1. Thegroundstatewavefunctionforahydrogenatomisψ 0 (r) =


 Phebe Freeman
 1 years ago
 Views:
Transcription
1 CHEM 5: Examples for chapter Thegroundstatewavefunctionforahydrogenatomisψ (r = 1 e r/a. πa (a What is the probability for finding the electron within radius of a from the nucleus? (b Two excited states of hydrogen atom are given by the following wavefunctions: Solution: ψ 1 (r = A(+λre r a and ψ (r = Brsin(θcos(φe r a Proceed in the following order: 1 obtain λ from the orthogonality requirement between ψ and ψ 1, use the normalization requirement sepratately for ψ 1 and ψ to get constants A and B, respectively. (a The probability P for finding the electron within a (a ball with radius a ; denoted by V below is: P = = 4 a V ψ ψ dτ = 1 πa a 5a e 4 a e r/a 4πr dr }{{} =dτ = 1 5e. Note that the 4π in the volume element above comes from the fact that the function in the integral does not depend on the angles θ and φ and therefore the angular part can be integrated independently to yield 4π (i.e. the original volume element dτ = r sin(θdθdφdr is then effectively just 4πr dr. Above we have used the following result from a tablebook: x e ax = aax a (x xa + a 1
2 This was applied in the following form: a e r/a r dr = e r/a /a = a e 5a e 4 ( r r + /a ( /a ( a +a + a + a 4 + a 4 = a 5a e 4 (b First recall the wavefunctions: ψ (r,θ,φ = 1 πa e r/a 1 /a ( /a ψ 1 (r,θ,φ = A(+λre r/(a,ψ (r,θ,φ = Bsin(θcos(φre r/(a First we calculate λ from the orthogonality: ψ 1ψ dτ = 4π ψ 1(rψ (rr dr = 4πA πa (+λre r/(a r dr = (+λre r/(a r dr = e r/(a r dr+λ 16a λ 81 a4 = e r/(a r dr = a +λa 4 = λ = 1 a Then determine A from the normalization condition:
3 π π ψ 1 (r,θ,φ r sin(θdrdθdφ r=θ= φ= = 4πA ( r a e r/a r dr r= = 4πA 4 r= r e r/a dr 4 a r= r e r/a dr+ 1 a r= r 4 e r/a dr = 4πA (4! (1/a 4! a (1/a + 1 4! 4 a (1/a 5 = 4πA ( 8a 4a +4a = πa a = A = = πa 4 πa B can also be determined from normalization: π π ψ (r,θ,φ r sin(θdrdθdφ r= θ=φ= π = B r 4 e r/a dr sin (θdθ π cos (φdφ r= ( 4 B (1/a 5 = 1 4 πa 5 θ= φ= 4 π = B πa 5 = 1 B = Above we have used the following integrals (tablebook: 1 πa 5
4 x n e ax = n! a n+1 sin (ax = cos(ax 1a cos (ax = x + sin(ax 4a cos(ax 4a. (a Which of the following functions are eigenfunctions of d/ and d / : exp(ikx, cos(kx, k, and exp(ax. (b Show that the function f(x,y,z = cos(axcos(bycos(cz is an eigenfunction of the aplacian operator ( and calculate the corresponding eigenvalue. (c Calculate the standard deviation r for the ground state of hydrogen atom ψ (r. (d Calculatetheexpectationvalueforpotentialenergy(V(r = e 4πǫ r in hydrogen atom ground state ψ (r. Express the result finally in the units of ev. Solution: (a d(eikx = ike ikx = constant original function. Thus this is an eigenfunction of the given operator. = d (ikeikx = k e ikx = constant original function. Thus this is an eigenfunction of the given operator. d (e ikx d(k = d (k =. This could be considered as en eigenfunction with zero eigenvalue. d(eax = axe ax constant original function, thus not an eigenfunction. d (e ax = d(axeax = a(ax +1e ax constant original function, thus not an eigenfunction. = k sin(kx. Not an eigenfunction. d(cos(kx 4
