PHYS3060 Quantum Mechanics Solution to Problem Set 6
|
|
- Ruth Robinson
- 6 years ago
- Views:
Transcription
1 PHYS3060 Quantum Mechanics Solution to Problem Set 6 Multi-Electron Atoms (a) Write down the time-independent Schrödinger equation for a lithium atom (Z=3; i.e. three electrons) and explain the purpose of every term in this equation (there should be 0). You may use the shorthand notation (e.g. H, V 2, see lecture notes). If you do, make sure you define the shorthand symbols that you use. In short hand notation, the time-independent Schrödinger equation for the lithium atom reads [Ĥ + Ĥ2 + Ĥ3 + V 2 + V 3 + V 23 ] ψ ( r, r 2, r 3 ) = Eψ ( r, r 2, r 3 ) () The three Ĥi operators are defined as follows Ĥ i = h2 2m 2 3e2 4πɛ 0 r i (2) in this, the first and second term represent the kinetic energy of the electron and the electron-nucleus Coulomb attraction, respectively. (Remember that the atomic number of lithium is Z=3). For the three electrons i=,2, and 3, there are a total of six terms (three kinetic energy terms and three electron-nucleus attraction terms). The electron-electron repulsion between pairs of electrons is represented by the three V ij terms, which are defined as follows: V ij = + e 2 4πɛ 0 r i r j (3) Note the + sign (this was a fairly common mistake!); electrons are negatively charged, and thus the Coulomb energy between two electrons must be positive. The last (0th) term on the right hand side of Eq. is the total energy term. (b) Assume that the three particle eigenfunction ψ(r,r 2,r 3 ) can be written as a product of three single electron eigenfunction, i.e. ψ(r, r 2, r 3 ) = ψ n (r )ψ n2 (r 2 )ψ n3 (r 3 ). (4) Assume further that the electron-electron repulsion experienced by electron i is simply a constant value U i dependent on the main quantum number (n i ) of the electron. Use this to simplify the Schrödinger equation. Use the method of separation of variables to separate this simplified equation into three Schroedinger equations; one for each electron. Be judicious in your choice of separation variables, such that E, E 2, and E 3 are the energies of the first, second, and third electron, respectively, and that E=E +E 2 +E 3. As per instructions, we simplify the Schrödinger equation and use single-index constants U i to represent the electron-electron repulsion. We write the three-electron eigenfunction ψ( r, r 2, r 3 ) as product of three singleelectron eigenfunctions ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ). With these changes, Eq. now reads [Ĥ + Ĥ2 + Ĥ3 + U + U 2 + U 3 ] ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) = Eψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) (5) In this form, all terms are one-index terms, that is, they depend on the position of one electron only, which allows us to apply the method of separation of variables. We will go through this here step-by-step. First evaluate the square bracket on the left hand side.
2 2 Ĥ ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + Ĥ2ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + Ĥ3ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + (6) U ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + U 2 ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + U 3 ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) = Eψ n ( r )ψ n2 ( r 2 )ψ n 3( r 3 ) It is important to recognize here that the Ĥi s are not just simple factors and that the ψ s on their right cannot simply be cancelled. Ĥ i is a differential operator (see definition in Eq. 2) that modify those ψ s that are associated with the coordinate r i ; it leaves the others unaffected. This means for the Ĥψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) term that we must leave ψ n ( r ) on the right hand side of Ĥ. However, we can pull over ψ n2 ( r 2 ) and ψ n3 ( r 3 ) to the left hand side of Ĥ (... and, will in due course, cancel these factors). Doing this for all terms, we get: ψ n2 ( r 2 )ψ n3 ( r 3 )Ĥψ n ( r ) + ψ n ( r )ψ n3 ( r 3 )Ĥ2ψ n2 ( r 2 ) + ψ n ( r )ψ n2 ( r 2 )Ĥ3ψ n3 ( r 3 ) + (7) U ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + U 2 ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) + U 3 ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) = Eψ n ( r )ψ n2 ( r 2 )ψ n 3( r 3 ) Division by ψ n ( r )ψ n2 ( r 2 )ψ n3 ( r 3 ) leads to ψ n ( r )Ĥψ n ( r ) + ψ n2 ( r 2 )Ĥ2ψ n2 ( r 2 ) + ψ n3 ( r 3 )Ĥ3ψ n3 ( r 3 ) + U + U 2 + U 3 = E (8) Reorder such that terms dependent on electron are on the left hand side and all other terms on the right hand side. ψ n ( r )Ĥψ n ( r ) + U = E U 2 U 3 ψ n2 ( r 2 )Ĥ2ψ n2 ( r 2 ) ψ n3 ( r 3 )Ĥ3ψ n3 ( r 3 ) (9) In this form, everything associated with electron is on one side, and everything associated with the other two electrons is on the other side. We can therefore separate the equation into two, introducing a separation variable that is conveniently called E n. The left and right hand side thus read respectively and ψ n ( r )Ĥψ n ( r ) + U = E n (0) E U 2 U 3 ψ n2 ( r 2 )Ĥ2ψ n2 ( r 2 ) ψ n3 ( r 3 )Ĥ3ψ n3 ( r 3 ) = E n () Multiplication by ψ n ( r ) turns the left-hand-side equation (Eq. 0) into the familiar form of a Schrödinger Equation for a single electron, with a single extra (potential energy) term associated with U. Ĥ ψ n ( r ) + U ψ n ( r ) = E n ψ n ( r ) (2) The right-hand-side equation (Eq. ) depends on the position of electrons 2 and 3. We reorder this equation so as to bring all terms associated with electron 2 to the left hand side, and all terms associated with electron 3 to the right hand side. This gives us U 2 + ψ n2 ( r 2 )Ĥ2ψ n2 ( r 2 ) = E E n U 3 ψ n3 ( r 3 )Ĥ3ψ n3 ( r 3 ) (3)
3 3 We can again separate this into two equations, using E n2 Eq. 3 we get as the separation variable. For the left hand side of U 2 + which we can rewrite into the same form as Eq. 2: ψ n2 ( r 2 )Ĥ2ψ n2 ( r 2 ) = E n2 (4) For the right hand side of Eq. 3 we obtain Ĥ 2 ψ n2 ( r 2 ) + U 2 ψ n2 ( r 2 ) = E n2 ψ n2 ( r 2 ) (5) which we can reorder into E E n U 3 ψ n3 ( r 3 )Ĥ3ψ n3 ( r 3 ) = E n2 (6) Ĥ 3 ψ n3 ( r 3 ) + U 3 ψ n3 ( r 3 ) = [E E n E n2 ]ψ n3 ( r 3 ) (7) This equation can be further simplified by introducing E n3 such that which gives us after substitution into Eq. 7) E = E n + E n2 + E n3 (8) Ĥ 3 ψ n3 ( r 3 ) + U 3 ψ n3 ( r 3 ) = E n3 ψ n3 ( r 3 ) (9) Together, Eqs. 2, 5, and 9 are the three separated one-electron equations for the lithium atom. These have the general form: Ĥ i ψ ni ( r i ) + U i ψ ni ( r i ) = E ni ψ ni ( r i ) i =, 2, 3 (20) (c) Consider the special case, where the e-e repulsion energy is approximated as U i =0. Show that in this case one can use the energy Bohr/Schrödinger energy expression for a single-electron atom E n = 3.6 ev Z2 n 2 to work out the energy levels of a lithium atom as a function of the main quantum numbers n,n 2 and n 3 of the three electrons. Work out the energy of the three lowest electronic states of lithium in this simplified model. Make sure your result takes Pauli s exclusion principle into account. With the approximation U i =0, the one-electron equations (Eq. 20) simplify to Ĥ i ψ ni ( r i ) = E ni ψ ni ( r i ) (2) This is the Schödinger equation for a hydrogen-like atom, and we know that the solutions to this equation are hydrogen-like eigenfunctions and the energy is a function of the main quantum number n i of the electron given by
4 4 E ni = 3.6eV Z2 (22) As introduced above (Eq. 8), the total energy E of the lithium atom is the sum of the three single electron energies E n, E n2, and E n3, thus ( E n,n 2,n 3 = = E n + E n2 + E n3 = 3.6eV Z 2 n 2 + n 2 + ) 2 n 2 3 (23) The question now asks for the energy of the three lowest electronic states. For this you have to find the three electron configurations (combinations of the three quantum numbers n, n 2, and n 3 ) that give the lowest energy according to Eq. 23. The combinations are further subject to the Pauli principle which dictates that any two electrons in an atom must differ in at least one of the four quantum numbers n i, l i, m l,i, m s,i. The upshot of this is that a level with quantum number n can accommodate a maximum of 2n 2 electrons (see the discussion of degeneracy in E&R 7.5); that is, we have a maximum of two electrons with n= (and a maximum 8 with n=2). This means the configuration where all three electrons are in the n= is forbidden. The three lowest energy configurations are: n = n 2 = n 3 = 2 E 2 = ev (24) n = n 2 = n 3 = 3 E 3 = ev (25) n = n 2 = n 3 = 4 E 4 = ev (26) Note that raising the one electron into successively higher levels is more favorable than raising two electrons from n= into n = 2, i.e. n = n 2 = 2 n 3 = 2 E 2 = 83.6 ev (27) (d) How does the energy expression change when U i adopts a finite value? Because U i was introduced as a constant; the energy equation becomes ( E n,n 2,n 3 = 3.6 ev Z 2 n 2 + n 2 + ) 2 n 2 + U + U 2 + U 3 (28) 3 This can be shown by rewriting the general single-electron solution Eq. 20 as follows and define F ni = [E ni U i ] which gives us Ĥ i ψ ni ( r i ) = [E ni U i ] ψ ni ( r i ) (29) Ĥ i ψ ni ( r i ) = F ni ψ ni ( r i ) (30) This too has the general form of a hydrogen-like Schrödinger equation, and the known energy solution for F n is F ni = 3.6eV Z2 (3)
