# The Central Equation

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 The Central Equation Mervyn Roy May 6, Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given wavevector, k, and band index, n, to find the energy, E nk, and the single particle wavefunction, ψ nk. In a crystal, the eigenstates, ψ nk, obey Bloch s theorem, and ψ nk (r) = e ik r u nk (r), () where u nk has the periodicity of the lattice. So u nk (r + T) = u nk (r) if the translation vector, T = n 1 a 1 + n a + n 3 a 3 is a sum over unit cell vectors, a 1, a, a 3, of the crystal. As the u nk have the periodicity of the lattice we can expand them as a Fourier series, u nk = c (k)e i r, so that ψ nk = c (k)e i(k+) r, (3) where are reciprocal lattice vectors of the crystal, and the c (k) are the expansion coefficients of the single particle wavefunction at each k. The potential, v s (r), must also have the periodicity of the lattice, so v s (r) = V e i r. (4) 1

2 Before substituting into the Schrödinger equation (Eq. (1)) we note that, and ψ nk = v s (r)ψ nk = k + c e i(k+) r, (5) V c Q (k)e i( +Q+k) r. (6) Q Then, in Eq. (1), we have ( ) k + E nk e i(k+) r + V c Q (k)e i( +Q+k) r = 0. (7) It is now advantageous to rearrange the double sum involving V so that the plane wave part in the double sum becomes e i(k+) r. To do this, we write Q in terms of another reciprocal lattice vector, = + Q and replace Q with. Because the sums are over an infinite set of reciprocal lattice vectors, for each fixed, we can replace the infinite sum over Q with an infinite sum over the equivalent set of vectors,. Then, ( ) k + E nk c (k)e i(k+) r + V c (k)e i(+k) r = 0, (8) or, { ( k ) + e i(k+) r E nk c (k) + } V c (k) = 0. (9) The left hand side of this equation can only be zero if, for each, the term inside the curly brackets {...} is zero. Hence, ( ) k + E nk c (k) + V c (k) = 0. (10) Equation (10) is called the central equation. The form above is similar to that given by Tanner (Introduction to the physics of electrons in solids). We can also rearrange the central equation in a number of different ways. To obtain the form used by Martin (Electronic Structure) we must rewrite the infinite sum over reciprocal lattice vectors in Eq. (10). We choose a new reciprocal lattice vector, =, and replace the infinite sum over with an equivalent infinite sum over. Then, ( ) k + E nk c (k) + V c (k) = 0. (11) Q

3 Because both terms on the left now contain a coefficient that looks like c Q we can place both terms within the sum over and factorise out the c. [( k + ) ] E nk δ( ) + V c (k) = 0, (1) where the delta function δ( ) ensures we only get the required = terms in the first term in the sum over. Solving the central equation From Eq. (1) it should be clear that the central equation is an infinite set of coupled simultaneous equations - with one equation for each possible reciprocal lattice vector. We can solve the central equation numerically or, in some very simple cases, analytically. In whatever scheme we use we must truncate the infinite sum over reciprocal lattice vectors at some point. When solving the central equation it is common to take V 0 = 0. The V 0 Fourier component is the average value of the potential and we can always set the average of the potential to zero by adding or subtracting a constant offset to the energies..1 Truncating the series: example with 3 terms To explicitly see how Eq. (1) becomes a set of coupled simultaneous equations let us imagine that the infinite sum over reciprocal lattice vectors is restricted to just three possibilities, g, 0, and g. In this case in Eq. (1) has three possible values ( g, 0, g) and, in the sum over reciprocal lattice vectors, must sum over the three possible values g, 0, g. The central equation will therefore give us three simultaneous equations each containing three terms. For example, if we select = g we can write the first of the three simultaneous equations as, [( k + ) ] E nk δ( g ) + V g c (k) = 0. (13) = g,0,g In the first term in the sum, = g and, because we have picked = g, we get a contribution from the part of the equation containing the delta 3