5 d (cos(kx = d( ksin(kx = k cos(kx. This is an eigenfunction (with eigenvalue k. (b f(x, y, z = cos(ax cos(by cos(cz. The partial derivatives are: and f(x,y,z x f(x,y,z y f(x,y,z z = a sin(ax cos(by cos(cz = b cos(ax sin(by cos(cz = c cos(ax cos(by sin(cz f(x,y,z = a cos(axcos(bycos(cz x f(x,y,z = b cos(axcos(bycos(cz y f(x,y,z = c cos(axcos(bycos(cz z Therefore f(x,y,z = (a + b + c f(x,y,z = constant original function and this is an eigenfunction with the corresponding eigenvalue (a +b +c. (c The standard deviation can be calculated as: ψ (r = 1 πa e r/a ˆr = 1 πa ˆr = 1 πa ˆr = 9a 4 ˆr ˆr = a 9a ˆr ˆr = e r/a r } 4πr {{ dr} = 4 a5 a dτ 4 = a e r/a r4πr }{{ dr} = a dτ 4 = 4a 4 a.87a 5
6 (d The potential energy expectation can be calculated as: ψ (r = 1 e r/a and V(r = e πa 4πǫ r ˆV = e 4π ǫ a r= e r/a 1 r } 4πr {{ dr} = e πǫ a dτ = e a πǫ a 4 = e 7. ev 4πǫ a r= e r/a rdr. Consider function ψ(x = ( π 1 4 e x /. Using (only this function, show that: (a Operators ˆx and ˆp x do not commute. (b Operators ˆx and the inversion operator Î commute (Îx = x. Note that consideration of just one function does not prove a given property in general. Solution: (a ( π ψ(x = 1/4e x / ˆp x = i h d and [ˆx,ˆp x]ψ(x = (ˆxˆp x ˆp xˆxψ(x To obtain the commutator, we need to operate with ˆx and ˆp x : ( π 1/4x d π 1/4x (ˆxˆp x ψ(x = i h / e x = i h( e x / ( π 1/4 d ( (ˆp xˆxψ(x = i h xe x / ( π 1/4x ( π 1/4e = i h e x / x i h / When these are subtracted, we get: 6
7 ( π 1/4e x (ˆxˆp x ψ(x (ˆp xˆxψ(x = i h / When the wavefunction ψ(x is removed, we have: [ˆx,ˆp x ] = i h Because the commutator between these operators is nonzero, it means that they are complementary. (b The commutator for the given function is: [ ] ˆx,Î ψ(x = ( ( π 1/4e ˆx Î Îˆx x / ( π 1/4e ( = x ( x π / ( x ( π 1/4e ( = x x π 1/4e / x x / = ˆx and Î commute for the given function. 1/4e ( x / 4. Aparticleisdescribedbythefollowingwavefunction: ψ(x = cos(χφ k (x + sin(χφ k (x where χ is a parameter (constant and φ k and φ k are orthonormalized eigenfunctions of the momentum operator with the eigenvalues + hk and hk, respectively. (a What is the probability that a measurement gives + hk as the momentum of the particle? (b What is the probability that a measurement gives hk as the momentum of the particle? (c What wavefunction would correspond to.9 probability for a momentum of + hk? (d Consider another system, for which ψ =.9ψ 1 +.4ψ + c ψ. Calculate c when ψ 1, ψ and ψ are orthonormal. Use the normalization condition for ψ. Solution: 7
8 (a The probability for measuring + hk is cos (χ. We have a superposition of the eigenfunctions of momentum and therefore the squares of the coefficients for each eigenfunction give the corresponding probability. (b The probability for measuring hk is sin (χ. (c Given cos (χ =.9 (hence cos(χ = ±.95 we use the normalization condition cos (χ + sin (χ = 1 where we solve for sin(χ = ±.. The overall sign for the wavefunction does not matter and therefore we have two possibilities for our wavefunction: ψ =.95e ikx ±.e ikx. (d Normalizationcondition: 1 = (.9 +(.4 +c. Thusc = ± Consider a particle in the following onedimensional infinitely deep