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Mathematics in Physical Chemistry Mathematics in Physical Chemistry Ali Nassimi [email protected] Chemistry Department Sharif University of Technology May 8, 2021 1/1 Aim Your most valuable asset is your learning ability. This course is a practice in learning and specially improves your deduction skills. This course provides you with tools applicable in and necessary for modeling many natural phenomena. \The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved." The first part of the course reviews Linear algebra and calculus while introducing some very useful notations. In the second part of the course, we study ordinary differential equations. End of semester objective: Become familiar with mathematical tools needed for understanding physical chemistry. 2/1 Course Evaluation Final exam 11 Bahman 3 PM 40% Midterm exam 24 Azar 9 AM 40% Quizz 10% Class presentation 10% Office hours: Due to special situation of corona pandemic office hour is not set, email for an appointment instead. 3/1 To be covered in the course Complex numbers, Vector analysis and Linear algebra Vector rotation, vector multiplication and vector derivatives. Series expansion of analytic functions Integration and some theorems from calculus Dirac delta notation and Fourier transformation Curvilinear coordinates. Matrices 4/1 To be covered in the course When we know the relation between change in dependent variable with changes in independent variable we are facing a differential equation. The laws of nature are expressed in terms of differential equations. For example, study of chemical kinetics, diffusion and change in a systems state all start with differential equations. Analytically solvable ordinary differential equations. Due to lack of time a discussion of partial differential equations and a discussion of numerical solutions to differential equations are left to a course in computational chemistry. 5/1 References Modern calculus and analytic geometry by Richard A. Silverman "Mathematical methods for physicists", by George Arfken and Hans Weber Ordinary differential equations by D. Shadman and B. Mehri (A thin book in Farsi) Linear Algebra, Second Edition, Kenneth Hoffman, Ray Kanze The Raman effect by Derek A Long 6/1 Set theory Set, class or family x 2 A, y 2= A A ⊂ B or equivalently B ⊃ A A = B iff A ⊂ B ^ B ⊂ A, otherewise A 6= B. Proper subset The unique empty set Ø B − A = fxjx 2 B ^ x 2= Ag Universal set, U. Ac = U − A Venn diagrams A [ B (A cup B) A \ B (A cap B) disjoint set. 7/1 Relation Symmetric difference, A∆B = (A − B) [ (B − A) Ordered n-tuples vs. sets. Cartesian product of A and B, A × B = f(a; b)ja 2 A; b 2 Bg A1 × A2 × · · · × An = f(a1; a2; ··· ; an)ja1 2 A1; a2 2 A2; ··· an 2 Ang Real numbers, R 2-space = R2 3-space = R3 A relationship is defined as a set of ordered pairs (x,y), i.e., a relation S from the set X to the set Y is a subset of the cartesian product X × Y . Where the sets X and Y are the same there are 3 types of relations. A many to one or one to one relation from a set X to a set Y is called a function from X to Y. 8/1 Function A function from the set of positive integers to an arbitrary set is called an infinite sequence or sequence, y = yn or y1; y2; ··· ; yn; ··· or yn (n = 1; 2; ··· ) or fyng Dom f = fxj(x; y) 2 f for some yg, Rng f = fyj(x; y) 2 f for some xg f ⊂ Dom f × Rng f ⊂ X × Y Arbitrary set X to R, real valued function R to R, real function of one real variable R to Rn, point valued function or vector function of one real variable. Rn to R, real function of sevral real variables Rn to Rp, coordinate transformation, each coordinate of the p point y = (y1; ··· ; yp) 2 R is a dependent variable Domain as the largest set of values for which the formula makes sense. 9/1 Complex numbers \Imaginary numbers are a fine and wonderful refuge of the divine spirit, almost an amphibian between being and non-being." Gottfried Leibnitz Fundamental theorem of algebra: "Every non-constant single-variable polynomial hasp at least one complex root." X 2 + 1 = 0 defines x = i = −1. Complex number x = a + bi = (a; b) = ceθi . Complex conjugate, Complex plane, summation, multiplication, division, and logarithm. Euler formula, "our jewel", eiα = cos(α) + i sin(α) for real α Proof by Taylor expansion eix +e−ix eix −e−ix cos x = 2 ; sin x = 2i . ey +e−y cosh(y) = cos(iy) = 2 , ey −e−y i sinh(y) = sin(iy) ! sinh y = 2 . 