Lecture 1: Complex Number

Transcription

1 Lecture 1: Complex Number Key points Definition of complex number, where Modulus and argument Euler's formular Polar expression where if, then and rotates a point on the complex plane by Imaginary unit Rectanular expression or Modulus or Argument or Real part or Imaginary part or Conjugate Polar expression or Conversion functions:,, 1 Imaginary unit Definition: Imaginary Unit: Common sense,,,, where These relations are used in various series expansions appeared in quantum mechanics See Lecture?? See the section of complex plane for graphical representation of these relations In, imaginary unit is I (capital I) I

2 Exercise 11 Compute by hand and confirm the result with Answer 2 Complex number in rectangle expression Complex number in rectangle expression where x and y are real numbers ( ) x real part, y imaginary part If, then is real If, then is pure imaginary Common sense For and, For, is not necessarily the real part of because and can be complex numbers See Exercise 22 Generally, there are multiple expressions to describe a mathematical expression Close to the original mathematical expression 3 4 Close to computer languages (commands or functions) 3 4 Exercise 12 Calculate by hand and confirm the result with

3 Answer Exercise 13 For Answer where and are complex, show that 3 Complex plane, modulus and argument Complex Plain ( plane) Modulus: Argument(Phase): Figure 1 Commen sense For, Popular points on a unit circle: (when, the point on complex plane make a complete rotation along the circle Section 1 )

4 means that when a imaginary unit is multiplied to a complex number, the point rotates by Modulus: 5 or 5 Argument: Use evalf to get the numerical value in decimal form (truncated at some significant figures) Symbolic calculation Exercise 14 Find the modulus and argument of by hand and confirm the result with Answer 4 Complex conjugate

5 Definition: For and its conjugate is See Figure 1 for geometric relation between and In physics, indicates complex conjugate of z In math books, is often used Common sense, If then is real ( If then is pure imaginary ( Complex Conjugate: or or 25 is equivalent to 25 5 Euler's formula and polar expression Euler's formula: Polar Expression: Multipying to a complex number rotates the point by angle Common sense

6 , (In general function evalc converts polar expression to rectangle expression polar expression can be specified by polar function convert changes the expression Exercise 15 Express in polar expression/ Answer 6 Real and Imaginary parts are independent degree of freedome If, then and For, if then, and Warning, solve may be ignoring assumptions on the input variables is not so smart! (1) Warning, solve may be ignoring assumptions on the input

7 variables With a little help by a human, can solve it (2) 7 Examples in physics - Traveling plane wave A plane wave expressed as is unbiqutus in physics (sound wave, electromagnetic wave, a free quantum particle, ) Assuming is real, we can visualize this wave as a circular motion at each position At, and do not change in time The factor makes a circular motion with angular and the starting point of the circular motion is determined by, the rotation speed is the same everywhere On the other hand, the starting point shifts by phase angle See Figure below Now we consider two souond waves A wave traveling in +x and -x direction can be expressed as respectively, where k and are wave vector and frequency A is a complex amplitude

8 Suppose that the wave is coming from x<0 toward x0 and reflected back to x<0, the total wave is Reflection by a soft interface: Reflection by a hard interface: Since, Many physical waves are measured in real number In such cases, we use the real part of complex wave as real wave (Actually, the imaginary part is equally valid as real wave) Writing the amplitrude in polar form Sound wave in an open pipe Sound wave in a closed pipe

9 Homework: Due 9/2 (Tue), 11am 11 Hyperbolic functions Using the Euler's Formula, prove the following relations (Assume that x is real) Quantum wave function The state of a free quantum particle is descrive by a complex function determined by the Schrödinger equation, which is where m is the mass of the particle 1 Show that a traveling wave is a solution to the Schrödinger equation 2 3 where Find the Schrödinger equation for the complex conjugate of In quantum mechanics, a real function plays an important role Show that