box potential: {, when x a V(x =, when x > a Note that the position of the potential was chosen differently than in the lectures. The following two wavefunctions are eigenfunctions of the Hamiltonian corresponding to this potential: ψ 1 (x = 1 a cos ( πx a and ψ (x = 1 a sin ( πx a with the associated eigenvalues are E 1 = 1 ev and E = 4 ev. Define a superposition state ψ as ψ = 1 (ψ 1 +ψ. (a What is the average energy of the above superposition state (e.g. < Ĥ >? (b Plot ψ 1 and ψ and determine the most probable positions for a particle in these states. (c What are the most probable positions for the particle given by wavefunction ψ (x = sin( πx where the box potential is now located between [, ]. Solution: 8
9 (a We calculate the expectation value (Ĥψ 1 = E 1 ψ 1 and Ĥψ = E ψ : Ĥ = ψ (xĥψ(x = 1 (ψ 1(x+ψ (xĥ(ψ 1(x+ψ (x = 1 (ψ 1(x+ψ(x(E 1 ψ 1 (x+e ψ (x = 1 (E 1 +E =.5 ev (b For a = 1 both ψ 1 (x (one maximum and ψ (x (maximum and minimum are shown below: The most probable values for position can be obtained from the squared wavefunctions: ψ 1 hasthemaximumvalueatr = whereasψ hastwomaximaat ±.5. Note that on average both will give an outcome of ˆr =. (c The most probable positions are given by the square of the wavefunction ψ (x = (/ 1/ sin(πx/. The probability function 9
10 is then ψ (x sin (πx/. The extremum points for this function can be obtained by: d ( sin (πx/ = sin(πx/cos(πx/ = πx x = n ( πx = nπ or = n+ 1 π n+1/ or x = Second derivatives can be used to identify the extrema: d ( sin (πx/ cos ( πx ( πx sin At x = n the values are positive which means that these correspond to (local minima. For x = n+1/ the values are nega tive and these points correspond to (local maxima. For example, when = the probability function looks like: 6. Calculate the uncertainty product p x for the following wavefunctions: (a ψ n (x = ( 1/sin a a with x (particle in a box (b ψ n (x = N v H v ( ( xexp Solution: x (harmonic oscillator 1
11 (a For momentum: ( 1/ ψ n (x = sin, where x ( 1/ ( ˆp x = sin i h d ( 1/ sin }{{}}{{}}{{} ψn (x ˆp x ψ n(x = i h sin d ( sin = πni h ˆp x = = h = n π h sin cos } {{ } = ( 1/ ( sin i h d } {{ } ψn sin Now we can calculate p = = }{{} ˆp x d sin sin sin } {{ } = ( 1/ sin }{{} ψ n = n π h ˆp x ˆp x = nπ h. For position 11
12 we have: ˆx = = ˆx = ( 1/ sin x xsin = / ( 1/ sin = (4n π 6 4π n ( 1/ sin ( 1/ x sin = x sin = (π n 6π n }{{} = ( 1 π n Now x = ˆx ˆx ( = 1 π n = Since we have both x and p, we can evaluate the uncertainty product: n π p x = h π n. Thesmallestvalueisobtainedwithn = 1: p x.568 h > h. (b First recall that: ( ψ n (x = N v H v x e x /, where N v = 1 ( 1/4 v v! π Also the following relations for Hermite polynomials are used (lecture notes & tablebook: H v+1 = yh v vh v 1 H v (yh v (ye y dy = {, if v v π v v!, if v = v 1
13 Denote y = x and hence dy =. Now ˆx is given by: ˆx = N v = N v = N v H v ( x e x / xh v ( x e x / H v (ye y / yh v (ye y / dy H v (yy H }{{} v (ye y dy = = 1 H v+1(y+vh v 1 (y The last two steps involved using the recursion relation for Hermite polynomials as well as their orthogonality property. Next we calculate ˆx : ˆx = N v / = N v / = π N v / (yh v (y e y dy ( 1 H v+1(y+vh v 1 (y e y dy ( v+1 (v +1! +v v 1 (v 1! 4 = 1 ((v +1+v = 1 ( v + 1 = h ( v + 1 µk (v Combiningtheabovecalculationsgives x = + 1 h µk. Next we calculate p: ˆp = i h d and dy =. ˆp = = i hn v N v H v (ye y /ˆp ( N v H v (ye y / dy H v (ye y / d ( H v (ye y / dy 1
