Dimensional analysis
|
|
- Julianna Abigayle Small
- 5 years ago
- Views:
Transcription
1 Dimensional analysis Calvin W. Johnson September 9, 04 Dimensional analysis Dimensional analysis is a powerful and important tool in calculations. In fact, you should make it a habit to automatically use dimensional analysis to check not only your answers but also your equations. Later on we will see we can even use it to check integrals! Let me give an example: the Schrödinger equation for free particles (with no potential energy). In one dimensional systems, the -dependent Schrödinger equation is h m x ψ = i h t ψ () while the -independent equation is h ψ = Eψ. () m x In the above, x is position, t is, m is mass, and E energy. h is Planck s constant (actually it is h/π) which has dimensions of action = momentum length = energy. Let s dimensionally analyze the second equation first; we can do that even without fully understanding it. Because both equations are linear in the wavefunction ψ (meaning ψ appears at most first-order, no ψ or ψ 3 ), the dimension of ψ is not relevant here; in other words, the dimensions of ψ cancel out. (Below I will discuss how we can find the dimension of ψ.) The right-hand side of the equation has (leaving off the dimension of ψ) the dimension of energy. The left hand side has (again leaving off ψ) dimensions [ h] [m][x] = mass length (3) where the square brackets [...] signify dimension of. The next step will likely take some trial and error; don t feel bad if it takes you a few, or many, tries. I m simply not writing down for you all the wrong steps I took. The fundamental dimensions (units, see below for more on the distinction) are mass, length, and, and others such as energy, momentum, and action,
2 are composites of these. But we have a hint: we have an equation. We do not expect something like mass = length; that would be illogical. So we take action = energy and substitute it in the left-hand side: mass length = energy mass length (Again, if you first try action = momentum length there is nothing wrong or bad about that, it may or may not lead you to the correct answer as quickly.) At first this doesn t look helpful, but what we do is take out one of the energies and then we remember, I hope, that energy = energy mass length mass length energy = mass length (which I remember from kinetic energy = (/)mv and velocity v = length/). We then see that energy length = energy = energy mass mass length mass length mass length = energy; (4) in other words, we get dimensions of energy on both the left-hand and right-hand sides, so both sides of the equation agree in dimensions. Hurray! What about the -dependent Schrödinger equation? The left-hand side remains the same. The right-hand side, ignoring ψ as usual, is [ h] [t] (5) (the unit imaginary, i, is dimensionless of course). This is easier, as [ h] = action = energy and we get [ h] [t] = energy = energy (6) so both side match once again. If we take the Schrödinger equation including potential energy, we have (for the -independent equation) h ψ + V (x)ψ = Eψ. (7) m x
3 so the potential energy, V, must have dimensions of energy, as it does. Let s explore a little further with potential energy. I ll take a couple of cases. The harmonic oscillator. This is a fundamental problem in quantum mechanics as well as in classical mechanics and during this course we will study it in detail. Here V (x) = (/)mω x, where ω is the frequency of the oscillator and hence has dimensions of (). Although we expect the dimensions to work out to be energy, we need to check! So: [ mω x ] = mass length (8) which are the dimensions for energy (again, remember kinetic energy). So far so good. We can do something more, however. Ultimately we want to solve h m x ψ + mω x ψ = Eψ. Now the energy E, which will be a number (in fact, an eigenvalue as we will see), so it cannot depend upon x, but it will, of course, have to have dimensions of energy. When we solve this particular equation, we will find that the solutions with have E proportional to hω. Does this make sense? Yes: [ hω] = action frequency = energy = energy. (9) Good! In fact, the energy has come out in terms if h, ω, and m because those are the only dimensionful constants in the problem. It just so happens here that the energy does not depend on mass m. The linear potential. Suppose V (x) = kx. What must the dimensions of k be? Well, we want energy = [k] length, (0) so k has the ungainly dimensions of energy/length. (It s okay if you break it down to, say, mass-length/ but I don t find this additionally helpful). When we solve the equation h ψ + kxψ = Eψ. m x the only dimensionful constants are h, m, and k, and E must be proportional to some combination of them. It takes some trial and error to figure it out. It helps to see that energy/mass = (length/). But [ h] = energy and [k] = energy/length, then [ hk] = energy length 3 ()