5 5 Substituting back F ni = [E ni U i ] gives us E ni = 3.6eV Z2 And for all three electrons in combination, we get Eq. 28 above. + U i (32) (e) Briefly explain why the product of single electron eigenfunctions (Eq. 4) does not satisfy the particle indistinguishability principle. Propose a better, antisymmetric, form for the three-electron eigenfunction ψ(r,r 2,r 3 ). Explain why this form does not allow two electrons to have the same quantum numbers. As a simple product of one-electron functions, Eq. 4 does not satisfy the indistinguishability principle. The exchange of two particles leads to a change in the probability density (see discussion E&R chapter 9-2). An improved form uses a determinant ψ ( r, r 2, r 3 ) = 6 ψ n ( r ) ψ n ( r 2 ) ψ n ( r 3 ) ψ n2 ( r ) ψ n2 ( r 2 ) ψ n2 ( r 3 ) ψ n3 ( r ) ψ n3 ( r 2 ) ψ n3 ( r 3 ) (33) In the determinant form, the exchange of two electrons is equivalent to the exchange of two columns, which turns the determinant into its negative as required for an antisymmetric wavefunction. If two electrons have the same quantum number (e.g. n 2 =n 3 ), then two of the rows of the determinant would be identical. This in turn means the determinant evaluates to zero. Note that this argument includes the spin quantum number. (f) With the improved, antisymmetric form of the three-electron eigenfunction, would it still be possible to use the method of separation of variables to separate the three-electrons Schrödinger equation into three single-electron equations? Provide a brief illustrative response, no need to go through the full algebra. The answer is no, separation of variables is not possible. Try it out (say, for the simpler case of a antisymmetric two electron eigenfunction)!
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
More informationAn 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
More informationCHAPTER 9 ATOMIC STRUCTURE AND THE PERIODIC LAW
CHAPTER 9 ATOMIC STRUCTURE AND THE PERIODIC LAW Quantum mechanics can account for the periodic structure of the elements, by any measure a major conceptual accomplishment for any theory. Although accurate
More information5.61 Physical Chemistry 25 Helium Atom page 1 HELIUM ATOM
5.6 Physical Chemistry 5 Helium Atom page HELIUM ATOM Now that we have treated the Hydrogen like atoms in some detail, we now proceed to discuss the next simplest system: the Helium atom. In this situation,
More informationCalculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1
Calculate Highest Common Factors(HCFs) & Least Common Multiples(LCMs) NA1 What are the multiples of 5? The multiples are in the five times table What are the factors of 90? Each of these is a pair of factors.
More informationKEY. Honors Chemistry Assignment Sheet- Unit 3
KEY Honors Chemistry Assignment Sheet- Unit 3 Extra Learning Objectives (beyond regular chem.): 1. Related to electron configurations: a. Be able to write orbital notations for s, p, & d block elements.
More informationElectron Arrangements
Section 3.4 Electron Arrangements Objectives Express the arrangement of electrons in atoms using electron configurations and Lewis valence electron dot structures New Vocabulary Heisenberg uncertainty
More information3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationPart I: Principal Energy Levels and Sublevels
Part I: Principal Energy Levels and Sublevels As you already know, all atoms are made of subatomic particles, including protons, neutrons, and electrons. Positive protons and neutral neutrons are found
More information1.6 The Order of Operations
1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative
More informationTIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points
TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS 3650, Exam 2 Section 1 Version 1 October 31, 2005 Total Weight: 100 points 1. Check your examination for completeness prior to starting.