14 Above differentiation of the eigenfunction changes parity and therefore the overall parity of the integrand is odd. Integral of odd function is zero and thus ˆp =. For ˆp we have: ˆp = N v H v (ye y /ˆp N v H v (ye y / dy Theoperatormustalsobetransformedfromxtoy: ˆp = ( i hd/ = ( i h d/. The above becomes now: ( N v H v (ye y / ( h d N dy v H v (ye y / dy = h N v = h N v = h N v ( H v (ye y / d H dy v (ye y / dy = h Nv ( v 1 H v (y (y 1H v (y yh v(y+h v(y e y dy }{{} = vh v(y H v (y [ (y 1H v (y vh v (y ] e y dy Hv(ye y dy+ }{{} = π v v! y Hv(y e }{{} =( 1 H v+1(y+vh v 1 (y y dy = h [ Nv ( v 1 π v v!+ 4 1 π v+1 (v +1!+v ] π v 1 (v 1! = h [ Nv π v v! v +1 v +1 v ] [ = h v + 1 ] = h [ mk v + 1 ] 14
15 Therefore we have p = x p = (v + 1 h mk [ v + 1 ]. Overall we then have h mk h mk [ v + ] 1 = h(v + 1 h. 7. Scanning tunneling microscopy (STM is a technique for visualizing samples at atomic resolution. It is based on tunneling of electron through the vacuum space between the STM tip and the sample. The tunneling current (I ψ where ψ is the electron wavefunction is very sensitive to the distance between the tip and the sample. Assume that the wavefunction for electron tunneling through the vacuum is given by ψ(x = Bexp Kx with K = m e (V E/ h and V E =. ev. What would be the relative change in the tunneling current I when the STM tip is moved from x 1 =.5 nm to x =.6 nm from the sample (e.g. I 1 /I =?. Solution: The current is proportional to the square the wavefunction ( ψ(x : I ψ(x = B e Kx I 1 = e K(x 1 x I K = m e (V E / h }{{} = ev ( ( kg(. ev( J/eV = ( Js K(x 1 x = K ( m 1.45 I 1 I 4. 1/ 1/ 8. Show that the sperhical harmonic functions a Y,, b Y, 1 and c Y, are eigenfunctions of the (threedimensional rigid rotor Hamiltonian. What are the rotation energies and angular momenta in each case? 15
16 Solution: Ĥψ(r,θ,φ = Eψ(r,θ,φ where Ĥ = h Λ = 1 sin (θ φ + 1 sin(θ θ sin(θ θ I Λ Since h is constant, it is sufficient to show that spherical harmonics I are eigenfunctions of Λ. We operate on the given spherical harmonics by Λ. (a Λ Y (θ,φ = Λ 1 = (no angular dependency. The rotation π energy is E = h =. Also I = h = (since ˆ = h Λ. (b Λ Y 1 (θ,φ =...differentiation... = 6Y 1 energy is E = h I (c Expression for Y (θ, φ. The rotation h ( 6 =. Also I = 6 h = 6 h. was not given in the lecture notes. We use Maxima to obtain the expression: Y (θ,φ = 5 7e φi sin (θ 8. Then we 5π needtoevaluate: Λ Y (θ,φ =...differentiation... = 1Y (θ,φ. From this we can get the rotational energy as E = 6 h. Also I = 1 h = 1 h. 9. Calculate the zcomponent of angular momentum and the rotational kinetic energy in planar (e.g. dimensional; rotation in xyplane rotation for the following wavefunctions (φ is the rotation angle with values between [,π[: (a ψ = e iφ (b ψ = e iφ (c ψ = cos(φ (d ψ = cos(χe iφ +sin(χe iφ Solution: 16
