Complex Numbers: Euler's Formula and Exponential Form

Introduction

Welcome to the first lesson in our journey toward understanding the Gaussian wave packet! Since the wave function, $Ψ (x, t)$ , which lies at the heart of quantum mechanics, is a complex-valued function, a solid command of complex numbers is essential. They aren't just a mathematical convenience; they are fundamental to the theory's structure, encoding information about both the probability of finding a particle and its phase.

In this lesson, we will focus on one of the most powerful tools for handling complex numbers: the exponential form. We will start with a brief review of complex numbers in their rectangular form, then introduce the polar and exponential representations. The centerpiece of this lesson is Euler's formula, a remarkable equation that connects exponential functions to trigonometry. By the end of this lesson, you will be able to confidently manipulate complex numbers in their exponential form, a skill that will be indispensable throughout this course.

A Quick Review of Complex Numbers

Let's start by refreshing some basic concepts. A complex number, $z$ , is a number that can be expressed in the form $z = x + i y$ , where $x$ and $y$ are real numbers and $i$ is the imaginary unit, satisfying $i^{2} = - 1$ .

Real Part: $Re (z) = x$
Imaginary Part: $Im (z) = y$ (Note that the imaginary part is the real number $y$ , not $i y$ ).
Complex Plane (Argand Diagram): A complex number $z = x + i y$ can be visualized as a point $(x, y)$ in a 2D Cartesian coordinate system. The horizontal axis is the "real axis" and the vertical axis is the "imaginary axis."
Complex Conjugate: The conjugate of $z$ , denoted $z ˉ$ or $z^{*}$ , is $z ˉ = x - i y$ . Geometrically, it's the reflection of $z$ across the real axis.
Modulus (Absolute Value): The modulus of $z$ , denoted $∣ z ∣$ , is its distance from the origin in the complex plane. It is calculated as $∣ z ∣ = x^{2} + y^{2}$ . An important identity is $∣ z ∣^{2} = z z ˉ$ .

For a detailed refresher on these foundational concepts, please review the first few sections of the following document.

Complex Numbers

Let's review the basics of complex numbers with the chapter 'Complex Numbers' from Mary L. Boas's textbook. It provides a clear and comprehensive introduction.

Please read Sections 1, 2, 3, and 4 (pages 46-50). Focus on the definitions of a complex number, the complex plane (Argand diagram), the modulus, and the complex conjugate.

Test your understanding!

Let $z = 4 - 3 i$ . Find:

$Re (z)$ and $Im (z)$
The complex conjugate, $z ˉ$
The modulus, $∣ z ∣$

Show answer

$Re (z) = 4$ and $Im (z) = - 3$ .
$z ˉ = 4 + 3 i$ .
$∣ z ∣ = 4^{2} + (- 3)^{2} = 16 + 9 = 25 = 5$ .

From Rectangular to Polar and Exponential Form

While the rectangular form $z = x + i y$ is great for addition and subtraction, it can be cumbersome for multiplication and division. A more elegant representation uses polar coordinates $(r, θ)$ , where $r$ is the modulus and $θ$ is the angle (or argument) the vector from the origin to the point $z$ makes with the positive real axis.

From basic trigonometry, we have:

x = r cos θ and y = r sin θ

Substituting these into the rectangular form gives the polar form:

z = r cos θ + i (r sin θ) = r (cos θ + i sin θ)

This is where one of the most beautiful formulas in mathematics comes into play.

Euler's Formula

Euler's formula provides a profound connection between the exponential function and the trigonometric functions:

e^{i θ} = cos θ + i sin θ

This formula is not just a definition; it can be derived. Given your preference for both physical intuition and mathematical rigor, let's explore it from two complementary perspectives.

1. The Intuitive, Dynamic View

What does it mean to raise $e$ to an imaginary power? A fantastic way to build intuition is to think about the differential equation that defines the exponential function. The following video from 3Blue1Brown provides a brilliant visual explanation.

The Physics of Euler's Formula | Laplace Transform Prelude

To understand why Euler's formula makes sense, let's watch 'The Physics of Euler's Formula' by 3Blue1Brown. It builds the concept from the ground up, starting with real exponents and moving into the complex plane.

Watch from timestamp 01:40 to 05:51. Pay close attention to how the property that the derivative of e^(kt) is k times itself leads to different behaviors: growth/decay for real k, and rotation for imaginary k.

The key insight is that for the function $f (t) = e^{i t}$ , its rate of change (velocity) is $f^{'} (t) = i \cdot e^{i t} = i \cdot f (t)$ . Since multiplication by $i$ corresponds to a 90-degree rotation in the complex plane, this means the velocity vector is always perpendicular to the position vector. The only motion that satisfies this condition is moving in a circle at a constant speed, which is exactly what $cos t + i sin t$ describes on the unit circle.

2. The Rigorous, Mathematical Proof

To formally prove Euler's formula, we can use the Maclaurin (Taylor) series expansions for $e^{x}$ , $cos x$ , and $sin x$ , which you may have encountered in your mathematics studies. We assume these series also hold for complex arguments.

The video below walks through this derivation step-by-step.

