMomentum expectation Momentum expectation value value for for infinite square well
Quantum Mechanics and Atomic Physics Lecture 9: The Uncertainty Principle and Commutators http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh Announcement Quiz in next class (Oct. 5): will cover Reed
More informationRecall the Goal. What IS the structure of an atom? What are the properties of atoms?
Recall the Goal What IS the structure of an atom? What are the properties of atoms? REMEMBER: structure affects function! Important questions: Where are the electrons? What is the energy of an electron?
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 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 informationChapter 28 Quantum Theory Lecture 24
Chapter 28 Quantum Theory Lecture 24 28.1 Particles, Waves, and Particles-Waves 28.2 Photons 28.3 Wavelike Properties Classical Particles 28.4 Electron Spin 28.5 Meaning of the Wave Function 28.6 Tunneling
More informationPhysics 486 Discussion 5 Piecewise Potentials
Physics 486 Discussion 5 Piecewise Potentials Problem 1 : Infinite Potential Well Checkpoints 1 Consider the infinite well potential V(x) = 0 for 0 < x < 1 elsewhere. (a) First, think classically. Potential
More informationspring mass equilibrium position +v max
Lecture 20 Oscillations (Chapter 11) Review of Simple Harmonic Motion Parameters Graphical Representation of SHM Review of mass-spring pendulum periods Let s review Simple Harmonic Motion. Recall we used
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 informationHydrogen atom energies. Friday Honors lecture. Quantum Particle in a box. Classical vs Quantum. Quantum version
Friday onors lecture Prof. Clint Sprott takes us on a tour of fractals. ydrogen atom energies Quantized energy levels: Each corresponds to different Orbit radius Velocity Particle wavefunction Energy Each
More informationSPARKS CH301. Why are there no blue fireworks? LIGHT, ELECTRONS & QUANTUM MODEL. UNIT 2 Day 2. LM15, 16 & 17 due W 8:45AM
SPARKS CH301 Why are there no blue fireworks? LIGHT, ELECTRONS & QUANTUM MODEL UNIT 2 Day 2 LM15, 16 & 17 due W 8:45AM QUIZ: CLICKER QUESTION Which of these types of light has the highest energy photons?
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 informationPhysics 1C. End of Chapter 30 Exam Preparationds
Physics 1C End of Chapter 30 Exam Preparationds Radioactive Decay Example The isotope 137 Cs is a standard laboratory source of gamma rays. The half-life of 137 Cs is 30 years. (a) How many 137 Cs atoms
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 informationProbability, Expectation Values, and Uncertainties
Chapter 5 Probability, Epectation Values, and Uncertainties As indicated earlier, one of the remarkable features of the physical world is that randomness is incarnate, irreducible. This is mirrored in
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 9, February 8, 2006
Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer Lecture 9, February 8, 2006 The Harmonic Oscillator Consider a diatomic molecule. Such a molecule
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 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 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 informationTutorial for PhET Sim Quantum Bound States
Tutorial for PhET Sim Quantum Bound States J. Mathias Weber * JILA and Department of Chemistry & Biochemistry, University of Colorado at Boulder With this PhET Sim, we will explore the properties of some
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 informationQuantum Mechanics. incorporate the phenomenon of quantum tunneling, which we explore in this theory for explaining the behavior of
chapter 41 Quantum Mechanics 41.1 The Wave Function 41. Analysis Model: Quantum Particle Under Boundary Conditions 41.3 The Schrödinger Equation 41.4 A Particle in a Well of Finite Height 41.5 Tunneling
More informationChapter 16 Waves. Types of waves Mechanical waves. Electromagnetic waves. Matter waves
Chapter 16 Waves Types of waves Mechanical waves exist only within a material medium. e.g. water waves, sound waves, etc. Electromagnetic waves require no material medium to exist. e.g. light, radio, microwaves,
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 informationMore on waves + uncertainty principle
More on waves + uncertainty principle ** No class Fri. Oct. 18! The 2 nd midterm will be Nov. 2 in same place as last midterm, namely Humanities at 7:30pm. Welcome on Columbus day Christopher Columbus
More informationStructure of the atom
Structure of the atom What IS the structure of an atom? What are the properties of atoms? REMEMBER: structure affects function! Important questions: Where are the electrons? What is the energy of an electron?