17 The Hamiltonian for D rotation is Ĥ = ˆ z = h d where the I I dφ rotation axis is denoted by z and φ is the angle of rotation. (a Operatefirstby ˆ z on e iφ : ˆ z e iφ = i h d dφ eiφ = he iφ. ThusĤeiφ = h I eiφ. (b The same reasoning gives: ˆz e iφ = i h d dφ e iφ = he iφ and Ĥe iφ = 4 h I e iφ. (c The given wavefunction is not an eigenfunction of ˆ z : ˆ z cos(φ = i h( sin(φ (expectation value is zero. However, it is an eigenfunction of Ĥ: Ĥcos(φ = h cos(φ = h cos(φ. I dφ d I (d Operation by ˆ z would change the plus sign in the middle of the wavefunction into a minus, hence this is not an eigenfunction of ˆ z. It is an eigenfunction of Ĥ: Ĥ(cos(χe iφ + sin(χe iφ = h I ( (i cos(χe iφ +( i sin(χe iφ = h I ( cos(χe iφ +sin(χe iφ. 17
I. The BornOppenheimer Separation of Electronic and Nuclear Motions
Chapter 3 The Application of the Schrödinger Equation to the Motions of Electrons and Nuclei in a Molecule Lead to the Chemists' Picture of Electronic Energy Surfaces on Which Vibration and Rotation Occurs
More informationBasic Quantum Mechanics
Basic Quantum Mechanics Postulates of QM  The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn)  To every observable (physical magnitude) there
More informationModule 1: Quantum Mechanics  2
Quantum Mechanics  Assignment Question: Module 1 Quantum Mechanics Module 1: Quantum Mechanics  01. (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization
More informationWrite your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.
UNIVERSITY OF LONDON BSc/MSci EXAMINATION June 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship
More informationChapter 24 Physical Pendulum
Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which 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 informationUNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Solution for take home exam: FYS311, Oct. 7, 11. 1.1 The Hamiltonian of a charged particle in a weak magnetic field is Ĥ = P /m q mc P A
More informationPDE and BoundaryValue Problems Winter Term 2014/2015
PDE and BoundaryValue Problems Winter Term 2014/2015 Lecture 15 Saarland University 12. Januar 2015 c Daria Apushkinskaya (UdS) PDE and BVP lecture 15 12. Januar 2015 1 / 42 Purpose of Lesson To show
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) 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
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationTriple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.
Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals
More informationProblems in Quantum Mechanics
Problems in Quantum Mechanics Daniel F. Styer July 1994 Contents 1 SternGerlach Analyzers 1 Photon Polarization 5 3 Matrix Mathematics 9 4 The Density Matrix 1 5 Neutral K Mesons 13 6 Continuum Systems
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationThe 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
More informationEngineering Math II Spring 2015 Solutions for Class Activity #2
Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the xaxis. Then find
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
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 information* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B. PPT No.