4 and squaring this (yes, I know it seems crazy, but you have to try crazy things once in a while) [( hk) ] = energy 4 length. Hmm. If we divide by mass m, [( hk) /m] = energy 3 energy mass length () and use energy/mass = (length/) then we have So finally has dimensions of energy. [( hk) /m] = energy 3 (3) ( h k m )/3 (4) I realize this kind of manipulation is far from trivial; you have to be willing to try different combinations and be persistent. The most important messages you should get from these notes is that you have to pay attention to dimensions (and in fact, dimension are more important than actual numbers, because if the dimensions are wrong, the numbers don t matter). As a matter of reflex you should try to always confirm the dimensionality of an answer. We will get some practice in this when we solve the harmonic oscillator and the hydrogen atom. You try: confirm directly that expression (4) has dimensions of energy. You try (): If we take the potential energy V (x) = λx 4, what are the dimensions of λ? (See below for solution) You try (): For the potential energy V (x) = λx 4, which of the following combinations of constants have dimensions of energy? (it is possible that some, all, or none are). (a) ( h λ (b) ( hm ( (c) (d) ) 3/ ) /3 λ ) /3 h 4 λ m ( h 4 λ m ) /3 (answer will be given in class) Note Although somes units and dimensions are used interchangeably (and even I do somes as well, which is lazy of me), technically units refer to a specific value, such as seconds, meters, kilograms, which dimensions refers to, length, mass, respectively. I will not nitpick on this distinction but you should be aware of it. 4
5 Answers to selected problems () [λ] = energy/length 4. 5
Harmonic Oscillator with raising and lowering operators. We write the Schrödinger equation for the harmonic oscillator in one dimension as follows:
We write the Schrödinger equation for the harmonic oscillator in one dimension as follows: H ˆ! = "!2 d 2! + 1 2µ dx 2 2 kx 2! = E! T ˆ = "! 2 2µ d 2 dx 2 V ˆ = 1 2 kx 2 H ˆ = ˆ T + ˆ V (1) where µ is
More informationMATH325 - QUANTUM MECHANICS - SOLUTION SHEET 11
MATH35 - QUANTUM MECHANICS - SOLUTION SHEET. The Hamiltonian for a particle of mass m moving in three dimensions under the influence of a three-dimensional harmonic oscillator potential is Ĥ = h m + mω
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 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 informationMinimal coupling and Berry s phase
Phys460.nb 63 5 Minimal coupling Berry s phase Q: For charged quantum particles (e.g. electrons), if we apply an E&M field (E or B), the particle will feel the field. How do we consider the effect E B
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 informationBrief review of Quantum Mechanics (QM)
Brief review of Quantum Mechanics (QM) Note: This is a collection of several formulae and facts that we will use throughout the course. It is by no means a complete discussion of QM, nor will I attempt
More informationPHY 407 QUANTUM MECHANICS Fall 05 Problem set 1 Due Sep
Problem set 1 Due Sep 15 2005 1. Let V be the set of all complex valued functions of a real variable θ, that are periodic with period 2π. That is u(θ + 2π) = u(θ), for all u V. (1) (i) Show that this V
More informationP3317 HW from Lecture 7+8 and Recitation 4
P3317 HW from Lecture 7+8 and Recitation 4 Due Friday Tuesday September 25 Problem 1. In class we argued that an ammonia atom in an electric field can be modeled by a two-level system, described by a Schrodinger
More informationCh 125a Problem Set 1
Ch 5a Problem Set Due Monday, Oct 5, 05, am Problem : Bra-ket notation (Dirac notation) Bra-ket notation is a standard and convenient way to describe quantum state vectors For example, φ is an abstract
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 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 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 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 informationNumerical Solution of a Potential Final Project