More informationChap. 3. Elementary Quantum Physics
Chap. 3. Elementary Quantum Physics 3.1 Photons - Light: e.m "waves" - interference, diffraction, refraction, reflection with y E y Velocity = c Direction of Propagation z B z Fig. 3.1: The classical view
More informationModern Physics notes Paul Fendley Lecture 6
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 6 Size of the atom A digression on hand-waving arguments Spectral lines Feynman, 2.4-5 Fowler, Spectra, The Bohr atom The size of the atom
More informationPhysics 43 Chapter 41 Homework #11 Key
Physics 43 Chapter 4 Homework # Key π sin. A particle in an infinitely deep square well has a wave function given by ( ) for and zero otherwise. Determine the epectation value of. Determine the probability
More informationModern Physics notes Paul Fendley Lecture 3
Modern Physics notes Paul Fendley fendley@virginia.edu Lecture 3 Electron Wavelength Probability Amplitude Which slit? Photons Born, IV.4 Feynman, 1.6-7, 2.1 Fowler, Rays and Particles The wavelength of
More informationLab 5: Measuring Magnetic Field of Earth
Dr. W. Pezzaglia Physics B, Spring 010 Page 1 Las Positas College Lab 5: Magnetic Field 010Mar01 Lab 5: Measuring Magnetic Field of Earth Mar 1, Monday: Lab 5 (today) Lab # due Video: Mechanical Universe
More informationHydrogen atom energies. From Last Time. Today. Another question. Hydrogen atom question. Compton scattering and Photoelectric effect
From ast Time Observation of atoms indicated quantized energy states. Atom only emitted certain wavelengths of light Structure of the allowed wavelengths indicated the what the energy structure was Quantum
More informationName Solutions to Test 3 November 7, 2018
Name Solutions to Test November 7 8 This test consists of three parts. Please note that in parts II and III you can skip one question of those offered. Some possibly useful formulas can be found below.
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 informationNo Lecture on Wed. But, there is a lecture on Thursday, at your normal recitation time, so please be sure to come!
Announcements Quiz 6 tomorrow Driscoll Auditorium Covers: Chapter 15 (lecture and homework, look at Questions, Checkpoint, and Summary) Chapter 16 (Lecture material covered, associated Checkpoints and
More informationPHYS Concept Tests Fall 2009
PHYS 30 Concept Tests Fall 009 In classical mechanics, given the state (i.e. position and velocity) of a particle at a certain time instant, the state of the particle at a later time A) cannot be determined
More informationIntro/Review of Quantum
Intro/Review of Quantum QM-1 So you might be thinking I thought I could avoid Quantum Mechanics?!? Well we will focus on thermodynamics and kinetics, but we will consider this topic with reference to the
More informationChem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 7, February 1, 2006
Chem 350/450 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 006 Christopher J. Cramer ecture 7, February 1, 006 Solved Homework We are given that A is a Hermitian operator such that
More information22.02 Intro to Applied Nuclear Physics
22.02 Intro to Applied Nuclear Physics Mid-Term Exam Solution Problem 1: Short Questions 24 points These short questions require only short answers (but even for yes/no questions give a brief explanation)
More informationIntro/Review of Quantum
Intro/Review of Quantum QM-1 So you might be thinking I thought I could avoid Quantum Mechanics?!? Well we will focus on thermodynamics and kinetics, but we will consider this topic with reference to the
More informationAppendix A. The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System
Appendix A The Particle in a Box: A Demonstration of Quantum Mechanical Principles for a Simple, One-Dimensional, One-Electron Model System Real quantum mechanical systems have the tendency to become mathematically
More informationChancellor Phyllis Wise invites you to a birthday party!
Chancellor Phyllis Wise invites you to a birthday party! 50 years ago, Illinois alumnus Nick Holonyak Jr. demonstrated the first visible light-emitting diode (LED) while working at GE. Holonyak returned
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 informationPhysics 1C. Class Review. "All good things must come to an end. --Proverb
Physics 1C Class Review "All good things must come to an end. --Proverb 1) Simple Harmonic Motion: For a mass on a spring, the force on the mass will be given by: In general, anything that exhibited simple
More informationAction Principles in Mechanics, and the Transition to Quantum Mechanics
Physics 5K Lecture 2 - Friday April 13, 2012 Action Principles in Mechanics, and the Transition to Quantum Mechanics Joel Primack Physics Department UCSC This lecture is about how the laws of classical
More informationWave Properties of Particles Louis debroglie:
Wave Properties of Particles Louis debroglie: If light is both a wave and a particle, why not electrons? In 194 Louis de Broglie suggested in his doctoral dissertation that there is a wave connected with
More information