What Information Does a Wavefunction Represent?

• Particle
• Curve of likelihood
• Dice

What you will learn

• State what a wavefunction tells you about a particle.
• Explain why the total probability must come to one.
• See that a wavefunction also gives probabilities for quantities other than position.

Before you start

No prior knowledge is needed. The article uses only the main idea of probability and no calculation.

What does a wavefunction tell you about a particle's position?

It tells you the probability of finding the particle somewhere. According to the University of Texas at Austin notes, after many false starts, physicists in the early twentieth century concluded that the only physical interpretation of a wavefunction consistent with experimental observations is probabilistic in nature.

Concretely, the squared magnitude of the wavefunction can be interpreted as the probability density of finding a particle at a certain position and time, according to the LibreTexts chapter on the meaning of the wavefunction. The probability of finding the particle between two points is the area under the probability-density curve between them.

• Curve
• Shaded area
• Two points

Why must the total probability be one?

Because if the particle exists, it is certain to be found somewhere. The University of Texas at Austin notes call this the normalization condition and say the probability interpretation is only tenable when it holds. The MIT notes put it briefly: the probability must sum to one.

For the single-particle position model considered here, the wavefunction is normalized. When the whole area under the probability-density curve is one, each piece of area can be read as a probability.

• Curve
• Whole area
• Normalized mark

Which quantities does a wavefunction describe?

Not only position. The Stanford Encyclopedia of Philosophy says that, given the state of the system, one can derive probability distributions for all of its physical quantities, such as position, momentum or energy.

So the wavefunction is a description of the state, and from it you read the probabilities for each quantity you care about.

• State
• First distribution
• Second distribution
• Third distribution

Real-world example

Imagine looking for a particle in non-overlapping regions. The area under the density curve in each region gives the probability of finding the particle there. Adding all regions that cover the whole space gives a total probability of one. Rain probabilities in different areas do not illustrate this sum, because rain can occur in several places at once.

This is only a comparison to help you remember.

Common misunderstandings

A wavefunction is not the path of the particle. It gives probability, not a trajectory.

A large wavefunction magnitude at a location corresponds to a large probability density there. The probability of finding the particle in a region depends on the area under the density curve over that region.

A wavefunction does not speak only about position. From the state one can derive distributions for many quantities.

Frequently asked questions

What information does a wavefunction represent?

Information about probability: its square is the probability density of finding the particle at a position and time.

What is normalization?

The condition that the total probability is one, because the particle is certain to be found somewhere.

Does a wavefunction tell you about energy?

Given the state, one can derive probability distributions for many quantities, energy among them.

Summary

• A wavefunction gives probability; its square is the probability density of finding the particle.
• The total probability must be one, called the normalization condition.
• From the state one can derive probabilities for many quantities, not only position.