The probability density function $|Ψ|²$

Question

Max Born stated that $|Ψ|²$ is the probability density of a particle, given its wave function to be $Ψ$. But why is this? Where does this come from?

Answer 1

As Farcher answered, it was a guess which seemed to explain many of the observations. However, there are different ways to motivate this reasoning.

Consider the Schrodinger equation of a particle: \begin{equation} i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi +V \end{equation} Multiplying this by the complex conjugate wavefunction $\psi^*$ (for more details look at [1]), we get the following equation: \begin{equation} \frac{\partial |\psi|^2}{\partial t} - i \frac{\hbar}{2m}\nabla \cdot(\psi^*\nabla\psi - \psi\nabla\psi^*) = 0 \tag{*} \end{equation} This looks a lot like continuity equation of the form \begin{equation} \frac{\partial \rho}{\partial t} + \nabla \cdot\vec{j} = 0 \end{equation} which usually signifies that the "charge" associated to $\rho$ is conserved. The equation * can be brought to this form which leads us to believe $|\psi|^2$ must be some kind of a probability density.

Another way to see this is to note that any wavefunction can be decomposed into energy eigenstates: \begin{equation} |\psi \rangle = \sum_n a_n |n \rangle \tag{**} \end{equation} where $a_n$ are the complex coefficients and $|n \rangle$ are the energy eigenstates. Note that because of equation *, $\langle \psi|\psi \rangle$ doesn't change in time so we could just set it to 1 for convenience. Then, the equation ** immediately implies the following \begin{equation} \sum_n |a_n|^2 = 1 \end{equation} which seems like the "probabilities add up to one" rule if we identify $p_n = |a_n|^2$ where $p_n$ is the probability of a particle being at state n.

Ref:

[1] https://www1.itp.tu-berlin.de/brandes/public_html/qm/umist_qm/node16.html

Answer 2

Born formulated his statistical/probabilistic interpretation of QM for two main reasons.

The first was the difficulty in maintaining de Broglie's point of view of wave-like particles. One of the observations made by Born in his very readable paper, where he proposed the statistical/probabilistic interpretation of the wavefunction, was that, for $N$ particles, they are waves in a $3N$-dimensional configuration space, not in the actual $3$-D space. The second was the analogy with statistical mechanics. Atomic-scale experiments and statistical mechanics share the problem of not knowing the starting configuration accurately. Then we can naturally introduce probabilities to cope with this limitation. Forces are not used for reproducing single trajectories but for determining the statistical distribution of measurement results.

Nowadays, we have direct evidence of the correctness of Born's interpretation when we see the outcome of single particle scattering experiments like in the famous Tonomura's one electron double slit experiment.