What Does the Wave Function Represent?

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The wave function allows us to calculate the probability of finding a particle.
But what does the wave function itself represent physically?
Let us explore this question.

Visualization of a wave function

The Shape of a Wave Function

Imagine a single electron moving through space. Did you picture it as a tiny sphere, perhaps about 1 cm across? In quantum mechanics, however, the state of an electron is described by a wave function, like the one shown above.

The illustration shows a one-dimensional wave function. The horizontal axis represents the direction of motion, the vertical axis represents the imaginary part, and the depth axis represents the real part. Because wave functions are complex-valued, we need three dimensions to visualize even this simple example. What may look mysterious at first is, in this picture, simply a helix in three-dimensional space.

The distance from the horizontal axis gives the magnitude, or absolute value, of the wave function. The angle around that axis gives its phase.

Some readers may not find this image mysterious at all. Many others, however, may think:

“This does not look like an electron.”

I think that reaction is completely natural.

The Copenhagen Interpretation

Experiments show that the radius of an electron is smaller than 1 femtometer (10⁻¹⁵ m), while the radius of an atomic nucleus is also on the order of 1 fm. Yet the electron’s wave function can extend across an entire atom, roughly 53,000 fm in radius. If the nucleus were scaled up to a sphere 1 mm across, the electron’s wave function would spread over a region about 53 m wide. When we measure the electron’s position, the wave function appears to “collapse,” and the particle is found in one particular place.

This apparent collapse raises a difficult question: does the wave function really disappear when we observe the particle? The Copenhagen interpretation addresses this mystery with two central ideas:

• Before measurement, a physical system generally does not have definite values for all observable quantities.
• Quantum mechanics predicts only the probabilities of different measurement outcomes.

Although this view is widely accepted, it can still feel unsatisfying. It is a little like saying, “The app was running normally, but the moment I opened it, it vanished.”

What Is the Physical Meaning of the Wave Function?

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

• The wave function extends to infinity. Why, then, does it seem to “cut off” when a measurement is made?
• Multiplying the wave function by a constant leaves all physical predictions unchanged. What, then, does its magnitude correspond to physically?
• Changing the overall phase of the wave function also leaves all physical predictions unchanged. What is the physical meaning of that phase?

We live in three-dimensional space, and as Einstein showed, space can be curved by mass and energy. Could the wave function itself be related to another kind of “space” that can be shaped or curved? Perhaps its magnitude corresponds to the “size” of such a space, and its phase to an “angle” within it.

• Spin in Quantum Mechanics: Is It Really a Rotation? (2014/9/1)
• Animation of a Rotating Electron Wave Function (2013/6/30)
• The Many-Worlds Interpretation of Quantum Mechanics (2013/6/22)

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