More informationSection 11.3 Atomic Orbitals Objectives
Objectives 1. To learn about the shapes of the s, p and d orbitals 2. To review the energy levels and orbitals of the wave mechanical model of the atom 3. To learn about electron spin A. Electron Location
More informationelectron configuration
electron configuration Electron Configuration Knowing the arrangement of electrons in atoms will better help you understand chemical reactivity and predict an atom s reaction behavior. We know when n=1
More informationDepartment of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI
Department of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VI Solving a System of Linear Algebraic Equations (last updated 5/19/05 by GGB) Objectives:
More informationFractions to decimals
Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of
More informationMATRIX 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 +
More informationSection 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5
Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.
More informationUNIT (2) ATOMS AND ELEMENTS
UNIT (2) ATOMS AND ELEMENTS 2.1 Elements An element is a fundamental substance that cannot be broken down by chemical means into simpler substances. Each element is represented by an abbreviation called
More informationMulti-electron atoms
Multi-electron atoms Today: Using hydrogen as a model. The Periodic Table HWK 13 available online. Please fill out the online participation survey. Worth 10points on HWK 13. Final Exam is Monday, Dec.
More information5.5. Solving linear systems by the elimination method
55 Solving linear systems by the elimination method Equivalent systems The major technique of solving systems of equations is changing the original problem into another one which is of an easier to solve
More informationWAVES AND ELECTROMAGNETIC RADIATION
WAVES AND ELECTROMAGNETIC RADIATION All waves are characterized by their wavelength, frequency and speed. Wavelength (lambda, ): the distance between any 2 successive crests or troughs. Frequency (nu,):
More informationSIMPLIFYING ALGEBRAIC FRACTIONS
Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is
More informationMATRIX 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
More information2 ATOMIC SYSTEMATICS AND NUCLEAR STRUCTURE
2 ATOMIC SYSTEMATICS AND NUCLEAR STRUCTURE In this chapter the principles and systematics of atomic and nuclear physics are summarised briefly, in order to introduce the existence and characteristics of
More information9.2 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
More informationObjectives 404 CHAPTER 9 RADIATION
Objectives Explain the difference between isotopes of the same element. Describe the force that holds nucleons together. Explain the relationship between mass and energy according to Einstein s theory
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationName Partners Date. Energy Diagrams I
Name Partners Date Visual Quantum Mechanics The Next Generation Energy Diagrams I Goal Changes in energy are a good way to describe an object s motion. Here you will construct energy diagrams for a toy
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More information= 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)
More informationBasic Nuclear Concepts
Section 7: In this section, we present a basic description of atomic nuclei, the stored energy contained within them, their occurrence and stability Basic Nuclear Concepts EARLY DISCOVERIES [see also Section
More informationBohr Model Calculations for Atoms and Ions
Bohr Model Calculations for Atoms and Ions Frank Riou Department of Chemistry College of St. nedict St. Johnʹs University St. Joseph, MN 56374 Abstract A debroglie Bohr model is described that can be used
More informationSection 3: Crystal Binding
Physics 97 Interatomic forces Section 3: rystal Binding Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride
More informationOrder of Operations More Essential Practice
Order of Operations More Essential Practice We will be simplifying expressions using the order of operations in this section. Automatic Skill: Order of operations needs to become an automatic skill. Failure
More informationLesson 3. Chemical Bonding. Molecular Orbital Theory
Lesson 3 Chemical Bonding Molecular Orbital Theory 1 Why Do Bonds Form? An energy diagram shows that a bond forms between two atoms if the overall energy of the system is lowered when the two atoms approach
More information19.1 Bonding and Molecules
Most of the matter around you and inside of you is in the form of compounds. For example, your body is about 80 percent water. You learned in the last unit that water, H 2 O, is made up of hydrogen and
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationSCPS Chemistry Worksheet Periodicity A. Periodic table 1. Which are metals? Circle your answers: C, Na, F, Cs, Ba, Ni
SCPS Chemistry Worksheet Periodicity A. Periodic table 1. Which are metals? Circle your answers: C, Na, F, Cs, Ba, Ni Which metal in the list above has the most metallic character? Explain. Cesium as the
More informationLaboratory 11: Molecular Compounds and Lewis Structures
Introduction Laboratory 11: Molecular Compounds and Lewis Structures Molecular compounds are formed by sharing electrons between non-metal atoms. A useful theory for understanding the formation of molecular
More informationMULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.