4 function. In this case, when g, we obviously only retain the terms containing the Fourier components of the potential V. Remembering that V 0 = 0 we can therefore write Eq. (13) as, ( ) k g E nk c g (k) + V g c 0 (k) + V g c g (k) = 0. (14) Similarly, we can write down another equation for the case when = 0. In this case we obviously do not get a contribution from the delta function term if = g or = g, and ( ) k V g c g (k) + E nk c 0 (k) + V g c g (k) = 0. (15) Finally, we get the third equation in the set by choosing = g, ( ) k + g V g c g (k) + V g c 0 (k) + E c g (k) = 0. (16) We can write this set of simultaneous equations as a matrix equation, ( 1 k ) g E nk ( V g V g V 1 ) c g (k) g k E nk ( V g V g V 1 g k + ) c 0 (k) = 0,(17) g E nk c g(k) and this only has non-trivial solutions if the determinant, ( 1 k ) g E nk V g V g ( V 1 ) g k E nk V g ( V g V 1 g k + ) = 0. (18) g E nk. eneral form In shorthand form we can write our matrix equation (Eq. (17)) as, [H EI] c = 0, (19) where I is the unit matrix and E are the eigenvalues E nk. From our previous example (with the sum restricted to 3 reciprocal lattice vectors) H is the hamiltonian matrix, ( 1 k g ) V g V g H = ( V 1 g k ) ( V g V g V 1 g k + g ) 4, (0)

5 and c is the eigenvector that contains the expansion coefficients, c g (k) c = c 0 (k). (1) c g (k) The notation in Eq. (19) is completely general. So, to find the single particle energies, we must always find the values of E so that the determinant, H EI = 0. () In our previous example H was a 3 3 matrix, but we could have restricted the infinite series in the central equation to any number of terms. For example, if we had included N reciprocal lattice vectors we would have an N N hamiltonian matrix..3 The origin of band gaps at the Brillouin zone boundary.3.1 The empty lattice If we set v s (r) = 0 then all of the off-diagonal terms in the hamiltonian matrix are zero and we get a simple set of uncoupled equations that immediately give us the allowed energies, E nk = 1 k + n. (3) These are essentially the familiar free electron solutions, E = p /m, but, in a periodic crystal wavevectors k and k + are equivalent. We can use this equivalence to simplify the way in which we represent the states: any wavevector k that is larger than a reciprocal lattice vector,, can be represented by a shorter wavevector inside the first Brillouin zone, k = k, because of the periodicity of the reciprocal lattice. In the reduced zone scheme we represent all of the possible E nk in terms of wavevectors within the first Brillouin zone. This is illustrated schematically in Fig Weak periodic potential: solution at k = g/ First we assume that the potential is weak and periodic such that v s (r) = V cos(g r), only the V g Fourier components of the potential are non-zero, and V g = V g. Then, as V is small, we might imagine that the wavefunctions ψ nk = c e i(k+) r will be similar to the free electron solutions, ψ nk = 5

6 3.5 g Energy (ev) g 0 -g -g -g/ Γ g/ g g Figure 1: Illustration of the reduced zone scheme. The parts of the E-k curve outside the first Brillouin zone are folded back into the first Brillouin zone by translating by an integer number of reciprocal lattice vectors. The dotted vertical lines denote the edges of the first Brillouin zone at g/ and g/. k e ik r and it should therefore be possible to restrict the infinite sum in Eq. (1) to just a few terms. As we are interested in solutions near the Brillouin zone boundary, where k = g/ we can restrict this sum to cover just two reciprocal lattice vectors. Near k = g/ we assume that the = 0 and = g components are the most important so that, ψ nk = c 0 e ik r + c g e i(k g) r. (4) The hamiltonian matrix is then just a matrix and, by retaining only the = g and = 0 terms in Eq. (17) we can find the allowed energies from, ( 1 k ) g E nk V g ( V 1 ) g k E nk = 0. (5) If we use the shorthand λ k = 1 k then the allowed energies are obtained by solving the resultant quadratic equation (λ k g E)(λ k E) Vg = 0. The solutions for the allowed energies near k = g/ are then, E = 1 (λ k + λ k g ) ± 1 [ ] 1/ (λ k λ k g ) + 4Vg. (6) At the zone boundary, k = g/ and λ k = λ k g = g /8. So, at the zone boundary there are two possible values of the energy, E = g /8 ± V g. (7) 6