Complex Numbers - Exponential Form or Euler's Form | ExamSolutions Maths Revision Tutorials

For a rigorous derivation, this video from ExamSolutions, 'Complex Numbers - Exponential Form or Euler's Form', shows how Euler's formula arises naturally from the series expansions of the exponential, cosine, and sine functions.

Watch from timestamp 02:56 to 09:19. Follow along as the series for e^(iθ) is separated into its real and imaginary parts, which then perfectly match the series for cos(θ) and sin(θ).

By substituting Euler's formula into the polar form, we arrive at the elegant and powerful exponential form of a complex number:

z = r e^{i θ}

Here is a helpful image summarizing the key aspects of Euler's formula.

Euler's Formula and Complex Numbers — A summary of Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, its properties, and its geometric interpretation on the unit circle in the complex plane.

Manipulating Complex Numbers in Exponential Form

The true power of the exponential form shines when performing multiplication, division, and finding powers or roots. The rules are identical to those for real exponents.

Let $z_{1} = r_{1} e^{i θ_{1}}$ and $z_{2} = r_{2} e^{i θ_{2}}$ .

Multiplication:
$z_{1} z_{2} = (r_{1} e^{i θ_{1}}) (r_{2} e^{i θ_{2}}) = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$
(To multiply, multiply the moduli and add the arguments.)
Division:
$\frac{z _{1}}{z _{2}} = \frac{r _{1} e ^{i θ_{1}}}{r _{2} e ^{i θ_{2}}} = \frac{r _{1}}{r _{2}} e^{i (θ_{1} - θ_{2})}$
(To divide, divide the moduli and subtract the arguments.)
Powers (De Moivre's Theorem):
$z^{n} = (r e^{i θ})^{n} = r^{n} e^{in θ}$
(This is De Moivre's theorem: $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$ .)
Complex Conjugate:
$z ˉ = r e^{i θ} = r e^{- i θ}$
(The conjugate has the same modulus but the negative of the argument.)

Example: Converting and Calculating

Let's calculate $(1 + i)^{8}$ .

Convert to exponential form:
First, find $r$ and $θ$ for $z = 1 + i$ .
- $r = ∣ z ∣ = 1^{2} + 1^{2} = 2$ .
- $θ = arctan (y / x) = arctan (1/1) = π /4$ (since it's in the first quadrant).
- So, $1 + i = 2 e^{iπ /4}$ .
Calculate the power:
$(1 + i)^{8} = (2 e^{iπ /4})^{8} = (2)^{8} e^{i (8 \cdot π /4)} = 16 e^{i 2 π}$
Convert back to rectangular form:
$16 e^{i 2 π} = 16 (cos (2 π) + i sin (2 π)) = 16 (1 + i \cdot 0) = 16$

This is much faster than multiplying $(1 + i)$ by itself eight times!

Test your understanding!

Use the exponential form to simplify the expression $z = \frac{1 + i 3}{1 - i}$ . Express your final answer in both exponential and rectangular forms.

Show answer

First, convert the numerator and denominator to exponential form.

Numerator: $z_{1} = 1 + i 3$

$r_{1} = 1^{2} + (3)^{2} = 1 + 3 = 2$
$θ_{1} = arctan (3 /1) = π /3$ (first quadrant)
So, $z_{1} = 2 e^{iπ /3}$

Denominator: $z_{2} = 1 - i$

$r_{2} = 1^{2} + (- 1)^{2} = 2$
$θ_{2} = arctan (- 1/1) = - π /4$ (fourth quadrant)
So, $z_{2} = 2 e^{- iπ /4}$

Now, perform the division:
$z = \frac{z _{1}}{z _{2}} = \frac{2 e ^{iπ /3}}{2 e ^{- iπ /4}} = \frac{2}{2} e^{i (π /3 - (- π /4))} = 2 e^{i (π /3 + π /4)} = 2 e^{i (7 π /12)}$
This is the answer in exponential form.

To get the rectangular form, we use Euler's formula:
$z = 2 (cos (\frac{7 π}{12}) + i sin (\frac{7 π}{12}))$
Using a calculator, $cos (7 π /12) \approx - 0.2588$ and $sin (7 π /12) \approx 0.9659$ .
$z \approx 2 (- 0.2588 + 0.9659 i) \approx - 0.366 + 1.366 i$
This is the answer in rectangular form.

Conclusion

In this lesson, we have laid the groundwork for the mathematical language of quantum mechanics. You've seen that complex numbers are more than just an algebraic curiosity and can be represented geometrically and exponentially, which greatly simplifies complex calculations.

Here are the key takeaways:

Complex numbers can be represented in rectangular $(x + i y)$ and exponential $(r e^{i θ})$ forms.
Euler's formula, $e^{i θ} = cos θ + i sin θ$ , is the bridge between these two representations.
The exponential form makes multiplication, division, powers, and roots of complex numbers intuitive and elegant: you just use the standard rules of exponents.

Mastering these manipulations is crucial. In our next lesson, we will tackle another essential mathematical tool: Gaussian integrals. These integrals appear frequently in quantum mechanics, particularly when we need to ensure that the total probability of finding a particle is 1—a process called normalization. You will find that complex numbers can sometimes make these tricky integrals easier to solve.

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