* Biot Savart s Law Statement, Proof Applications of Biot Savart s Law * Magnetic Field Intensity H * Divergence of B * Curl of B PPT No. 17 Biot Savart s Law A straight infinitely long wire is carrying
More informationIntroduction to Schrödinger Equation: Harmonic Potential
Introduction to Schrödinger Equation: Harmonic Potential ChiaChun Chou May 2, 2006 Introduction to Schrödinger Equation: Harmonic Potential TimeDependent Schrödinger Equation For a nonrelativistic particle
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationHarmonic Oscillator and Coherent States
Chapter 5 Harmonic Oscillator and Coherent States 5. Harmonic Oscillator In this chapter we will study the features of one of the most important potentials in physics, it s the harmonic oscillator potential
More informationCHEM6085: Density Functional Theory Lecture 2. Hamiltonian operators for molecules
CHEM6085: Density Functional Theory Lecture 2 Hamiltonian operators for molecules C.K. Skylaris 1 The (timeindependent) Schrödinger equation is an eigenvalue equation operator for property A eigenfunction
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
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 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 informationChapter 13. Gravitation
Chapter 13 Gravitation 13.2 Newton s Law of Gravitation In vector notation: Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G = 6.67
More informationMolecular Symmetry 1
Molecular Symmetry 1 I. WHAT IS SYMMETRY AND WHY IT IS IMPORTANT? Some object are more symmetrical than others. A sphere is more symmetrical than a cube because it looks the same after rotation through
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationPhysics 9 Fall 2009 Homework 7  Solutions
Physics 9 Fall 009 Homework 7  s 1. Chapter 33  Exercise 10. At what distance on the axis of a current loop is the magnetic field half the strength of the field at the center of the loop? Give your answer
More informationLab M1: The Simple Pendulum
Lab M1: The Simple Pendulum Introduction. The simple pendulum is a favorite introductory exercise because Galileo's experiments on pendulums in the early 1600s are usually regarded as the beginning of
More informationProperties of Legendre Polynomials
Chapter C Properties of Legendre Polynomials C Definitions The Legendre Polynomials are the everywhere regular solutions of Legendre s Equation, which are possible only if x 2 )u 2xu + mu = x 2 )u ] +
More informationPhysics 1120: Simple Harmonic Motion Solutions
Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured
More information221A Lecture Notes on Spin
22A Lecture Notes on Spin This lecture note is just for the curious. It is not a part of the standard quantum mechanics course. True Origin of Spin When we introduce spin in nonrelativistic quantum mechanics,
More information1 Introduction to quantum mechanics
ntroduction to quantum mechanics Quantum mechanics is the basic tool needed to describe, understand and devise NMR experiments. Fortunately for NMR spectroscopists, the quantum mechanics of nuclear spins
More informationExamination paper for Solutions to Matematikk 4M and 4N
Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
More informationTHE PRIME NUMBER THEOREM
THE PRIME NUMBER THEOREM NIKOLAOS PATTAKOS. introduction In number theory, this Theorem describes the asymptotic distribution of the prime numbers. The Prime Number Theorem gives a general description
More informationPart 3: The MaxwellBoltzmann Gas
PHYS393 Statistical Physics Part 3: The MaxwellBoltzmann Gas The Boltzmann distribution In the previous parts of this course, we derived the Boltzmann distribution: n j = N Z e ε j kt, (1) We applied
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationPHY411. PROBLEM SET 3
PHY411. PROBLEM SET 3 1. Conserved Quantities; the RungeLenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationMath 504, Fall 2013 HW 3
Math 504, Fall 013 HW 3 1. Let F = F (x) be the field of rational functions over the field of order. Show that the extension K = F(x 1/6 ) of F is equal to F( x, x 1/3 ). Show that F(x 1/3 ) is separable
More informationChapter 9. Gamma Decay
Chapter 9 Gamma Decay As we have seen γdecay is often observed in conjunction with α or βdecay when the daughter nucleus is formed in an excited state and then makes one or more transitions to its ground
More informationQuantum Mechanics: Postulates
Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, xray, etc.) through
More information4.1: Angles and Radian Measure
4.1: Angles and Radian Measure An angle is formed by two rays that have a common endpoint. One ray is called the initial side and the other is called the terminal side. The endpoint that they share is
More informationBead moving along a thin, rigid, wire.
Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing
More information1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!