Numerical Solution of a Potential Final Project 1 Introduction The purpose is to determine the lowest order wave functions of and energies a potential which describes the vibrations of molecules fairly
More informationChemistry 532 Problem Set 7 Spring 2012 Solutions
Chemistry 53 Problem Set 7 Spring 01 Solutions 1. The study of the time-independent Schrödinger equation for a one-dimensional particle subject to the potential function leads to the differential equation
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 information1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2
15 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: d 2 H = h2 2mdx + 1 2 2 kx2 (15.1) where k is the force
More informationCreation and Destruction Operators and Coherent States
Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function state harmonic oscillator wave function, We first rewrite the ground < x 0 >= ( π h )1/4 exp( x2 a 2 h )
More informationP3317 HW from Lecture and Recitation 7
P3317 HW from Lecture 1+13 and Recitation 7 Due Oct 16, 018 Problem 1. Separation of variables Suppose we have two masses that can move in 1D. They are attached by a spring, yielding a Hamiltonian where
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More informationPhysics 202 Laboratory 5. Linear Algebra 1. Laboratory 5. Physics 202 Laboratory
Physics 202 Laboratory 5 Linear Algebra Laboratory 5 Physics 202 Laboratory We close our whirlwind tour of numerical methods by advertising some elements of (numerical) linear algebra. There are three
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationThe Bohr Magneton and Bohr's second and third biggest mistakes
The Bohr Magneton and Bohr's second and third biggest mistakes by Miles Mathis Abstract: I will show several problems with the derivation of the Bohr Magneton. Using that analysis, I will look again at
More informationREVIEW: The Matching Method Algorithm
Lecture 26: Numerov Algorithm for Solving the Time-Independent Schrödinger Equation 1 REVIEW: The Matching Method Algorithm Need for a more general method The shooting method for solving the time-independent
More informationExpansion of Terms. f (x) = x 2 6x + 9 = (x 3) 2 = 0. x 3 = 0
Expansion of Terms So, let s say we have a factorized equation. Wait, what s a factorized equation? A factorized equation is an equation which has been simplified into brackets (or otherwise) to make analyzing
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 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 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 informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More information5.1 Classical Harmonic Oscillator
Chapter 5 Harmonic Oscillator 5.1 Classical Harmonic Oscillator m l o l Hooke s Law give the force exerting on the mass as: f = k(l l o ) where l o is the equilibrium length of the spring and k is the
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationPhys 172 Modern Mechanics Summer 2010
Phys 172 Modern Mechanics Summer 2010 r r Δ p = F Δt sys net Δ E = W + Q sys sys net surr r r Δ L = τ Δt Lecture 14 Energy Quantization Read:Ch 8 Reading Quiz 1 An electron volt (ev) is a measure of: A)
More informationThe Schroedinger equation
The Schroedinger equation After Planck, Einstein, Bohr, de Broglie, and many others (but before Born), the time was ripe for a complete theory that could be applied to any problem involving nano-scale
More informationHarmonic Oscillator I
Physics 34 Lecture 7 Harmonic Oscillator I Lecture 7 Physics 34 Quantum Mechanics I Monday, February th, 008 We can manipulate operators, to a certain extent, as we would algebraic expressions. By considering
More informationSolution to Proof Questions from September 1st
Solution to Proof Questions from September 1st Olena Bormashenko September 4, 2011 What is a proof? A proof is an airtight logical argument that proves a certain statement in general. In a sense, it s
More informationP3317 HW from Lecture and Recitation 10
P3317 HW from Lecture 18+19 and Recitation 10 Due Nov 6, 2018 Problem 1. Equipartition Note: This is a problem from classical statistical mechanics. We will need the answer for the next few problems, and
More informationExplanations of animations
