Why is probability density = |wavefunction|^2?

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SUMMARY

The probability density of a quantum particle is defined as the square of its wavefunction, represented mathematically as |wavefunction|^2. This relationship is a postulate known as the Born rule, which is motivated by analogies drawn from classical wave phenomena, such as the intensity of light in a double slit experiment being proportional to the square of the wave amplitude. The discussion highlights that while there is no formal derivation of this rule, it is supported by experimental observations where the wave nature of particles is described by solutions to the wave equation.

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DuckAmuck
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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.
 
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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).
 

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