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Professor Justin Lorieau

2. Complex Numbers & Phase

Why You Can't Master Physics Without Complex Numbers
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Why You Can't Master Physics Without Complex Numbers

⏱ 14:18YouTube 4K / 1080p✏️ Practice Problems ↗

1. The Geometry of Imaginary Numbers

Section titled “1. The Geometry of Imaginary Numbers”

Calling complex numbers “imaginary” is an unfortunate historical accident. Positive and negative signs encode one-dimensional orientation (+1 is forward, -1 is reverse). Multiplying by -1 is a 180∘180^\circ flip along a single line.

Multiplying by ii represents a 90∘90^\circ counter-clockwise rotation in a two-dimensional number plane:

i×1=i(90∘)i \times 1 = i \quad (90^\circ) i×i=i2=−1(180∘)i \times i = i^2 = -1 \quad (180^\circ)

2. Euler’s Formula and Polar Form

Section titled “2. Euler’s Formula and Polar Form”

Any complex number z=x+iyz = x + iy can be decomposed into an absolute magnitude (radius rr) and a phase angle (θ\theta):

z=reiθ=r(cos⁡θ+isin⁡θ)z = r e^{i\theta} = r (\cos\theta + i\sin\theta)

Where:

  • Magnitude: r=∣z∣=x2+y2r = |z| = \sqrt{x^2 + y^2}
  • Phase: θ=arctan⁡(yx)\theta = \arctan\left(\frac{y}{x}\right)

In quantum mechanics, a wavefunction’s state is an element of a complex Hilbert space:

Ψ(x,t)=ψ(x)e−iEt/ℏ\Psi(x, t) = \psi(x) e^{-i E t / \hbar}

The complex phase factor e−iEt/ℏe^{-iEt/\hbar} drives interference phenomena while preserving probability normalization:

P(x)=∣Ψ(x,t)∣2=Ψ∗Ψ=∣ψ(x)∣2P(x) = |\Psi(x, t)|^2 = \Psi^* \Psi = |\psi(x)|^2

Practice Problems

Section titled “Practice Problems”

Problem 1: Euler Form Conversion

Convert the complex number z=1+i3z = 1 + i\sqrt{3} into exponential polar form reiθr e^{i\theta}.

💡 View Hint
Compute r using the Pythagorean theorem and determine theta from the ratio y/x.
📝 Worked Solution & Video Walkthrough
  1. Calculate magnitude:
r=12+(3)2=1+3=2r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2
  1. Calculate phase angle:
θ=arctan⁡(31)=π3(60∘)\theta = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3} \quad (60^\circ)

Therefore:

z=2eiπ/3z = 2 e^{i\pi/3}