1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with
More information1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes
Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.
More informationFree Electron Fermi Gas (Kittel Ch. 6)
Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 5.111 Principles of Chemical Science, Fall 2005 Please use the following citation format: Sylvia Ceyer and Catherine Drennan, 5.111 Principles of Chemical Science,
More informationPREPARATION FOR MATH TESTING at CityLab Academy
PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST
More informationCHEM 1411 Chapter 5 Homework Answers
1 CHEM 1411 Chapter 5 Homework Answers 1. Which statement regarding the gold foil experiment is false? (a) It was performed by Rutherford and his research group early in the 20 th century. (b) Most of
More informationMATH10212 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
More informationA pure covalent bond is an equal sharing of shared electron pair(s) in a bond. A polar covalent bond is an unequal sharing.
CHAPTER EIGHT BNDING: GENERAL CNCEPT or Review 1. Electronegativity is the ability of an atom in a molecule to attract electrons to itself. Electronegativity is a bonding term. Electron affinity is the
More informationIONISATION ENERGY CONTENTS
IONISATION ENERGY IONISATION ENERGY CONTENTS What is Ionisation Energy? Definition of t Ionisation Energy What affects Ionisation Energy? General variation across periods Variation down groups Variation
More informationClick on the links below to jump directly to the relevant section
Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is
More informationA.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents
Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify
More information13- What is the maximum number of electrons that can occupy the subshell 3d? a) 1 b) 3 c) 5 d) 2
Assignment 06 A 1- What is the energy in joules of an electron undergoing a transition from n = 3 to n = 5 in a Bohr hydrogen atom? a) -3.48 x 10-17 J b) 2.18 x 10-19 J c) 1.55 x 10-19 J d) -2.56 x 10-19
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More information2.3. Finding polynomial functions. An Introduction:
2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationAtomic Structure Ron Robertson
Atomic Structure Ron Robertson r2 n:\files\courses\1110-20\2010 possible slides for web\atomicstructuretrans.doc I. What is Light? Debate in 1600's: Since waves or particles can transfer energy, what is
More informationDepartment of Physics and Geology The Elements and the Periodic Table
Department of Physics and Geology The Elements and the Periodic Table Physical Science 1422 Equipment Needed Qty Periodic Table 1 Part 1: Background In 1869 a Russian chemistry professor named Dmitri Mendeleev
More informationSample Problems. Practice Problems
Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these
More informationWhat does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.
PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of
More informationLinear Programming. March 14, 2014
Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1
More informationAnswers to Basic Algebra Review
Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract
More informationAll the examples in this worksheet and all the answers to questions are available as answer sheets or videos.
BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents
More informationQuestion: Do all electrons in the same level have the same energy?
Question: Do all electrons in the same level have the same energy? From the Shells Activity, one important conclusion we reached based on the first ionization energy experimental data is that electrons
More informationMathematical goals. Starting points. Materials required. Time needed
Level A3 of challenge: C A3 Creating and solving harder equations equations Mathematical goals Starting points Materials required Time needed To enable learners to: create and solve equations, where the
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationMolecular-Orbital Theory
Molecular-Orbital Theory 1 Introduction Orbitals in molecules are not necessarily localized on atoms or between atoms as suggested in the valence bond theory. Molecular orbitals can also be formed the
More informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
More information9/13/2013. However, Dalton thought that an atom was just a tiny sphere with no internal parts. This is sometimes referred to as the cannonball model.