1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating
More informationPhysics 41 HW Set 1 Chapter 15
Physics 4 HW Set Chapter 5 Serway 8 th OC:, 4, 7 CQ: 4, 8 P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59, 67, 74 OC CQ P: 4, 5, 8, 8, 0, 9,, 4, 9, 4, 5, 5 Discussion Problems:, 57, 59,
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationMeasuring the Earth s Diameter from a Sunset Photo
Measuring the Earth s Diameter from a Sunset Photo Robert J. Vanderbei Operations Research and Financial Engineering, Princeton University rvdb@princeton.edu ABSTRACT The Earth is not flat. We all know
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More informationClass XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34
Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in
More informationfound between x and x + dx is then equal to j^(x)[ dx, provided that tjkx) is normalized in the usual way: X  I (6.1)
VI. OPERATORS & EXPECTATION VALUES 1. The Calculation of Average Values The description of quantum states in terms of probability amplitudes at once implies the possibility of using these probabi lity
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationLecture L13  Conservative Internal Forces and Potential Energy
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L13  Conservative Internal Forces and Potential Energy The forces internal to a system are of two types. Conservative forces, such as gravity;
More informationCHAPTER 13 MOLECULAR SPECTROSCOPY
CHAPTER 13 MOLECULAR SPECTROSCOPY Our most detailed knowledge of atomic and molecular structure has been obtained from spectroscopy study of the emission, absorption and scattering of electromagnetic radiation
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More information2m dx 2 = Eψ(x) (1) Total derivatives can be used since there is but one independent variable. The equation simplifies to. ψ (x) + k 2 ψ(x) = 0 (2)
CHAPTER 3 QUANTUM MECHANICS OF SOME SIMPLE SYSTEMS The Free Particle The simplest system in quantum mechanics has the potential energy V equal to zero everywhere. This is called a free particle since it
More informationPreCalculus Review Problems Solutions
MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry PreCalculus Review Problems Solutions Problem 1. Give equations for the following lines in both pointslope and slopeintercept form. (a) The
More informationQuantum Chemistry Lecture notes to accompany Chemistry University of Houston and Eric R. Bittner All Rights Reserved.
Quantum Chemistry Lecture notes to accompany Chemistry 6321 Copyright @19972001, University of Houston and Eric R. Bittner All Rights Reserved. November 21, 2001 Contents 0 Introduction 7 0.1 Essentials........................................
More informationWASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS
Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,
More informationNotes on Quantum Mechanics. Finn Ravndal Institute of Physics University of Oslo, Norway
Notes on Quantum Mechanics Finn Ravndal Institute of Physics University of Oslo, Norway email: finnr@fys.uio.no v.4: December, 2006 2 Contents 1 Linear vector spaces 7 1.1 Real vector spaces...............................
More informationUnits and Vectors: Tools for Physics
Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff 1.1.1 The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the
More informationSection 3.2 Polynomial Functions and Their Graphs
Section 3.2 Polynomial Functions and Their Graphs EXAMPLES: P(x) = 3, Q(x) = 4x 7, R(x) = x 2 +x, S(x) = 2x 3 6x 2 10 QUESTION: Which of the following are polynomial functions? (a) f(x) = x 3 +2x+4 (b)
More informationSolutions to Vector Calculus Practice Problems
olutions to Vector alculus Practice Problems 1. Let be the region in determined by the inequalities x + y 4 and y x. Evaluate the following integral. sinx + y ) da Answer: The region looks like y y x x
More informationChapter 18 Friedmann Cosmologies
Chapter 18 Friedmann Cosmologies Let us now consider solutions of the Einstein equations that may be relevant for describing the largescale structure of the Universe and its evolution following the big
More informationDERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS
MISN0106 DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS Force Center (also the coordinate center) satellite DERIVATION OF ORBITS IN INVERSE SQUARE LAW FORCE FIELDS by Peter Signell 1. Introduction..............................................
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationMINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFTINVARIANT SUBSPACES OF L 2 (R n )
MINIMAL GENERATOR SETS FOR FINITELY GENERATED SHIFTINVARIANT SUBSPACES OF L 2 (R n ) MARCIN BOWNIK AND NORBERT KAIBLINGER Abstract. Let S be a shiftinvariant subspace of L 2 (R n ) defined by N generators
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationAngular Momentum, Hydrogen Atom, and Helium Atom
Chapter Angular Momentum, Hydrogen Atom, and Helium Atom Contents.1 Angular momenta and their addition..................4. Hydrogenlike atoms...................................38.3 Pauli principle, Hund
More informationTimebin entanglement of quasiparticles in semiconductor devices. Luca Chirolli, Vittorio Giovannetti, Rosario Fazio, Valerio Scarani
Timebin entanglement of quasiparticles in semiconductor devices Luca Chirolli, Vittorio Giovannetti, Rosario Fazio, Valerio Scarani The concept of energytime entanglement τ 1 τ 2 total energy well defined
More informationMATHEMATICS CLASS  XII BLUE PRINT  II. (1 Mark) (4 Marks) (6 Marks)
BLUE PRINT  II MATHEMATICS CLASS  XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants
More informationPhysics in the Laundromat
Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized
More informationLecture L303D Rigid Body Dynamics: Tops and Gyroscopes
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L303D Rigid Body Dynamics: Tops and Gyroscopes 3D Rigid Body Dynamics: Euler Equations in Euler Angles In lecture 29, we introduced
More informationQuantum Mechanics I. Peter S. Riseborough. August 29, 2013
Quantum Mechanics I Peter S. Riseborough August 9, 3 Contents Principles of Classical Mechanics 9. Lagrangian Mechanics........................ 9.. Exercise............................. Solution.............................3
More informationEuler Angle Formulas. Contents. David Eberly Geometric Tools, LLC Copyright c All Rights Reserved.