Explanations of animations This directory has a number of animations in MPEG4 format showing the time evolutions of various starting wave functions for the particle-in-a-box, the free particle, and the
More information1 Planck-Einstein Relation E = hν
C/CS/Phys C191 Representations and Wavefunctions 09/30/08 Fall 2008 Lecture 8 1 Planck-Einstein Relation E = hν This is the equation relating energy to frequency. It was the earliest equation of quantum
More informationQUANTUM CHEMISTRY PROJECT 1
Chemistry 460 Fall 2017 Dr. Jean M. Standard September 11, 2017 QUANTUM CHEMISTRY PROJECT 1 OUTLINE This project focuses on applications and solutions of quantum mechanical problems involving one-dimensional
More informationPhysics Nov Bose-Einstein Gases
Physics 3 3-Nov-24 8- Bose-Einstein Gases An amazing thing happens if we consider a gas of non-interacting bosons. For sufficiently low temperatures, essentially all the particles are in the same state
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 informationSolving with Absolute Value
Solving with Absolute Value Who knew two little lines could cause so much trouble? Ask someone to solve the equation 3x 2 = 7 and they ll say No problem! Add just two little lines, and ask them to solve
More informationPhilosophy of QM Eighth lecture, 23 February 2005
Philosophy of QM 24.111 Eighth lecture, 23 February 2005 Strategy for analysis 1. Break the experimental situation up into steps. 2. At each step, look for easy cases. 3. Describe hard cases as linear
More informationChapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)
Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction
More informationSecond Quantization Method for Bosons
Second Quantization Method for Bosons Hartree-Fock-based methods cannot describe the effects of the classical image potential (cf. fig. 1) because HF is a mean-field theory. DFF-LDA is not able either
More informationProblem 1: A 3-D Spherical Well(10 Points)
Problem : A 3-D Spherical Well( Points) For this problem, consider a particle of mass m in a three-dimensional spherical potential well, V (r), given as, V = r a/2 V = W r > a/2. with W >. All of the following
More informationQuantum Mechanics in One Dimension. Solutions of Selected Problems
Chapter 6 Quantum Mechanics in One Dimension. Solutions of Selected Problems 6.1 Problem 6.13 (In the text book) A proton is confined to moving in a one-dimensional box of width.2 nm. (a) Find the lowest
More information1.3 Harmonic Oscillator
1.3 Harmonic Oscillator 1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total Hamiltonian looks like: H = h2 d 2 2mdx + 1 2 2 kx2 (1.3.1) where k is the force
More informationStatistical Interpretation
Physics 342 Lecture 15 Statistical Interpretation Lecture 15 Physics 342 Quantum Mechanics I Friday, February 29th, 2008 Quantum mechanics is a theory of probability densities given that we now have an
More informationPH 253 Final Exam: Solution
PH 53 Final Exam: Solution 1. A particle of mass m is confined to a one-dimensional box of width L, that is, the potential energy of the particle is infinite everywhere except in the interval 0
More informationPDEs in Spherical and Circular Coordinates
Introduction to Partial Differential Equations part of EM, Scalar and Vector Fields module (PHY2064) This lecture Laplacian in spherical & circular polar coordinates Laplace s PDE in electrostatics Schrödinger
More informationOrder of Magnitude Estimates in Quantum
Order of Magnitude Estimates in Quantum As our second step in understanding the principles of quantum mechanics, we ll think about some order of magnitude estimates. These are important for the same reason
More informationUNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS. BSc and MPhys Undergraduate Programmes in Physics LEVEL HE2
Phys/Level /1/9/Semester, 009-10 (1 handout) UNIVERSITY OF SURREY FACULTY OF ENGINEERING AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Undergraduate Programmes in Physics LEVEL HE PAPER 1 MATHEMATICAL,
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More informationQFT. Unit 1: Relativistic Quantum Mechanics
QFT Unit 1: Relativistic Quantum Mechanics What s QFT? Relativity deals with things that are fast Quantum mechanics deals with things that are small QFT deals with things that are both small and fast What
More informationVibrations and Rotations of Diatomic Molecules
Chapter 6 Vibrations and Rotations of Diatomic Molecules With the electronic part of the problem treated in the previous chapter, the nuclear motion shall occupy our attention in this one. In many ways
