Theoretical Classical Physics

Lecture 1 Introduction Complex numbers Sep 6, 2018 Dr. Caio Licciardi, F-532 [email protected]

Theoretical

Classical Physics

PHYS-3466

1.

Course introduction

2.

The puzzling number 𝑖 = −1

o Class hours and location:

 Monday @ 11:30 am – 12:50 pm, F-536  Thursday @ 11:30 am – 12:50 pm, F-536 o Instructor:

 Caio Licciardi, F-532 – Phone: 705-675-1151 x2309  [email protected] o Office hours:  Wednesday @ 1:00 pm – 3:00 pm, F-532 o Website:  https://clicciardi.ca/teaching/phys3466-f18/

Practical info

• Sep 19, Oct 17 and Nov 21

Textbooks

o Main textbook:

 Essential Mathematical Methods for Physicists, H. J. Weber and G. B. Arfken

o Alternative:

 Mathematics for physicists, P. Dennery and A. Krzywicki o Supplementaries:

 An imaginary tale, P. J. Nahin

Schedule

Grading

o Your final grade will be based out of a total 100 points:  Final exam (FE) = 40 points

 Mid-term (MT) = 20 points

 Homework (HW) = 10 points (1 point / assignment)  Class-work (CW) = 30 points

• Quizzes = 5 points (~1 every other tutorial) • Board solutions / elective project = 25 points

𝐺 = 𝐹𝐸 + 𝑀𝑇 + 𝐻𝑊 + 𝐶𝑊

10 marks

o Quizzes :

 10-min multiple-choice test o Board solutions :

 selected problems in the list of exercises for the tutorial o Elective project :

•

Natural numbers:

1, 2, 3 …

•

Positive numbers:

0, ½ , 3, 𝜋

• Can be physically interpreted

• Segment lengths in a continuous line

•

Negative numbers

• Introduces the notion of direction around zero

• Need abstract interpretations: deficit, inverse, etc.

Negative numbers

Negative numbers require additional “algebraic tools”

• Need a symbol to represent it: “-”

• The negative of 𝑥 is 𝑥_𝑛 such that 𝑥 + 𝑥_𝑛 = 0

• ⟹ 𝑥_𝑛 ≡ −𝑥 because 𝑥 + −𝑥 = 0

• Multiplication rule: − × − = +

• 𝑥 × 𝑦 = 𝑥𝑦

• −𝑥 × 𝑦 + 𝑥𝑦 = −𝑥 + 𝑥 × 𝑦 = 0 ⇒ −𝑥 × 𝑦 = −(𝑥𝑦)

• −𝑥 × −𝑦 = − 𝑥 −𝑦 = −(− 𝑥𝑦 ) ⇒ −𝑥 × (−𝑦) = 𝑥𝑦

•

Summary of introducing the negative numbers

• (2) Negative numbers require an additional symbol to represent a number

• (3) And an additional rule to multiply numbers

The square root of a number 𝑥 is a number such that

𝑟2 = 𝑥 ⟺ 𝑟 = 𝑥 Geometrically understood!

Constructing the square root of a line: 1. Line = 𝐺𝐻

2. Construct 𝐹𝐺 = 1

3. 𝐾 = bi-section of FH

4. Draw semi-circle of radius 𝐹𝐾 = 𝐾𝐻

5. I = intersection of perpendicular to FH in G and

semi-circle 𝐹𝐺 + 𝐺𝐻 = 2 𝐼𝐾 𝐹𝐺 + 𝐺𝐾 = 𝐼𝐾 𝐼𝐺2 + 𝐺𝐾2 = 𝐼𝐾2 𝐼𝐺2 = 𝐺𝐻 What if 𝐹𝐺 ≠ 1?

Square root of positive numbers

•

John Wallis

• English mathematician (1616-1703)

• Using the notion of direction around zero, suggested that

GHpositive because it extends to the right from 𝐺 to 𝐻

• Showed that extending to the left of 𝐺 into 𝐻

• 𝐼𝐻 = (𝐺𝐻)(𝐻𝐹)

Square root of negative numbers ?

