Chemistry 330

1. 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 P(x,y,z) y x

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

5. The Transformation • To convert spherical polar to Cartesian coordinates

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

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 y x

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 I y Z = x + yi  x R - Z = x - yi

20. The Complex Conjugate • Suppose we had a complex number • C = a + bi • The complex conjugate of c • C*= a – bi • Note • (C C*) = (a2 + b2) • 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 I y Z = x + yi r  x R X = r cos  z =r cos+ i sin  = r e i y = r sin 

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 (4x2) = 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