7 A periodic potential, no matter how small, therefore opens up a gap at the Brillouin zone boundary of V g. This is illustrated in figure. Energy 3g /8 g /8 V g Figure : E(k) relation in the presence of a weak periodic potential (solid red line). The faint dotted line shows the empty lattice solutions. The weak periodic potential introduces a gap of V g at the Brillouin zone boundary where k = g/. Γ g/ k.4 Numerical solution In principle, from Eq. (1), the hamiltonian matrix is infinite in extent, but to solve the central equation we must always truncate the sum over reciprocal lattice vectors at some point. By doing this we restrict the reciprocal lattice vectors in our Fourier expansion of the single-particle wavefunction (Eq. 3) to be less than some maximum length max, ψ nk = c (k)e i(k+) r. (8) < max As with any Fourier series, we can include more and more terms in the expansion until the wavefunction is as accurate as we need it to be. How do we know how many terms to include? As always, we can use the variational theorem to help us decide. The variational theorem tells us that each E nk will decrease as max increases and our approximate form for the wavefunction improves. We can therefore examine the convergence of E nk with max to determine how many reciprocal lattice vectors we must include to obtain the single-particle energies to a given accuracy (see, for example, figure 3). Many computer codes have been developed to solve systems of equations, like the central equation, numerically. To solve Eq. (1) we must essentially give our computer code just two pieces of information. First we need to specify how many reciprocal lattice vectors we would like to include in our 7

8 expansion of the single particle wavefunction (this sets the size of the N N hamiltonian in Eq. (19)). Second we need to supply the Fourier components of the potential, V, that set each element of the hamiltonian matrix in Eq. (19). There exist many sophisticated numerical routines which will then diagonalise this matrix - essentially, calculate and report the values of E that satisfy H EI = Link to density functional theory In a density functional theory (DFT) calculation the single particle Schrödinger equations (Eq. (1)) are the equations for the fictitious Kohn-Sham orbitals. These orbitals give the real electron density of the system by n(r) = nk ψ nk, where the sum is over the occupied single particle energy levels. This density can then be used to calculate the total energy, E = T s [n] + v s [n] = T s [n] + v(r)n(r)dr + E H [n] + E XC [n], (9) where T s is the kinetic energy functional, E H is the Hartree energy, and E XC is the exchange and correlation energy. As the variational theorem tells us that the total energy, E, must decrease as we increase max, we can examine the convergence of E with max to determine how many Fourier components we need to retain in the expansion Eq. (3). In a DFT calculation it is common to use a plane wave cut-off energy, E cut, to specify the maximum number of reciprocal lattice vectors to be included in the calculation at each k point. The cut-off energy, E cut sets max according to, E cut = 1 k + max. (30) 8

9 -8.58 Total Energy, E (H) E(E cut )-E(4) E cut (H) E cut (H) Figure 3: Example calculation showing total energy, E in Hartrees (H) against the plane wave cut-off energy, E cut, which determines the number of plane waves in the expansion, Eq. (3). The faint horizontal line denotes the converged energy of E = H at E cut = 4. The inset shows the difference between the total energy calculated at each E cut and the converged energy calculated at E cut = 4 H. 9

### The nearly-free electron model

Handout 3 The nearly-free electron model 3.1 Introduction Having derived Bloch s theorem we are now at a stage where we can start introducing the concept of bandstructure. When someone refers to the bandstructure

