# Why is probability density = |wavefunction|^2?

• DuckAmuck
In summary, the "rule" that the wavefunction of a particle times its complex conjugate is a probability density is essentially a postulate. It has been motivated by analogy, specifically in the case of a double slit experiment with light. However, there is no derivation of the rule yet and it is not clear why the probability density is not equal to |wavefunction|^4 or some other even number. Further analysis and research may provide a better understanding of the Born rule.
DuckAmuck
I have looked around for an answer to this. People just call it a "rule". So is it just assumed that the wavefunction of a particle times its complex conjugate is a probability density, or is there some way to show this? For instance, why isn't probability density equal to |wavefunction|^4 or some other even number?
Thanks.

There is no derivation of the rule, at least not yet. It is essentially a postulate, though I've seen it motivated by analogy. In particular, in a double slit experiment with light, the intensity of the interference pattern is proportional to the wave amplitude squared. So, by analogy, in single photon/single electron/etc experiments where the peaks build up one particle at a time, the peaks are in some sense a probability density. Since the "wave" nature of the particle is guided by the solutions to the wave equation, one might interpret the square amplitude of the wave function as the probability density.

There may be more careful analyses that motivate the Born rule better, but I'm not aware of them. See the wikipedia page for more information (and links to more).

## 1. What is probability density?

Probability density is a measure of the likelihood that a certain event will occur. In the context of quantum mechanics, it refers to the probability of finding a particle in a particular location within a given space.

## 2. What is a wavefunction?

A wavefunction is a mathematical representation of the state of a quantum system. It describes the probability amplitude of a particle at any given point in space and time.

## 3. Why is probability density equal to the absolute value of the wavefunction squared?

This is a fundamental principle of quantum mechanics known as the Born rule. It states that the probability of finding a particle at a specific location is equal to the square of the magnitude of the wavefunction at that point.

## 4. How does the wavefunction relate to the physical properties of a particle?

The wavefunction contains information about the position, momentum, and other physical properties of a particle. It can be used to calculate the probability of a particle having a certain position or momentum at a given time.

## 5. Can the wavefunction change over time?

Yes, according to the Schrödinger equation, the wavefunction can change over time as the quantum system evolves. This allows us to make predictions about the behavior of particles in quantum systems.

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