1 cos(x) · cos(y) = 2 [cos(x + y) + cos(x − y)], cos(x + y) = cos x cos y − sin x sin y, sin(x + y) = sin x cos y + cos x sin y. 10/1 Coordinate System Rectangular cartesian coordinate system is a one to one correspondence between ordered sets of numbers and points of space. Ordinate (vertical) vs. abscissa (horizontal) axes. Round or curvilinear coordinate system Curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved, e.g., rectangular, spherical, and cylindrical coordinate systems. Coordinate surfaces of the curvilinear systems are curved. Plane polar coordinate system, x = r cos θ; y = r sin θ; dS = rdrdθ, Spherical polar coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ; dV = r 2 sin θdrdφdθ Rectangular coordinates 11/1 Coordinate System Figure: Polar coordinates taken from The Raman Effect by DA Long 12/1 Vector analysis Scalar quantities have magnitude vs. vector quantities which have magnitude and direction. Triangle law of vector addition. Parallelogram law of vector addition (Allows for vector subtraction), further it shows commutativity and associativity. 13/1 Vector analysis: direction cosines Figure: Direction Cosines taken from The Raman Effect by DA Long Direction cosines, projections of A~. Geometric or algebraic representation. 14/1 Vector analysis: direction cosines Consider two vectors of unit length r1 = (x1; y1; z1) = (lr1x ; lr1y ; lr1z ) and r2 = (x2; y2; z2) = (lr2x ; lr2y ; lr2z ) For the angle between r1 and r2, cos θ = x1x2 + y1y2 + z1z2 = lr1x lr2x + lr1y lr2y + lr1z lr2z For orthogonal unit vectors r and r', lrx lr 0x + lry lr 0y + lrz lr 0z = 0 2 2 2 Also, lrx + lry + lrz = 1 2 2 2 2 2 2 Thus, lx0x + lx0y + lx0z = 1 ly 0x + ly 0y + ly 0z = 2 2 2 1 lz0x + lz0y + lz0z = 1. Further, lx0x lx0y + ly 0x ly 0y + lz0x lz0y = 0 lx0Y lx0Z + ly 0y ly 0z + lz0y lz0z = 0 lx0z lx0x + ly 0z ly 0x + lz0z lz0x = 0. In summary, lρ0ρlσ0ρ = δρ0σ0 lρ0ρlρ0σ = δρσ 15/1 Vector analysis Unit vectors, A~ = Ax x^ + Ay y^ + Az z^. Expansion of vectors in terms of a set of linearly independent basis allow algebraic definition of vector addition and subtraction, i.e., A~ ± B~ =x ^(Ax ± Bx ) +y ^(Ay ± By ) +z ^(Az ± Bz ). jAj, Norm for scalars and vectors. Ax = jAj cos α; Ay = jAj cos β; Az = jAj cos γ Pythagorean theorem, 2 2 2 2 2 2 2 jAj = Ax + Ay + Az ; cos α + cos β + cos γ = 1. Vector field: An space to each point of which a vector is associated. Direction of vector r is coordinate system independent. 16/1 Rotation of the coordinate axes x0 = x cos φ + y sin φ y 0 = −x sin φ + y cos φ Since each vector can be represented by a point in space a vector field A is defined as an association of vectors to points of space such that 0 0 Ax = Ax cos φ + Ay sin φ Ay = −Ax sin φ + Ay cos φ 17/1 N-dimensional vectors x0 cos φ sin φ x = . y 0 − sin φ cos φ y x ! x1; y ! x2; z ! x3 0 PN 0 xi = j=1 aij xj ; i = 1; 2; ··· ; N; aij = cos(xi ; xj ): In Cartesian coordinates, 0 0 0 P xi = cos(xi ; x1)x1 + cos(xi ; x2)x2 + ··· = j aij xj thus 0 @xi aij = : @xj By considering primed coordinate axis to rotate by −φ, P 0 0 P 0 0 P 0 xj = i cos(xj ; xi )xi = i cos(xi ; xj )xi = i aij xi resulting in @xj aji = 0 = aij . @xi A is the matrix whose effect is the same as rotating the coordinate axis, whose elements are aij . 18/1 Vectors and vector space Orthogonality condition for A: AT A = I or 0 0 0 X X @xi @xi X @xj @xi @xj aij aik = = 0 = = δjk @xj @xk @x @xk @xk i i i i By depicting a vector as an n-tuple, B = (B1; B2; ··· ; Bn), define: Vector equality. Vector addition Scalar multiplication Unique Null vector Unique Negative of vector Addition is commutative and associative. Scalar multiplication is distributive and associative. 19/1 Algebraic structures: Group A group is a set, G, together with an operation * (called the group law of G) that combines any two elements a and b to form another element, denoted a*b or ab. Closure: For all a, b in G, the result of the operation, a*b, is also in G. Associativity: For all a, b and c in G, (a*b)*c = a*(b*c).
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