### An introduction to the electric conduction in crystals

An introduction to the electric conduction in crystals Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 23, 2010 1 Velocity of band electrons The electronic states

### PHY 140A: Solid State Physics. Solution to Homework #8

HY 140A: Solid State hysics Solution to Homework #8 Xun Jia 1 December 11, 006 1 Email: jiaxun@physics.ucla.edu roblem #1 Sqrare lattice, free electron energies. (a). Show for a simple square lattice (two

### Introduces the bra and ket notation and gives some examples of its use.

Chapter 7 ket and bra notation Introduces the bra and ket notation and gives some examples of its use. When you change the description of the world from the inutitive and everyday classical mechanics to

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

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

### Seminar Topological Insulators

Seminar Topological Insulators The Su-Schrieffer-Heeger model 1 These slides are based on A Short Course on Topological Insulators by J. K. Asbóth, L. Oroszlány, A. Pályi; arxiv:1509.02295v1 2 Outline

### Hermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)

CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system

### The Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933

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

### Plate waves in phononic crystals slabs

Acoustics 8 Paris Plate waves in phononic crystals slabs J.-J. Chen and B. Bonello CNRS and Paris VI University, INSP - 14 rue de Lourmel, 7515 Paris, France chen99nju@gmail.com 41 Acoustics 8 Paris We

### 1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as

Chapter 3 (Lecture 4-5) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series

### DFT in practice : Part I. Ersen Mete

plane wave expansion & the Brillouin zone integration Department of Physics Balıkesir University, Balıkesir - Turkey August 13, 2009 - NanoDFT 09, İzmir Institute of Technology, İzmir Outline Plane wave

### Electronic Structure Methods. by Daniel Rohr Vrije Universiteit Amsterdam

Electronic Structure Methods by Daniel Rohr Vrije Universiteit Amsterdam drohr@few.vu.nl References WFT Szaboo & Ostlund Modern Quantum Chemistry Helgaker, Jørgensen & Olsen Molecular Electronic-Structure

### CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

### Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following

Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following form in two and three dimensions: k (r + R) = e 2 ik

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

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

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

### 1.5 Light absorption by solids

1.5 Light absorption by solids Bloch-Brilloin model L e + + + + + allowed energy bands band gaps p x In a unidimensional approximation, electrons in a solid experience a periodic potential due to the positively

### Bloch waves and Bandgaps

Bloch waves and Bandgaps Chapter 6 Physics 208, Electro-optics Peter Beyersdorf Document info Ch 6, 1 Bloch Waves There are various classes of boundary conditions for which solutions to the wave equation

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### A Beginner s Guide to Materials Studio and DFT Calculations with Castep. P. Hasnip (pjh503@york.ac.uk)

A Beginner s Guide to Materials Studio and DFT Calculations with Castep P. Hasnip (pjh503@york.ac.uk) September 18, 2007 Materials Studio collects all of its files into Projects. We ll start by creating

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### Some general points about bandstructure

Handout 5 Some general points about bandstructure 5.1 Comparison of tight-binding and nearly-free-electron bandstructure Let us compare a band of the nearly-free-electron model with a one-dimensional tight-binding

### 4. The Infinite Square Well

4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

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

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

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### A comprehensive theory of conductivity

A comprehensive theory of conductivity Balázs Hetényi Department of Physics Bilkent University Ankara, Turkey Topology and physics Quantum Hall effect (integer & fractional ) Topological insulators Skyrmions

### Lecture 3: Electron statistics in a solid

Lecture 3: Electron statistics in a solid Contents Density of states. DOS in a 3D uniform solid.................... 3.2 DOS for a 2D solid........................ 4.3 DOS for a D solid........................

