The Mystery of Wave Functions

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The wave function describes the probability of finding a particle.
What does a wave function represent physically?
Let’s explore this question.

Wave function visualization

The Shape of a Wave Function

For example, imagine a single electron in flight. Did you picture a sphere about 1 cm in diameter? We actually describe the electron’s state with a wave function, as shown above.

The illustration represents a one-dimensional wave function. The horizontal axis shows the direction of motion, the vertical axis represents the imaginary component, and the depth axis corresponds to the real component. Since wave functions are complex-valued, we need three dimensions to visualize them. What may seem mysterious is simply a helix in three-dimensional space.

The radial distance from the horizontal axis is the magnitude (absolute value) of the wave function, and the angle around that axis is its phase.

Some readers may not find the image mysterious, but many likely think:

“This picture doesn’t look like an electron at all.”

I believe that’s a completely natural reaction.

Copenhagen Interpretation

Experiments show an electron’s radius is smaller than 1 femtometer (10⁻¹⁵ m), while an atomic nucleus is about 1 fm. Yet the electron’s wave function extends across the atom—roughly 53,000 fm in radius. If you scaled the nucleus to a 1 mm sphere, the electron would spread over a 53 m region. When we measure the electron’s position, the wave function appears to “collapse,” localizing the particle.

This collapse raises a puzzling question: does the wave function truly vanish upon observation? The Copenhagen interpretation addresses this mystery with two key ideas:

• Physical systems generally do not have definite properties before measurement.
• Quantum mechanics can only predict the probabilities of different measurement outcomes.

While this view is widely accepted, it can feel unsatisfying—like saying, “The app was running fine, but the moment I opened it, it vanished!”

What Does a Wave Function Physically Represent?

What is the true physical significance of the wave function? What does it actually describe?

• The wave function extends to infinity. Why then does it appear to “cut off” upon measurement?
• Multiplying the wave function by a constant leaves all predictions unchanged. What does its magnitude correspond to physically?
• Rotating the wave function’s phase by a fixed angle also leaves predictions unchanged. What is the physical interpretation of the phase?

We inhabit a three-dimensional space, which—as Einstein taught us—is a fabric that can be curved by mass and energy. Could the wave function itself be a kind of “space” we curve? Its magnitude might correspond to the “size” of this space, and its phase to the “angle” within it.

• Spin in Quantum Mechanics: Explaining the mysterious properties of spin (2014/9/1)
• Electron Wave Functions in Quantum Mechanics: An Electron’s Rotational Animation (2013/6/30)
• Many-Worlds Interpretation: Explaining the idea of parallel universes (2013/6/22)

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