More informationSystematic Uncertainty Max Bean John Jay College of Criminal Justice, Physics Program
Systematic Uncertainty Max Bean John Jay College of Criminal Justice, Physics Program When we perform an experiment, there are several reasons why the data we collect will tend to differ from the actual
More informationQuantum Theory of Matter
Imperial College London Department of Physics Professor Ortwin Hess o.hess@imperial.ac.uk Quantum Theory of Matter Spring 014 1 Periodic Structures 1.1 Direct and Reciprocal Lattice (a) Show that the reciprocal
More informationChapter 1: Useful definitions
Chapter 1: Useful definitions 1.1. Cross-sections (review) The Nuclear and Radiochemistry class listed as a prerequisite is a good place to start. The understanding of a cross-section being fundamentai
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More informationNotes 11: OLS Theorems ECO 231W - Undergraduate Econometrics
Notes 11: OLS Theorems ECO 231W - Undergraduate Econometrics Prof. Carolina Caetano For a while we talked about the regression method. Then we talked about the linear model. There were many details, but
More informationBasic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi
Basic Quantum Mechanics Prof. Ajoy Ghatak Department of Physics Indian Institute of Technology, Delhi Module No. # 07 Bra-Ket Algebra and Linear Harmonic Oscillator II Lecture No. # 02 Dirac s Bra and
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 informationChemistry 432 Problem Set 4 Spring 2018 Solutions
Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and
More informationLecture 12. The harmonic oscillator
Lecture 12 The harmonic oscillator 107 108 LECTURE 12. THE HARMONIC OSCILLATOR 12.1 Introduction In this chapter, we are going to find explicitly the eigenfunctions and eigenvalues for the time-independent
More informationDavid J. Starling Penn State Hazleton PHYS 214
All the fifty years of conscious brooding have brought me no closer to answer the question, What are light quanta? Of course today every rascal thinks he knows the answer, but he is deluding himself. -Albert
More information129 Lecture Notes Relativistic Quantum Mechanics
19 Lecture Notes Relativistic Quantum Mechanics 1 Need for Relativistic Quantum Mechanics The interaction of matter and radiation field based on the Hamitonian H = p e c A m Ze r + d x 1 8π E + B. 1 Coulomb
More informationSample Quantum Chemistry Exam 1 Solutions
Chemistry 46 Fall 217 Dr Jean M Standard September 27, 217 Name SAMPE EXAM Sample Quantum Chemistry Exam 1 Solutions 1 (24 points Answer the following questions by selecting the correct answer from the
More informationTime-Independent Perturbation Theory
4 Phys46.nb Time-Independent Perturbation Theory.. Overview... General question Assuming that we have a Hamiltonian, H = H + λ H (.) where λ is a very small real number. The eigenstates of the Hamiltonian
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 informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More informationASTRO 114 Lecture Okay. We re now gonna continue discussing and conclude discussing the entire
ASTRO 114 Lecture 55 1 Okay. We re now gonna continue discussing and conclude discussing the entire universe. So today we re gonna learn about everything, everything that we know of. There s still a lot
More informationProblem Set 7: Solutions
University of Alabama Department of Physics and Astronomy PH 53 / LeClair Fall 010 Problem Set 7: Solutions 1. a) How many different photons can be emitted by hydrogen atoms that undergo transitions from
More information3. Quantum Mechanics in 3D
3. Quantum Mechanics in 3D 3.1 Introduction Last time, we derived the time dependent Schrödinger equation, starting from three basic postulates: 1) The time evolution of a state can be expressed as a unitary
More informationStudent Exploration: Air Track
Name: Date: Student Exploration: Air Track Vocabulary: air track, approach velocity, conservation of energy, conservation of momentum, elasticity, kinetic energy, momentum, separation velocity, velocity
More informationMITOCW watch?v=fxlzy2l1-4w
MITOCW watch?v=fxlzy2l1-4w PROFESSOR: We spoke about the hydrogen atom. And in the hydrogen atom, we drew the spectrum, so the table, the data of spectrum of a quantum system. So this is a question that