John Dalton was an English scientist who lived in the early 1800s. Dalton s atomic theory served as a model for how matter worked. The principles of Dalton s atomic theory are: 1. Elements are made of
More informationBonding & Molecular Shape Ron Robertson
Bonding & Molecular Shape Ron Robertson r2 n:\files\courses\1110-20\2010 possible slides for web\00bondingtrans.doc The Nature of Bonding Types 1. Ionic 2. Covalent 3. Metallic 4. Coordinate covalent Driving
More informationChapter 7. Electron Structure of the Atom. Chapter 7 Topics
Chapter 7 Electron Structure of the Atom Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 7 Topics 1. Electromagnetic radiation 2. The Bohr model of
More information5.61 Fall 2012 Lecture #19 page 1
5.6 Fall 0 Lecture #9 page HYDROGEN ATOM Consider an arbitrary potential U(r) that only depends on the distance between two particles from the origin. We can write the Hamiltonian simply ħ + Ur ( ) H =
More informationHFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationPart 1 Expressions, Equations, and Inequalities: Simplifying and Solving
Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words
More informationTo give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
More informationis identically equal to x 2 +3x +2
Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3
More informationSolving Rational Equations
Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,
More informationYears after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540
To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration
More information1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
More informationElements in the periodic table are indicated by SYMBOLS. To the left of the symbol we find the atomic mass (A) at the upper corner, and the atomic num
. ATOMIC STRUCTURE FUNDAMENTALS LEARNING OBJECTIVES To review the basics concepts of atomic structure that have direct relevance to the fundamental concepts of organic chemistry. This material is essential
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationChapter 22 The Hamiltonian and Lagrangian densities. from my book: Understanding Relativistic Quantum Field Theory. Hans de Vries
Chapter 22 The Hamiltonian and Lagrangian densities from my book: Understanding Relativistic Quantum Field Theory Hans de Vries January 2, 2009 2 Chapter Contents 22 The Hamiltonian and Lagrangian densities
More informationMolecular Models & Lewis Dot Structures
Molecular Models & Lewis Dot Structures Objectives: 1. Draw Lewis structures for atoms, ions and simple molecules. 2. Use Lewis structures as a guide to construct three-dimensional models of small molecules.
More informationName period AP chemistry Unit 2 worksheet Practice problems
Name period AP chemistry Unit 2 worksheet Practice problems 1. What are the SI units for a. Wavelength of light b. frequency of light c. speed of light Meter hertz (s -1 ) m s -1 (m/s) 2. T/F (correct
More informationBasic Concepts in Nuclear Physics
Basic Concepts in Nuclear Physics Paolo Finelli Corso di Teoria delle Forze Nucleari 2011 Literature/Bibliography Some useful texts are available at the Library: Wong, Nuclear Physics Krane, Introductory
More informationELECTRON CONFIGURATION (SHORT FORM) # of electrons in the subshell. valence electrons Valence electrons have the largest value for "n"!
179 ELECTRON CONFIGURATION (SHORT FORM) - We can represent the electron configuration without drawing a diagram or writing down pages of quantum numbers every time. We write the "electron configuration".
More informationAtoms and Elements. Outline Atoms Orbitals and Energy Levels Periodic Properties Homework
Atoms and the Periodic Table The very hot early universe was a plasma with cationic nuclei separated from negatively charged electrons. Plasmas exist today where the energy of the particles is very high,
More informationChapter Five: Atomic Theory and Structure
Chapter Five: Atomic Theory and Structure Evolution of Atomic Theory The ancient Greek scientist Democritus is often credited with developing the idea of the atom Democritus proposed that matter was, on
More informationSolutions of Linear Equations in One Variable
2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools
More informationSolution. Problem. Solution. Problem. Solution
4. A 2-g ping-pong ball rubbed against a wool jacket acquires a net positive charge of 1 µc. Estimate the fraction of the ball s electrons that have been removed. If half the ball s mass is protons, their
More informationChapter 3. Cartesian Products and Relations. 3.1 Cartesian Products
Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing
More informationNumerical analysis of Bose Einstein condensation in a three-dimensional harmonic oscillator potential
Numerical analysis of Bose Einstein condensation in a three-dimensional harmonic oscillator potential Martin Ligare Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837 Received 24
More information5 Homogeneous systems
5 Homogeneous systems Definition: A homogeneous (ho-mo-jeen -i-us) system of linear algebraic equations is one in which all the numbers on the right hand side are equal to : a x +... + a n x n =.. a m
More informationDefinition of derivative
Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example
More informationIONISATION ENERGY CONTENTS
IONISATION ENERGY IONISATION ENERGY CONTENTS What is Ionisation Energy? Definition of t Ionisation Energy What affects Ionisation Energy? General variation across periods Variation down groups Variation
More informationFYS3410 - Vår 2016 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v16/index.html
FYS3410 - Vår 2016 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v16/index.html Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18,
More informationSecond Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients. y + p(t) y + q(t) y = g(t), g(t) 0.
Second Order Linear Nonhomogeneous Differential Equations; Method of Undetermined Coefficients We will now turn our attention to nonhomogeneous second order linear equations, equations with the standard
More information