Presentation on theme: "The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University 化學數學（一）"— Presentation transcript:
The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University 化學數學（一）
Chapter 5 Differential Equations Simple Ordinary Differential Equations (ODE) Kinetics of Chemical Reactions Partial Differential Equations (PDE) Chemical Thermodynamics Gamma Functions Beta Functions Hermite Functions Legendre Functions Laguerre Functions Bessel Functions Contents Covered in Chapters 11-14
Overview of Differential Equations (DE) Ordinary DE (ODE): One variable First-order ODE, Second-order ODE, … Constant coefficient ODE, Variable coefficient ODE Partial DE (PDE): Multi-variable DE: Equations that contains (partial) derivatives.
Examples ODE PDE First order: Second order: constant coefficients Second order: variable coefficients
Some First- and Second-order ODEs First order rate process (growth/decay) Second-order rate process Free falling of an object Classical harmonic oscillator One-dimensional Vibration of atomic bonds
Solving A DE Find the function(s) (of one or more variables) that satisfy the ODE/PDE. This step normally involves integration and/or series expansion. Initial or boundary conditions are usually required to specify the solution. Therefore, both equations and initial/boundary conditions are equally important in solving a specific practical problem.
I. First Order ODE Examples: First order rate process (growth/decay) Second-order rate process Initial condition: y=10 when x=0
Classroom Exercise Find the general and particular solutions of the following equation with the given initial condition:
Solving First Order ODE Separable Equations: First-order linear equations: + initial conditions
Example: Electric Circuit Three sources of electric potential drop ( drop of voltage): R L E For constant electromotive force: E=E 0 Initial condition, I(0)=0: Inductive time constant:
II. Second-Order ODE: Constant Coefficients Inhomogeneous, linear, variable coefficients: Homogeneous and linear, variable coefficients: Homogeneous, linear and constant coefficients: Inhomogeneous, linear and constant coefficients:
Principle of Superposition: Example Particular solutions Linearly independent (not related by a proportional coefficient)
Principle of Superposition (for Homogeneous Linear DEs) The linear combination of two (particular) solutions of a homogeneous DE is also a solution of the DE.
The general solution (constant coefficients) (characteristic equation or auxiliary equation) guess
Example The two particular solutions being linearly independent, the general solution is
General Solution The general solution of an inhomogeneous linear ODE is the sum of the general solution of the corresponding homogeneous equation and a particular solution of the inhomogeneous equation.
Some Important Particular Solutions The determination of the coefficient(s) in y p is obtained by substituting it back to the inhomogeneous equation. However, if y p is already in y h then the general solution should be: where the choice of c(x): If the characteristic equation of the corresponding homogeneous equation has two (real or complex) roots, then c(x) =x, or else, c(x)=x 2. If r(x) is the sum of terms given in above table, the total y p (x) is the sum of respective y p of all terms. [This leads to a method of series expansion for general r(x) ]
The determination of the coefficient(s) in y p is obtained by substituting it back to the inhomogeneous equation. However, if y p is already in y h then the general solution should be: where the choice of c(x): If the characteristic equation of the corresponding homogeneous equation has two (real or complex) roots, then c(x) =x, or else, c(x)=x 2. If r(x) is the sum of terms given in above table, the total y p (x) is the sum of respective y p of all terms. [This leads to a method of series expansion for general r(x) ]
Orthogonality and Normalization: The Associated Legendre Functions
The Hermite Equation The recurrence relation: Hermite polynomials:
Hermite Functions The Hermite functions: Orthogonality: Its solution:
Classroom Exercise Write down the normalized Hermite functions: which satisfy orthonormal condition:
Example Show that the Hermite function is a solution of the Schrödinger equation for the harmonic oscillator Let if
The Laguerre Equation n: real number Laguerre polynomials: Recurrence relation:
Associated Laguerre Functions The associated Laguerre equation It’s solution is associated Laguerre polynomials: they arise in the radial part of the wavefunctions of hydrogen atom in the form of associated Laguerre functions: which satisfy: and are orthogonal with respect to the weight function x 2 in the interval [0,∞]:
Bessel Functions The Bessel equation: Therefore, it can be solved by Frobenius method.
Bessel Functions for Integer n Bessel functions of the first kind of order n: Examples:
Bessel Functions of Half-Integer Order Bessel functions of half-integral order can be expressed in terms of elementary functions. All others can be obtained with the recurrence relation: Examples:
Spherical Bessel Functions Spherical Neumann Functions These functions are important in treating scattering processes (which are always useful in dynamics of molecules, atoms, nucleons and more elementary particles ).
General Solutions The general solution of 1D wave equation: Yeah! Both F and G are arbitrary functions! The general solution of an ODE contains an arbitrary constant, the general solution of a PDE may contain a number of arbitrary functions.
Example Verify that the function is a solution of the 1D wave equation. The above solution can be written as
Classroom Exercise Verify that the function is a solution of the 1D wave equation.
Effective potential energy Very different close to the nucleus but similar far from it Solutions of wavefunction and energy for the two cases are very different close to the nucleus but similar to each other at far distances. S orbitals Non-s orbitals
Laguerre Equation and Laguerre Polynomials Normalization factor Laguerre polynomails Bohr radius=0.053 nm Bound state
An Illustration Calculate (1) the probability density for a 1s- electron at the nucleus and (2) the probability of finding a 2s-electron in a sphere with the nucleus at the center and radius of 0.053 nm. For a 1s-electron: n=1,l=0,m l =0, the wavefunction is Probability density is At the nucleus, r=0,
For a 2s-electron, The probability of finding a 2s-electron inside the sphere is
Structure of a Hydrogenic Atom Principal quantum number n determines energy Orbital quantum number l gives the angular momentum Magnetic quantum number m l gives the “z”-component of angular momentum
Energy Levels unbound state=free state Ground state Rydburg constant Ionization energy For hydrogen atom, E 1 =13.6 eV
Spectroscopic Measurement of Ionization Energy
Shells and subshells Principal quantum numbers shells Orbital quantum numbers subshells
The behavior differences between s, p, d and f orbitals near the nucleus s: has big probability amplitude near the nucleus p: probability amplitude ~r near the nucleus d: probability amplitude ~r near the nucleus 2 f: probability amplitude ~r near the nucleus 3
Quiz 1. Write the recurrence relation for Legendre, Hermite and Laguerre polynomials, respectively. 2. Write the zeroth order and first order Lagendre and Hermite polynomials. 3. Give the approximate expression of the state of a 3p electron in an atom.