G H

F

I

Idea of direction implies 𝐻𝐹 > 0and 𝐺𝐻 < 0

“Ingenious, but scarcely convincing”

G

H F

I

• Let’s look at the line of ”real” numbers with different eyes: • Instead of the symbols +/- ⟹ let’s assign an angle w.r.t. zero : ∠

• Positive numbers:+𝑥 ≡ 𝑥∠0°or+𝑥 ≡ 𝑥∠0 (rad)

• Negative numbers: −𝑥 ≡ 𝑥∠180°or−𝑥 ≡ 𝑥∠𝜋

• Multiplication rule:

• 𝑥 × −𝑦 ≡ 𝑥∠0 × 𝑦∠𝜋 = − 𝑥𝑦 ≡ 𝑥𝑦∠𝜋 = 𝑥𝑦 ∠ 0 + 𝜋

• −𝑥 × −𝑦 ≡ 𝑥∠𝜋 × 𝑦∠𝜋 = 𝑥𝑦 ∠ 𝜋 + 𝜋 = 𝑥𝑦∠0 ≡ (𝑥𝑦), since2𝜋 ≡ 0

• It is not un-natural to think of a general number: 𝑥 = 𝑟∠𝜃

• Extending the “real line”into the “complex plane”

Extending the set of imaginary numbers …

⟹ sum of the angles

𝑥 = 𝑟∠𝜃 ℂomplex plane real axis im a g in a ry a xi s

Unit in the perpendicular axis:

1∠𝜋₂ ≡ 𝑖 𝑖2 = 𝑖 × 𝑖 ≡ 1∠𝜋₂ × 1∠𝜋₂ = 1∠𝜋 ≡ −1 Caspar Wessel (1745 – 1818) 𝑖

• The complex plane is a geometric representation of the complex numbers

• Established by the real axis on the horizontal and the imaginary on the vertical

• A.K.A. the Argand plane

Complex plane

ℂ

• Complex number: 𝑧 = 𝑥 + 𝑖𝑦 = 𝑥, 𝑦

• Ordered pair of two real numbers : 𝑥, 𝑦 ∈ ℝ2

• Real part: ℜ𝑒 𝑧 = 𝑥 = 𝑟 cos 𝜃

• Imaginary part: ℑ𝑚 𝑧 = 𝑦 = 𝑟 sin 𝜃

• Modulus (magnitude): 𝑟 = 𝑥2 _{+ 𝑦}2 • Argument (phase): tan 𝜃 = Τ𝑦 𝑥

• Addition: • 𝑧1+ 𝑧2 = 𝑥1+ 𝑥2 + 𝑖 𝑦1+ 𝑦2 = (𝑥1 + 𝑥2, 𝑦1 + 𝑦2) • Multiplication: • 𝑧₁𝑧₂ = 𝑥₁𝑥₂ − 𝑦₁𝑦₂ + 𝑖 𝑥₁𝑦₂ + 𝑥₂𝑦₁ = 𝑥₁𝑥₂ − 𝑦₁𝑦₂, 𝑥₁𝑦₂ + 𝑥₂𝑦₁ • Division: • 𝑧1 𝑧₂ = 𝑥₁+𝑖𝑦₁ 𝑥₂+𝑖𝑦₂ = (𝑥₁+𝑖𝑦₁)(𝑥₂−𝑖𝑦₂) (𝑥₂+𝑖𝑦₂)(𝑥₂−𝑖𝑦₂) = (𝑥₁𝑥₂+𝑦₁𝑦₂)+𝑖(𝑥₂𝑦₁−𝑥₁𝑦₂) 𝑥₂2+𝑦₂2 • Complexconjugate: ҧ𝑧 = 𝑥 − 𝑖𝑦 = (𝑥, −𝑦) • Application: 𝑧 ҧ𝑧 = 𝑥 + 𝑖𝑦 𝑥 − 𝑖𝑦 = 𝑥2 + 𝑦2 = 𝑟2

Complex algebra

ℂ

• Consider the points 1∠𝜃 and 1∠𝛼 in the unit circle

• 1∠𝜃 = cos 𝜃 + 𝑖 sin 𝜃

• 1∠𝛼 = cos 𝛼 + 𝑖 sin 𝛼

• Multiplication : 1∠𝜃 × 1∠𝛼 = 1∠(𝜃 + 𝛼)

• 1∠(𝛼 + 𝜃) = cos(𝛼 + 𝜃) + 𝑖 sin(𝛼 + 𝜃)

• = (cos 𝜃 + 𝑖 sin 𝜃) × (cos 𝛼 + 𝑖 sin 𝛼)