Euler Angle Formulas David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: December 1, 1999 Last Modified: March 10, 2014 Contents 1 Introduction
More informationChapter 7: Superposition
Chapter 7: Superposition Introduction The wave equation is linear, that is if ψ 1 (x, t) ψ 2 (x, t) satisfy the wave equation, then so does ψ(x, t) = ψ 1 (x, t) + ψ 2 (x, t). This suggests the Principle
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More informationChapter 3 Vectors 3.1 Vector Analysis Introduction to Vectors Properties of Vectors Cartesian Coordinate System...
Chapter 3 Vectors 3.1 Vector Analsis... 1 3.1.1 Introduction to Vectors... 1 3.1.2 Properties of Vectors... 1 3.2 Cartesian Coordinate Sstem... 5 3.2.1 Cartesian Coordinates... 6 3.3 Application of Vectors...
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationSolid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16
Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 MaxPlanck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm. 02.001 Luruper Chaussee 149
More informationPolynomials and the Fast Fourier Transform (FFT) Battle Plan
Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and pointvalue representation
More informationThomson and Rayleigh Scattering
Thomson and Rayleigh Scattering In this and the next several lectures, we re going to explore in more detail some specific radiative processes. The simplest, and the first we ll do, involves scattering.
More informationChapter 6 Electromagnetic Radiation and the Electronic Structure of the Atom
Chapter 6 In This Chapter Physical and chemical properties of compounds are influenced by the structure of the molecules that they consist of. Chemical structure depends, in turn, on how electrons are
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationth ey are.. r.. ve.. an t.. bo th for th e.. st ru c.. tur alan d.. t he.. quait.. ta ive un de \ centerline {. Inrodu \quad c t i o na r s
   I  ˆ    q I q I ˆ I q R R q I q q I R R R R    ˆ @ & q k 7 q k O q k 8 & q & k P S q k q k ˆ q k 3 q k ˆ A & [ 7 O [8 P & S & [ [ 3 ˆ A @ q ˆ U q  : U [ U φ : U D φ φ Dφ A ψ A : I SS N :
More informationMATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 BASIC INTEGRATION
MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL 1 ASIC INTEGRATION This tutorial is essential prerequisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning
More informationThe nearlyfree electron model
Handout 3 The nearlyfree 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
More informationCOHERENT STATES IN QUANTUM MECHANICS
COHERENT STATES IN QUANTUM MECHANICS Moniek Verschuren July 8, 2011 Bachelor thesis Radboud University Nijmegen Supervisor: Dr. W.D. van Suijlekom Abstract In this thesis we construct coherent states
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 11 THREE DIMENSIONAL GEOMETRY 111 Overview 1111 Direction cosines of a line are the cosines of the angles made by the line with positive directions of the coordinate axes 111 If l, m, n are the
More informationQuantum Mechanics. Dr. N.S. Manton. Michælmas Term 1996. 1 Introduction 1
Quantum Mechanics Dr. N.S. Manton Michælmas Term 1996 Contents 1 Introduction 1 The Schrödinger Equation 1.1 Probabilistic Interpretation of ψ...................................1.1 Probability Flux and
More information