More informationchmy361 Lec42 Tue 29nov16
chmy361 Lec42 Tue 29nov16 1 Quantum Behavior & Quantum Mechanics Applies to EVERYTHING But most evident for particles with mass equal or less than proton Absolutely NECESSARY for electrons and light (photons),
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More information1 Schrödinger s Equation
Physical Electronics Quantum Mechanics Handout April 10, 003 1 Schrödinger s Equation One-Dimensional, Time-Dependent version Schrödinger s equation originates from conservation of energy. h Ψ m x + V
More informationApplications of Diagonalization
Applications of Diagonalization Hsiu-Hau Lin hsiuhau@phys.nthu.edu.tw Apr 2, 200 The notes cover applications of matrix diagonalization Boas 3.2. Quadratic curves Consider the quadratic curve, 5x 2 4xy
More informationInteracting Fermi Gases
Interacting Fermi Gases Mike Hermele (Dated: February 11, 010) Notes on Interacting Fermi Gas for Physics 7450, Spring 010 I. FERMI GAS WITH DELTA-FUNCTION INTERACTION Since it is easier to illustrate
More informationQuantum Mechanics II
Quantum Mechanics II Prof. Boris Altshuler March 8, 011 1 Lecture 19 1.1 Second Quantization Recall our results from simple harmonic oscillator. We know the Hamiltonian very well so no need to repeat here.
More informationPhysics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I
Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular
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 informationUnits, limits, and symmetries
Units, limits, and symmetries When solving physics problems it s easy to get overwhelmed by the complexity of some of the concepts and equations. It s important to have ways to navigate through these complexities
More informationLecture 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 informationOne-dimensional Schrödinger equation
Chapter 1 One-dimensional Schrödinger equation In this chapter we will start from the harmonic oscillator to introduce a general numerical methodology to solve the one-dimensional, time-independent Schrödinger
More informationHypothesis testing I. - In particular, we are talking about statistical hypotheses. [get everyone s finger length!] n =
Hypothesis testing I I. What is hypothesis testing? [Note we re temporarily bouncing around in the book a lot! Things will settle down again in a week or so] - Exactly what it says. We develop a hypothesis,
More information1 Basics of Quantum Mechanics
1 Basics of Quantum Mechanics 1.1 Admin The course is based on the book Quantum Mechanics (2nd edition or new international edition NOT 1st edition) by Griffiths as its just genius for this level. There
More informationLecture 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 information26 Group Theory Basics
26 Group Theory Basics 1. Reference: Group Theory and Quantum Mechanics by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian.
More informationMolecular Physics. Attraction between the ions causes the chemical bond.
Molecular Physics A molecule is a stable configuration of electron(s) and more than one nucleus. Two types of bonds: covalent and ionic (two extremes of same process) Covalent Bond Electron is in a molecular
More information2 Introduction to perturbation methods
2 Introduction to perturbation methods 2.1 What are perturbation methods? Perturbation methods are methods which rely on there being a dimensionless parameter in the problem that is relatively small: ε
More informationS.E. of H.O., con con t
Quantum Mechanics and Atomic Physics Lecture 11: The Harmonic Oscillator: Part I http://www.physics.rutgers.edu/ugrad/361 Prof. Sean Oh The Classical Harmonic Oscillator Classical mechanics examples Mass
More informationMATH240: Linear Algebra Review for exam #1 6/10/2015 Page 1
MATH24: Linear Algebra Review for exam # 6//25 Page No review sheet can cover everything that is potentially fair game for an exam, but I tried to hit on all of the topics with these questions, as well
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationName Final Exam December 7, 2015
Name Final Exam December 7, 015 This test consists of five parts. Please note that in parts II through V, you can skip one question of those offered. Part I: Multiple Choice (mixed new and review questions)
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 informationONE AND MANY ELECTRON ATOMS Chapter 15
See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum.
More information