𝚿 Complex plane wave

About

🎯 Build intuition for complex waves, phase propagation, and the role of $k$ and $\omega$
🧠 3D visualization found in Visualizing Quantum Mechanics with Python
🧠 2D visualization based on SinusoidalWave.html by Daniel V. Schroeder
🐍 A 3D VPython demo is available as well, see plane_wave.py
👉 More physics software by Daniel V. Schroeder can be found here
👉 Wave function of a free particle:

$$\psi(x, t) = Ae^{i(k x - \omega t)}$$

Live demo

About the 3D visualization

Each arrow represents the complex value of the wave function at a fixed position $x$. The arrow rotates in the complex plane as time evolves:

• z-direction: real part of $\psi$
• y-direction: imaginary part of $\psi$

The length of the arrows is constant, showing that the magnitude $|\psi|$ of a plane wave does not change in space or time. The color encodes the phase, making the spatial and temporal phase structure visible.

About the 2D visualization

This is an animated visualization of the behavior of a pure sinusoidal wavefunction in one dimension, representing a free quantum particle with a precise momentum that is inversely proportional to the wavelength. There is no potential energy and the particle is nonrelativistic, so the phase velocity is directly proportional to the momentum.

The real part is shown in orange, the imaginary part in blue. Alternatively, the probability density and phase are drawn, with the phase represented by hues going from

• red (pure real and positive) to
• light green (pure imaginary and positive) to
• cyan (pure real and negative) to
• purple (pure imaginary and negative) and finally back to red.

For a pure sinusoidal wavefunction, the probability density is the same everywhere. — Paraphrased from instructions at SinusoidalWave.html

Guided Exploration

Use the controls to explore the properties of the plane wave.

---

Exercise 1 — Spatial phase

1. Set $\omega = 0$.
2. Change the wave number $k$.

Questions

• What happens to the phase difference between neighboring arrows?
• How does this relate to the wavelength $\lambda = 2\pi/k$?

---

Exercise 2 — Temporal evolution

1. Fix $k$.
2. Increase $\omega$.

Questions

• What changes visually?
• What does $\omega$ control physically?
• Does the probability density change?

---

Exercise 3 — Direction of propagation

1. Set $k > 0$.
2. Observe how the phase moves in space.
3. Repeat for $k < 0$.

Questions

• In which direction does the phase propagate?
• How is this related to the sign of the momentum?

---

Exercise 4 — Real vs imaginary parts

Focus on a single arrow and track its motion.

Questions

• How are the real and imaginary parts related?
• Why can neither part alone represent the full wave?

---

Exercise 5 — Interpretation

A plane wave is not normalizable.

Questions

• What does that mean physically?
• Why are plane waves still useful in quantum mechanics?
• How might you build a localized wave packet from plane waves?

---

Optional challenge

Predict what would happen if **two plane waves with slightly different $k$ ** were added together. What new structure would you expect to see?

Complex Plane Waves in Quantum Mechanics

In quantum mechanics, the state of a free particle with definite momentum is described by a plane wave

$$\psi(x,t) = A e^{i(kx - \omega t)}.$$

This visualization shows the wave function as a geometric object rather than a real-valued curve.

At each position $x$:

• the arrow represents the complex value of $\psi(x,t)$
• the arrow rotates in the complex plane with angular frequency $\omega$
• the spatial phase advance is controlled by the wave number $k$

The real and imaginary parts are shown along orthogonal axes. The constant arrow length illustrates that

$$|\psi(x,t)|^2 = |A|^2$$

is uniform in space and time.

This emphasizes an important quantum-mechanical point:

A plane wave does not represent a localized particle, but a state with definite momentum and completely delocalized position.

The animation separates:

• phase evolution (rotation of arrows)
• from probability density (constant magnitude)

which is often obscured in standard textbook plots.

Concise derivation of the Schrödinger equation

According to De Broglie we have:

$$p = \dfrac{h}{\lambda} = \dfrac{h}{2\pi} \dfrac{2\pi}{\lambda} = \hbar k \Rightarrow \hbar k = \hbar \dfrac{\partial}{\partial x} \psi(x,t) = p \psi(x, t) \Rightarrow p = \hbar \dfrac{\partial}{\partial x}$$

The Kinetic energy can be expressed as:

$$K = \dfrac{p^2}{2m} = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} \psi(x,t)$$

The total energy is given by the Planck-Einstein relation:

$$E = hf = \dfrac{h}{2\pi}\dfrac{2\pi}{T} = \hbar \omega \Rightarrow -i\hbar\dfrac{\partial}{\partial t} \psi(x,t) = E \psi(x,t) \Rightarrow E = -i\hbar\dfrac{\partial}{\partial t}$$

From this we arrive at the Schrödinger equation:

$$(KE + PE)\Psi(x,,t) = E\Psi(x,t) = -i\hbar \dfrac{\partial}{\partial t}\Psi(x, t) = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} \Psi(x,t) + V(x)\Psi(x,t)$$