### Lecture 2: Angular momentum and rotation

Lecture : Angular momentum and rotation Angular momentum of a composite system Let J and J be two angular momentum operators. One might imagine them to be: the orbital angular momentum and spin of a particle;

### Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

### Ground State of the He Atom 1s State

Ground State of the He Atom s State First order perturbation theory Neglecting nuclear motion H m m 4 r 4 r r Ze Ze e o o o o 4 o kinetic energies attraction of electrons to nucleus electron electron repulsion

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### 1 of 43. Simultaneous Equations

1 of 43 Simultaneous Equations Simultaneous Equations (Graphs) There is one pair of values that solves both these equations: x + y = 3 y x = 1 We can find the pair of values by drawing the lines x + y

### 7. DYNAMIC LIGHT SCATTERING 7.1 First order temporal autocorrelation function.

7. DYNAMIC LIGHT SCATTERING 7. First order temporal autocorrelation function. Dynamic light scattering (DLS) studies the properties of inhomogeneous and dynamic media. A generic situation is illustrated

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### From Fourier Series to Fourier Integral

From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Dynamics. Figure 1: Dynamics used to generate an exemplar of the letter A. To generate

Dynamics Any physical system, such as neurons or muscles, will not respond instantaneously in time but will have a time-varying response termed the dynamics. The dynamics of neurons are an inevitable constraint

### 8.3. Solution by Gauss Elimination. Introduction. Prerequisites. Learning Outcomes

Solution by Gauss Elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated

### 3.4. Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes

Solving Simultaneous Linear Equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### A. The wavefunction itself Ψ is represented as a so-called 'ket' Ψ>.

Quantum Mechanical Operators and Commutation C I. Bra-Ket Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates

### Lecture 18 Time-dependent perturbation theory

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

### The Essentials of Quantum Mechanics

The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008-Oct-22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum

### Social Media Mining. Network Measures

Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

### Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

### Lecture 2: Semiconductors: Introduction

Lecture 2: Semiconductors: Introduction Contents 1 Introduction 1 2 Band formation in semiconductors 2 3 Classification of semiconductors 5 4 Electron effective mass 10 1 Introduction Metals have electrical

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

PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,

### 3.4. Solving simultaneous linear equations. Introduction. Prerequisites. Learning Outcomes

Solving simultaneous linear equations 3.4 Introduction Equations often arise in which there is more than one unknown quantity. When this is the case there will usually be more than one equation involved.

### Notes on wavefunctions

Notes on wavefunctions The double slit experiment In the double slit experiment, a beam of light is send through a pair of slits, and then observed on a screen behind the slits. At first, we might expect

### Mixed states and pure states

Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements

### Foundations of Chemical Kinetics. Lecture 20: The master equation

Foundations of Chemical Kinetics Lecture 20: The master equation Marc R. Roussel Department of Chemistry and Biochemistry Transition rates Suppose that Ps (t) is the probability that a system is in a state

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

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

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

### Particle in a Box : Absorption Spectrum of Conjugated Dyes

Particle in a Box : Absorption Spectrum of Conjugated Dyes Part A Recording the Spectra and Theoretical determination of λ max Theory Absorption bands in the visible region of the spectrum (350-700 nm)

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### 0% (0 out of 5 correct) The questions marked with symbol have not been graded.

Page 1 of 15 0% (0 out of 5 correct) The questions marked with symbol have not been graded. Responses to questions are indicated by the symbol. 1. Which of the following materials may form crystalline

### 1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### For the case of an N-dimensional spinor the vector α is associated to the onedimensional . N

1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. on-hermitian

### Mathematical Notation and Symbols

Mathematical Notation and Symbols 1.1 Introduction This introductory block reminds you of important notations and conventions used throughout engineering mathematics. We discuss the arithmetic of numbers,

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Linear Systems and Gaussian Elimination

Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### Biggar High School Mathematics Department. National 4 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 4 Learning Intentions & Success Criteria: Assessing My Progress Expressions and Formulae Topic Learning Intention Success Criteria I understand this Algebra

