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How Do Probability Amplitudes and Probabilities Differ?

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A beam splitter cube and circular optical filters on an optical bench, with amber reflections.

Here, probability amplitude means the value of the wavefunction in position space. Its squared magnitude gives a probability density; the probability in a region is calculated from that density. An amplitude can be a complex number; its magnitude is needed for this probability interpretation.

Contents
For this illustrative real amplitude, the squared magnitude turns both negative and positive parts into a nonnegative density.
  • Probability amplitude
  • Probability density

What you will learn

Before you start

It helps to read the articles on the wavefunction and on quantum probability. This article repeats the main idea where needed.

What is a probability amplitude?

A probability amplitude is the name for the wavefunction itself in Born's interpretation. According to the LibreTexts physical chemistry text, the Born interpretation, which is generally accepted today, calls the wavefunction the probability amplitude.

A wavefunction can in general be a complex function, so the physical meaning is not found in the function itself.

An amplitude can carry different signs and phases, so its value is not read directly as a probability.
  • Amplitude
  • Positive value
  • Negative value

Where does the probability come from?

The physical meaning lies in the product of the wavefunction with its complex conjugate, the absolute square, according to the same text. That square is called the probability density.

The MIT notes say the squared magnitude of the wave function can be interpreted as the probability density of finding a particle at a certain position and time. The probability of finding the particle between two points is worked out from this density.

The LibreTexts text adds that the probability density times a volume element gives the probability.

The squared magnitude gives probability density; the area under the density curve over an interval gives its probability.
  • Amplitude
  • Probability density
  • Interval area and probability

Why square it?

Because only the squared magnitude has physical meaning, while the wavefunction itself is complex. LibreTexts compares it to the way the intensity of a light wave depends on the square of the electric field: here, too, a square is used.

The University of Texas at Austin notes add that the probability interpretation is only tenable when the total probability is one, called the normalization condition.

The bars illustrate probabilities for different intervals, calculated from the same wavefunction.
  • Wavefunction
  • Probability pieces

Real-world example

Think of measuring the intensity of light. What we sense is brightness, not the value of the electric field at each instant. Brightness depends on the square of the electric field.

This is only a comparison to help you remember. A wavefunction is not a light wave.

Common misunderstandings

A probability amplitude is not a probability. The probability comes from the squared magnitude of the amplitude.

An amplitude is not necessarily positive. A wavefunction can be complex.

A probability density is not yet a probability. You multiply by a volume element or take the area over an interval.

Frequently asked questions

What is a probability amplitude?

The name for the wavefunction in Born's interpretation.

How is a probability worked out from an amplitude?

The squared magnitude of the amplitude gives the probability density, which times a volume element gives the probability.

Why not use the amplitude directly?

Because a wavefunction can in general be complex, so the physical meaning lies in its squared magnitude.

Summary

Terms in this article

English and Vietnamese terms, with the meanings used here. Hover, tap or Tab to a term with a dotted underline for an explanation as you read.

probability amplitudebiên độ xác suất
The name for the wavefunction in Born's interpretation.
probabilityxác suất
The value read from the squared magnitude of the amplitude, times a volume element.
probability densitymật độ xác suất
The absolute square of the wavefunction.
wavefunctionhàm sóng
The function describing a particle state, generally a complex function.
complex functionhàm phức
A function with complex values, so it has no direct physical meaning by itself.
normalizationchuẩn hóa
The condition that the total probability equals one.

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Sources

These sources are written for older readers. If they feel hard, read them with an adult.