Chapter 23 ODE (Ordinary Differential Equation)

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# Chapter 23 ODE (Ordinary Differential Equation) - PowerPoint PPT Presentation

Chapter 23 ODE (Ordinary Differential Equation) . Speaker: Lung-Sheng Chien. Reference: [1] Veerle Ledoux, Study of Special Algorithms for solving Sturm-Liouville and Schrodinger Equations.

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## Chapter 23 ODE (Ordinary Differential Equation)

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### Chapter 23 ODE (Ordinary Differential Equation)

Speaker: Lung-Sheng Chien

Reference: [1] Veerle Ledoux, Study of Special Algorithms for solving Sturm-Liouville and Schrodinger Equations.

[2] 陳振臺, university physics, lecture 5 [3] Harris Benson, university physics, chapter 15

Hook’s Law (虎克定律) [1]

Hook’s Law (虎克定律) [2]

Hook’s Law (虎克定律) [3]

dimension analysis

We must match dimension of each term in the equation

Period (周期)

Definition:

guess

or

We need two constraints to determine unknown constant

1

Initial position:

2

Initial velocity:

where

Hook’s Law (虎克定律) [4]

Simple Harmonic Oscillation:

is shifted by phase constant

is phase (相位)

the argument

Question 1: why do we guess

in equation

Question 2: where do two constant A, B (or C,D) come from?

Question 3: is the solution unique?

ODE [1]

First order linear ODE:

1

is first derivative.

First order : highest degree of differential operation

2

, then operator is linear, say

Linear : define operator

integration along curve

Observation1: when initial time is determined and initial condition

is unique

is given, then solution

However when initial condition is specified, then initial time is determined at once, hence we say

is one-parameter family, with parameter

First order linear ODE system of dimension two:

3

ODE system: more than one equation

4

system of dimension two: two equations

solution is

since system is de-couple

ODE [2]

has unique solution if two initial conditions are specified

for square matrix A

we define

implies

let

, then

Observation 2: ODE system of dimension two needs two initial conditions

, then

is unique solution of

Observation 3: if we write

we may guess that

for any square matrix A

is unique solution of

ODE [3]

Simple Harmonic Oscillation:

where

since initial velocity would also affect

Intuition: we CANNOT determine

by given initial position

It is well-known when you are in junior high school, not only free fall experiment but sliding car experiment.

Transformation between second order ODE and first order ODE system of dimension two

Define velocity

, then combine Newton’ second Law

, we have

we need two initial condition to have unique solution

From Symbolic toolbox in MATLAB, we can diagonalize matrix

MATLAB code

where

ODE [4]

Formal deduction:

with initial condition

Definition: fundamental matrix

then solution can be expressed by fundamental matrix and initial condition

From Symbolic toolbox in MATLAB,

we can compute fundamental matrix easily

ODE [5]

is composed of two fundamental solutions

fundamental matrix

is solution of

with initial condition

is solution of

with initial condition

with initial condition

is linear combination of fundamental solutions

solution of

The space of solutions of

is

The dimension of solution space is two,

ODE [6]

In order to achieve uniqueness, we need to specify two integration constant

Integral equation:

differential equation:

Existence and uniqueness (Contraction mapping principle)

Let

be continuous space equipped with norm

1

is complete under norm

2

by

define a mapping

then

Existence and uniqueness

is a contraction mapping if

ODE [7]

(ignore buoyancy 浮力)

recover force is opposite to displacement

resistive force is also opposite to displacement

Newton second’s Law:

1

what is equivalent ODE system of

2

what is fundamental matrix of this ODE system, use symbolic toolbox in MATALB

3

what is solution of

with initial condition

4

Can you use “contraction mapping principle” to prove existence and uniqueness?

ODE [7]

general solution

We have two choices to determine unknown constants A and B

1

initial condition

2

boundary condition

period

1

become discrete?

Why does angular frequency

What is physical meaning of discrete angular frequency?

2

We have still a constant B not be determined, why?

Schrodinger equation [1]

energy

is angular frequency

photon (光子):

is wave number

momentum

http://en.wikipedia.org/wiki/Matter_wave

Louis de Broglie in 1924 in his PhD thesis claims that “matter (object)” has the same relation as photon

de Broglie wavelength

de Broglie frequency of the wave

Fundamental of quantum mechanic:

matter wave (物質波) is described by wave length

and wave frequency

http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

Erwin Schrödinger in 1926 proposed a differential equation (called Schrodinger equation) to describe atomic systems.

total energy of a particle

matter wave

and

time-dependent Schrodinger equation

Schrodinger equation [2]

Question: physical interpretation of matter wave

probability finding particle on interval

in time

total number of particles in time

Example: plane wave

particle has the same probability found in any position, not physical

time-dependent Schrodinger equation

by

remove t-dependence, replace

time-independent Schrodinger equation

(one-dimensional)

(three-dimensional)

Objective of time-independent Schrodinger equation: find a stationary solution satisfying

with proper boundary conditions

Schrodinger equation [3]

Example: Hydrogen atom (氫原子)

parameter

B.C.

http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

discrete energy level:

http://www.touchspin.com/chem/SWFs/pt2k61012.swf

Schrodinger equation [4]

Time-independent Schrodinger equation

From dimensional analysis, we extract dimensionless quantities in this system

1

2

3

4

dimensionless form

Once characteristic length of the system is determined, for example

Then characteristic energy is

, this is near

Finite Difference Method [1]

First we consider one-dimensional problem (dimensionless form)

Finite Difference Method (有限差分法): divide interval

into N+1 uniform segments

labeled as

We approximate ODE on finite points, say we want to find a vector

satisfying

for

We don’t need to ask equation on end points, since equation only holds in interior

1

How to approximate second derivative

on grid points

2

How to achieve “solvability” though we know solution indeed exists in continuous sense

3

Finite Difference Method [2]

Definition: standard 3-point centered difference formula

we have

for some

is called local truncation error (LTE) since it is truncation from Taylor’s expansion

Finite Difference Method [4]

for

Example: n = 4

is real symmetric, then A is diagonalizable since eigenvalue of A is real

Finite Difference Method [5]

model problem: no potential ( V = 0 )

FDM

Question: can we find analytic formula for eigen-pair in this simple model problem?

solution is

solution is

FDM

for

Conjecture: we guess that eigenvector is

satisfies

(boundary condition)

for

Finite Difference Method [6]

model problem: no potential ( V = 0 )

eigen-function:

Finite Difference Method on model problem:

for

eigen-pair:

Question: How accurate are numerical eigen-value?

Exact wave number:

Numerical wave number:

Taylor expansion:

where

Question: eigenvalue is second order accuracy,

, is this reasonable, why?

Question: why does error of eigenvalue increase as wave number k increases?

Finite Difference Method [7]

model problem: no potential ( V = 0 )

Exercise 1: model problem (high order accuracy )

Finite Difference Method on model problem:

for

eigen-pair:

Question: how can we improve accuracy of eigenvalue of model problem?

Step 1: deduce 4-order centered finite difference scheme for second order derivative

Step 2: can you transform continuous equation to discrete equation?

for

Exercise 2: singular potential

effective potential:

finite difference with uniform mesh platform: MATLAB

bound state

There are only six bound state (energy E < 0 )

Definition: bound state means

Exercise: plot first 10 eigen-function and interpret why only first lowest six eigenvalue corresponds to “bound state”