Lecture 2: Series Expansion

Transcription

1 Lecture 2: Series Expansion Key points Maclaurin series : For, Taylor expansion: Maple commands series convert taylor, Student[NumericalAnalysis][Taylor] 1 Maclaurin series of elementary functions for for Maple

2 Visual understanding of the expansion Common sense (approximations for small x) For,

3 Exercise 21 Find a series expansion of using the series expansion of and Verify the result using Maple series command Exercise 22 Expand the folloiwing expression in Maclaurin series and find the terms up to the order of (1) (2) (3) 2 Dimensionless arguments in f(x) must be dimensionless If has a dimension of length ( ), what is the dimension of which is not possible Common sense A wave is expressed as where has the dimension of inverse length ( ) Henec, has no dimension An oscillation is often expressed as where is angular frequency Since the frequency has dimension of inverse time, is dimensionless The number of radio active neuclea decreses as where is a mean lifetime Note that is dimensionless Remark: If has a dimension, does cancell the dimension make a sense? It is OK as long as there is another log function that If and have the same dimension, is fine since, the dimension cancelled out This is very confusing Therefore, you should always combine logarithmic functions so that the argument becomes dimensionless

4 Exercise 23 have the same dimension Does expression sense? make a 3 Complex arguments For, where complex number (5) Common sense For, Exercise 24 Show that [Euler's formula] using the expansion of exponential function 4 Taylor expansion Example: Taylor expansion of around Example: Taylor expansion of around

5 Common sense Maple Exercise 25 Show that and Exercise 26 Expand in a Taylor series about Explain why only odd power appears 5 Examples in physics 1 Small amplitude oscillations Consider a particle of mass moving in a potential which has a stable equilibrium position at which force vanishes The equation of motion is given by The particle oscillates around the equilibrium position You can find the equilibrium position by solving When the amplitude of the oscillation is not very large, the potential can be approximated with the first few term in Taylor expansion around the equilibrium position: Then, the corresponding force is given by Introducing and, we have an approximate equation of motion which is nothing but a harmonic oscillator Pendulum Potential: ] Equation of motion: where Equilibrium position: or ( is an unstable equilibrium position) Taylor expansion:

6 For small angle radw at 5 digits degree, 1st order approx 2nd order approx (harmonic oscillator) (a kind of Duffing oscillator) Lennard-Jones potential Interatomic potential between rare gas atoms is often modeled with the Lennard-Jones potential (7) where is a distance between the atoms and is a constant The force is then given by (8) The force is zero at

7 (9) (10) Exact potential and harmonic approximation near the minimum Agreement is good only near the minimum 2 Non-relativistic limit A particle of rest mass has relativistic energy To find energy in the nonrelativistic limit, we expand (11) (12) 3 Infinitesimal rotations

8 Consider a rotation about x axis by angle (13) and another rotation about y axis by angle (14) In general these two rotations do not commute as shown here (15) If then and Substituting these approximation, we find the commutation relation (16) Hence, if the rotation angle is sufficiently small (order of ) the two rotations commute Now, if we take into account the second order,

9 (17) 4 Functions of an operator Taylor expansion can be used to define a function of operator Homework: Due 9/4, 11am 21 Small amplitude oscillation Another popular interatomic potential is Morse potential Find an equilibrium position and derive a harmonic potential about the equilibrium position Then, find the corresponding spring constant Compare the harmonic potential with the exact Morse potential by plotting them (Use, and ) 22 Taylor expansion and an operator function Using the Taylor expansion of exponential function, prove the following equality: 23Fermi-Dirac distribution function At temperature, the probability that a fermion occupies an energy state is given by the Fermi- Dirac distribution function where and are chemical potential and Boltzmann constant, respectively In metal, many physical phenomena are generated by electrons (fermion) whose energy is close to the chemical potential ( ) Therefore, we want to know how behaves near the chemical potential Expand about up to the order of