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

Assignment P.260: 38,40,42 P.286: 24,27,28 P.302: 7,9,12,15 PP : 2, 8, 10

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: Separable First-Order ODE

Classroom Exercise: Separable First-Order ODE

Reduction to Separable Form: Homogeneous Equations For n=0: Example:

Example: Separation of a Homogeneous Equation Check:

Chemical Kinetics

Rate of Reaction

Rate Constant and Order A  products first order 2A  products second order A + B  products second order

A  products: first order process -k ln[A] 0 ln[A] t 0

2A  products: second order process 2k 1/[A] 0 1/[A] t 0

A + B  products: second order process

First-Order Linear Equations: The Homogeneous Case

First-Order Linear Equations: The inhomogeneous Case

Example: Linear Equation

Classroom Exercise: Linear Equation

Chemical Kinetics Example

B A C B A C B A C

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

Three Cases

Example: Double root

Example: Complex roots

Classroom Exercise Find the general solution of the following ODE:

Classroom Exercise Find the general solution of the following ODE:

Particular Solutions Solutions with initial or boundary conditions.

Boundary Conditions

Example: The particle in a 1D box Two distinct regions: well and wall The microscopic entity cannot be outside of the well: Within the well, the particle is a free particle:

Boundary Conditions To ensure

Quantization of Energy Where there is constraint, there is quantization Only some energies are allowed: n: quantum numbers

Normalization for

First five normalized wavefunctions Where there is constraint, there is quantization Standing wave

Orthogonality

Example: The particle in a ring Choosing c 2 =0 (because n can take both positive and negative values) and normalizing the wavefunction:

Probability distribution

Orthogonality

Quiz Solve the following ODEs:

Inhomogeneous, Linear ODE

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.

Example

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) ]

The Method of Undetermined Coefficients

Classroom Exercise Double roots of homo. eq. Check above table, we findBut it’s one solution of homo eq.

Forced Oscillations Harmonic forceDamping force (friction) External periodic force When no friction is considered and the external force is electric force on a charge e:

General Solution Using method of undetermined coefficients:

Resonance On resonance, x p is a solution of the homo. eq., therefore the correct x p should be

III. Second-Order ODE: Special Cases of Variable Coefficients It’s hard or impossible to obtain the solution of a general second-order ODE Inhomogeneous, linear, variable coefficients:

The Power-Series Method can be determined.

Example

The Frobenius Method r: indicial parameter. (Euler-Cauchy Eq.) (Indicial equation)

Examples

Solutions from Frobenius Method Case 1: distinct roots not differing by an integer Case 2: Double root Case 3: roots differing by an integer

Examples

The Legendre Equation The power-series solution of the equation is therefore

The Convergence Condition For -1<x<1, the series is convergent.

The Legendre Polynomials

Example Show that the Legendre function of order 3 satisfies the Legendre equation

The Recurrence Relation of Legendre Functions

Example Use the recurrence relation to derive Take l =1:

The Associated Legendre Functions Under conditions: The particular solutions are associated Legendre functions: The associated Legendre equation

Example Use to derive

Orthogonality: The Legendre Functions

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:

Zeros of Bessel Functions x 1 0 J0J0 J1J1 10

An Approximate Expression

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 ).

IV. Partial Differential Equations 1-D wave equation Time-dependent Schrödinger equation 1-D diffusion equation 3-D Laplace equation 3-D Poisson equation Time-independent Schrödinger equation

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.

Separation of Variables

Motion in two and high dimensions

Separation of Variables A 2D problem reduced to two 1D problems! Reason?

Eigenfunctions of a particle on a surface

Some wavefunctions of a particle on a surface

Picture of the Interactions in an Atom

Hamiltonian of Hydrogenic Atoms Only one term Only two terms

Magic! 玩個小魔術 Reduced mass

How a two-body problem is turned into a one-body problem Center of mass

Classical Treatment Free motion of the center of mass One-body (with reduced mass) in a potential

Quantum Treatment

Motion of the center of massRelative motion

Free particle sucks, let’s forget it! Two-body problem  Free particle (Center of mass) + one-body problem

Radial 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

Hydrogenic radial wavefunctions Orbitaln l mlml 1s1 0 2s 2 0 2p 21 3s3 0 3p3 1 3d32

The radial wavefunctions of the first few hydrogenic atoms of atomic number Z

1s radial wavefunction

2s radial wavefunction

3s radial wavefunction

2p radial wavefunction

3p radial wavefunction

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 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

Shells and subshells

Atomic Orbitals:General Considerations Appropriate balance between potential and kinetic energy (c) (a): electron tends to escape (b): Electron tends to fall into the nucleus

Some typical atomic orbitals

Node( 節點 / 節面 )

Boundary Surface The probability of finding the electron inside the sphere is 90%

Mean radius of hydrogenic atoms For 1s

Radial Distribution Functions

Most probable radius For 1s, Its maximum:

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

2p Orbitals Nodal plane

3d Orbitals

Where are the nodal planes?

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.