Chemistry 330 The Mathematics Behind Quantum Mechanics

2 Coordinate Systems Function of a coordinate system locate a point (P) in space Describe a curve or a surface in space Types of co-ordinate systems Cartesian Spherical Polar Cylindrical Elliptical

3 Cartesian Coordinates The familiar x, y, z, axis system Point P - distances along the three mutually perpendicular axes (x,y,z). z x y P(x,y,z)

4 Spherical Coordinates Point P is based on a distance r and two angles (  and  ). z x y P(r, ,  ) r  

5 The Transformation To convert spherical polar to Cartesian coordinates

6 Cylindrical Coordinates Point P is based on two distances and an angle (  ). z x y P(r, ,z) r 

7 The Transformation To convert cylindrical to Cartesian coordinates

8 Differential Volume Elements Obtain d  for the various coordinate systems Cartesian coordinates

9 Differential Volume Elements Spherical polar coordinates

10 Differential Volume Elements Cylindrical coordinates

11 Vectors and Vector Spaces Vector – used to represent a physical quantity Magnitude (a scalar quantity – aka length) Direction Normally represent a vector quantity as follows or

12 Components of a Vector A unit vector – vector with a length of 1 unit. Three unit vectors in Cartesian space z x y

13 Vector Magnitude Magnitude of the vector is defined in terms of its projection along the three axes!! Magnitude of

14 Vectors (cont’d) Any vector can be written in terms of its components - projection Vectors can be added or subtracted Graphically Analytically Note – vector addition or subtraction yields another vector

15 Vector Multiplication Scalar Product – yields a number

16 Vector Multiplication Cross Product – yields another vector

17 The Complex Number System Let’s assume we wanted to take the square root of the following number. Define the imaginary unit

18 Imaginary Versus Complex Numbers A pure imaginary number = bi b is a real number A complex number C = a + bi Both a and b are real numbers

19 The Complex Plane Plot a complex number on a ‘modified x-y’ graph. Z = x + yi y x Z = x - yi  -- R I

20 The Complex Conjugate Suppose we had a complex number C = a + bi The complex conjugate of c C * = a – bi Note (C C * ) = (a 2 + b 2 ) A real, non-negative number!!

21 Other Related Quantities For the complex number Z = x +yi Magnitude Phase

22 Complex Numbers and Polar Coordinates The location of any point in the complex plane can be given in polar coordinates y x Z = x + yi  R I r X = r cos  y = r sin  z =r cos  + i sin  = r e i 

23 Differential Equations Equations that contain derivatives of unknown functions There are various types of differential equations (or DE’s) First order ordinary DE – relates the derivative to a function of x and y. Higher order DE’s contain higher order derivatives

24 Partial DE’s In 3D space, the relationship between the variables x, y, and z, takes the form of a surface. Function Derivatives

25 Partial DE’s (cont’d) For a function U(x,y,z) A partial DE may have the following form

26 Other Definitions Order of a DE Order of the derivatives in it. Degree of the DE The number of the highest exponent of any derivative

27 Methods of Solving DE’s Find the form of the function U(x,y,z)that satisfies the DE Many methods available (see math 367) Separation of variable is the most often used method in quantum chemistry

28 Operators An operator changes one function into another according to a rule. d/dx (4x 2 ) = 8x The operator – the d/dx The function f(x) is the operand Operators may be combined by Addition Multiplication

29 Operators (cont’d) Operators are said to commute iff the following occurs

30 The Commutator Two operators will commute if the commutator of the operators is 0! If  = 0, the operators commute!!

31 Operators (the Final Cut) The gradient operator (  - del) The Laplacian operator (  2 – del squared)

32 The Laplacian in Spherical Coordinates The Laplacian operator is very important in quantum mechanics. In spherical coordinates

33 Eigenvalues and Eigenfunctions Suppose an operator operates on a function with the following result P is an eigenvalue of the operator f(x,y) is an eigenfunction of the operator

34 Eigenfunctions (cont’d) Operators often have more than one set of eigenfunctions associated with a particular eigenvalue!! These eigenfunctions are degenerate

35 Linear Operators Linear operators are of the form Differential and integral operators are linear operators

36 Symmetric and Anti-symmetric Functions For a general function f(x), we change the sign of the independent variable If the function changes sign – odd If the sign of the function stays the same – even Designate as Symmetric – even Antisymmetric – odd

37 Integrating Even and Odd Functions Integrate a function over a symmetric interval (e.g., -x  t  +x) if f(x) is odd if f(x) is even

38 Mathematical Series Taylor Series When a = 0, this is known as a McLaren Series!

39 Periodic Functions Sin(x) and cos(x) are example of periodic functions! Real period functions are generally expressed as a Fourier series

40 Normalization A function is said to be normalized iff the following is true N – normalization constant

41 Orthogonal Functions Two functions (f(x) and g(x) are said to be orthogonal iff the following is true Orthogonal – right angles!!

42 The Kronecker Delta (  fg ) If our functions f(x) and g(x) are normalized than the following condition applies f(x) = g(x),  fg = 1 f(x)  g(x),  fg = 0