Where does the Born rule come from?

Question

As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives and imaginary numbers, but why is it the square, not the fourth or some higher power?

Answer 1

Suppose you want to describe the quantum mechanical behaviour of a system, building from scratch the wave equation it should satisfy. Consider first the diffraction pattern obtained with a double slit by a monochromatic light beam and compare it to the one by a monoenergetic beam of electrons.

In optics, the total amplitude $\Phi$ for two coherent incident light beams on a plane is the sum of the individual amplitudes, $\Phi=\Phi_1+\Phi_2=A_1e^{i\theta_1}+A_1e^{i\theta_2}$ and the intensity $I$ of the beam will be proportional to $|\Phi|^2$, $$I\sim|\Phi|^2=A_1^2+A_2^2+2A_1A_2\cos(\theta_1-\theta_2)$$ which is real and positive, and in fact by the definition of $I$ it is proportional to the number of photons in each point of the screen.

The pattern with the beam of electrons is entirely similar, so you can define a complex amplitude $\psi$ with the properties

1. It may satisfy a wave equation.
2. The density of electrons $\rho(x)$ is proportional to $|\psi|^2=\psi^*\psi$ in each point.

This way you guarantee that the density of particles will be positive and that it may manifest interference by means of superposition of amplitudes. Now, let's denote $A$ the factor of proportionality in property 2, then the total number of particles $N$ is given by $$N=\int\rho\,dx=A\int|\psi|^2dx\;\Longrightarrow\;\int|\psi|^2dx=\frac{N}{A}$$ Now, the number $N$ in general is big, unknown and knowing it is irrelevant, also as may be seen, the wave equation is homogeneous, so that $\psi$ is determined up to an arbitrary constant. This way, it is accustomed to take $A=N$, i.e. to take $|\psi|$ as a normalized function, $$\int\psi^*\psi\,dx=1$$ So you have that $\rho=N|\psi|^2$, then you can define $$\tilde{\rho}\equiv\frac{\rho}{N}$$ as a relative density of particles, that tells you what fraction of the total of particles is contained in the element $dx$, from here then $$\int\tilde{\rho}\,dx=1$$ So here it is: suppose you do the experiment with just one electron. Then $\tilde{\rho}\,dx$ may be interpreted as the probability that the electron is contained in the element $dx$ and $\int\tilde{\rho}\,dx=1$ tells you that the particle is somewhere in space with all confidence.

This is why in general, $\tilde{\rho}=\psi^*\psi$ may be interpreted as a probability density for localization of particles that hence implies conservation of probability.

Answer 2

For an electromagnetic wave, the Energy is proportional to the Electric/Magnetic Field (i.e. the wave) squared. This is a classical result which can be derived from the Maxwell equations.

When photons were discovered, the intensity of photons, or number of photons arriving at a certain place (for example, on a screen behind a double slit) was seen to be proportional to this squared field. However, it was now given a probabilistic interpretation: The intensity of the light is the probability of a photon impinging on that location.

The extension to "electron waves" is of course a wild guess that then proved to be true.

Answer 3

The "square" follows from the Schroedinger equation in a scattering problem setup. For a given flux of incident particles $J$ it gives the number of scattered particles per second per unit of solid angle $\frac{d^2N}{dtd\Omega}\propto J$, and thus for one particle it is a probability per second per unit of solid angle.