### 2 Creation and Annihilation Operators

Physics 195 Course Notes Second Quantization 030304 F. Porter 1 Introduction This note is an introduction to the topic of second quantization, and hence to quantum field theory. In the Electromagnetic

### Algebra Revision Sheet Questions 2 and 3 of Paper 1

Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### FURTHER VECTORS (MEI)

Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

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

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

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### Stochastic Doppler shift and encountered wave period distributions in Gaussian waves

Ocean Engineering 26 (1999) 507 518 Stochastic Doppler shift and encountered wave period distributions in Gaussian waves G. Lindgren a,*, I. Rychlik a, M. Prevosto b a Department of Mathematical Statistics,

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

### The Classical Linear Regression Model

The Classical Linear Regression Model 1 September 2004 A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y 1,

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture No. # 31 Recursive Sets, Recursively Innumerable Sets, Encoding

### Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

### There are four common ways of finding the inverse z-transform:

Inverse z-transforms and Difference Equations Preliminaries We have seen that given any signal x[n], the two-sided z-transform is given by n x[n]z n and X(z) converges in a region of the complex plane

### The Quantum Harmonic Oscillator Stephen Webb

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

### A Simple Introduction to Finite Element Analysis

A Simple Introduction to Finite Element Analysis Allyson O Brien Abstract While the finite element method is extensively used in theoretical and applied mathematics and in many engineering disciplines,

### The Application of Density Functional Theory in Materials Science

The Application of Density Functional Theory in Materials Science Slide 1 Outline Atomistic Modelling Group at MUL Density Functional Theory Numerical Details HPC Cluster at the MU Leoben Applications

### Solving simultaneous equations. Jackie Nicholas

Mathematics Learning Centre Solving simultaneous equations Jackie Nicholas c 2005 University of Sydney Mathematics Learning Centre, University of Sydney 1 1 Simultaneous linear equations We will introduce

### 2. Introduction and Chapter Objectives

Real Analog - Circuits Chapter 2: Circuit Reduction 2. Introduction and Chapter Objectives In Chapter, we presented Kirchoff s laws (which govern the interactions between circuit elements) and Ohm s law

### The Not-Formula Book for C1

Not The Not-Formula Book for C1 Everything you need to know for Core 1 that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes

### FOURIER TRANSFORM METHODS IN GEOPHYSICS David Sandwell, January, 2013

1 FOURIER TRANSFORM METHODS IN GEOPHYSICS David Sandwell, January, 2013 1. Fourier Transforms Fourier transform are use in many areas of geophysics such as image processing, time series analysis, and antenna

### Rotational and Vibrational Transitions

Rotational and Vibrational Transitions Initial questions: How can different molecules affect how interstellar gas cools? What implications might this have for star formation now versus star formation before

### Localized orbitals and localized basis sets

Localized orbitals and localized basis sets STUDENT CHALLENGE Mike Towler The Towler Institute via del Collegio 22 Vallico Sotto Email: mdt26@cam.ac.uk 1 Linear scaling QMC 0.0 0.2 0.5 0.7 1.0 1.2 1.4

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### MATHEMATICAL BACKGROUND

Chapter 1 MATHEMATICAL BACKGROUND This chapter discusses the mathematics that is necessary for the development of the theory of linear programming. We are particularly interested in the solutions of a

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Section 1.1 Real numbers. Set Builder notation. Interval notation

Section 1.1 Real numbers Set Builder notation Interval notation Functions a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function

### Foundation. Scheme of Work. Year 10 September 2016-July 2017

Foundation Scheme of Work Year 10 September 016-July 017 Foundation Tier Students will be assessed by completing two tests (topic) each Half Term. PERCENTAGES Use percentages in real-life situations VAT

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### 8.3. Gauss elimination. Introduction. Prerequisites. Learning Outcomes

Gauss elimination 8.3 Introduction Engineers often need to solve large systems of linear equations; for example in determining the forces in a large framework or finding currents in a complicated electrical