• Expanding the last line

• = cos 𝜃 cos 𝛼 − sin 𝜃 sin 𝛼 + 𝑖 sin 𝜃 cos 𝛼 + cos 𝜃 sin 𝛼

• Two complex numbers are equal only if their real and imaginary parts are separately equal:

cos(𝜃 + 𝛼) = cos 𝜃 cos 𝛼 − sin 𝜃 sin 𝛼 sin(𝜃 + 𝛼) = sin 𝜃 cos 𝛼 + cos 𝜃 sin 𝛼

Trigonometry

• 𝑀-times multiplication : 1∠_𝑚𝜃 × … × 1∠_𝑚𝜃 = 1∠𝜃 • 1∠𝜃 𝑚 𝑚 = cos 𝜃 𝑚 + 𝑖 sin 𝜃 𝑚 𝑚 • 1∠𝜃 = cos(𝜃) + 𝑖 sin(𝜃)

• Turning around this statement by taking the m-th root • cos(𝜃) + 𝑖 sin(𝜃) 1Τ𝑚 _{= cos} 𝜃

𝑚 + 𝑖 sin 𝜃 𝑚 • Some applications: • cos3𝜃 = 3 4cos 𝜃 + 1 4cos(3𝜃) • cos6𝜃 = 1 32cos(6𝜃) + 3 16cos(4𝜃) + 15 32cos 2𝜃 + 5 16

De Moivre’s theorem

• Question: can we raise a number to a complex? 𝑖𝑖 ? • - Functional rule for multiplication of complex numbers:

• 𝑓 𝛼 𝑓 𝜃 = 𝑓(𝛼 + 𝜃)

• - The De Moivre’s theorem means: • 𝑓 𝜃 𝑛 = 𝑓(𝑛𝜃)

• Let’s think of an actual function f that has these properties: • Turns multiplications into sums: 𝑓 𝜃 = 𝑒𝐾𝜃

• Differentiate 1∠𝜃 = cos 𝜃 + 𝑖 sin 𝜃 = 𝑒𝐾𝜃 ⇒ 𝑓′ 𝜃 = 𝑖𝑓 𝜃 ⇒ 𝐾 = 𝑖

• Power of exponential is a simple multiplication: • 𝑖𝑖 = 𝑓( Τ𝜋

2)𝑖 = (𝑒𝑖𝜋/2)𝑖= 𝑒−𝜋/2 = 0.2078 …

• In 1740, Euler states two solutions to the differential equation • 𝑑2𝑦

𝑑𝑥2+ 𝑦 = 0, 𝑦 0 = 2 and 𝑦

′ _{0 = 0}

• 𝑦 𝑥 = 2 cos(𝑥)and𝑦 𝑥 = 𝑒𝑥 −1 + 𝑒−𝑥 −1

• From where we can conclude • 2 cos(𝑥) = 𝑒𝑖𝑥 + 𝑒−𝑖𝑥

• Similarly, he also knew that: 2𝑖 𝑠𝑖𝑛(𝑥) = 𝑒𝑖𝑥 − 𝑒−𝑖𝑥

• In 1748, Euler published the explicit formula:

• 𝑒±𝑖𝑥 = cos(𝑥) ± 𝑖 sin(𝑥) • Consequently: • cos 𝑥 = 1 − 1 2!𝑥 2 ₊ 1 4!𝑥 4_{+ ⋯} • sin(𝑥) = 𝑥 − 1 3!𝑥 3₊ 1 5!𝑥 5_{+ ⋯} • 𝑒𝑖𝜋 = −1 Polar notation: 𝑟∠𝜃 ≡ 𝑟𝑒𝑖𝜃

Euler’s identity

• Euler was the first to observe that 𝑖𝑖 is real and has an infinity number of values • 𝑖𝑖 = 𝑒− Τ𝜋 2 is just one possible value

• 𝑖𝑖 = 𝑒𝑖 12𝜋+2𝜋𝑛 𝑖 = 𝑒− 12+2𝑛 𝜋, • • Similarly: • 1𝜋 = cos 2𝜋2𝑛 + 𝑖 sin(2𝜋2𝑛), 𝑛 = 0, ±1, ±2, …

• Logarithm is also a multi-valued function:

• ln(𝑎 + 𝑖𝑏) = 1₂ln(𝑎2+ 𝑏2) + 𝑖 tan−1 𝑏_𝑎 + 2𝜋𝑛 , 𝑛 = 0, ±1, ±2, …

• Example:

• Principal value of ln(1 + 𝑖) = 1₂ln 2 + i Τ𝜋 4 = 0.346573 + 𝑖0.785398

Many-valued functions

𝑛 = 0is customary called the principal value

Angle 0∘ 180∘ 90∘ 270∘ 𝑛 = 0 (0,0) (3, −3) Radius ∝ darkness