Max Born's statistical interpretation of the wave function How did Max Born derive the probability of finding a particle between $x$ and  $x+dx$ at instant $t$?:
$$ \left |\psi(x,t)\right|^2dx$$
Was this result mathematically derived?  Or is it just a postulate, like the Schrödinger equation itself?
 A: This statement is equivalent to the statement that $|\psi(x,t)|^2$ is the probability density function for the particle, because the definition of the probability density $p(x)$ is such that $p(x)dx$ is the probability of an event occurring between $x$ and $x+dx$.
The statement that $|\psi(x,t)|^2$ is the probability density function is known as the Born rule, and in most mainstream quantum mechanics interpretations, it's considered to be a fundamental postulate. Several lesser-known interpretations claim to be able to derive it or otherwise intuit it using statistical arguments (for example, quantum Bayesianism considers the Born rule to be an extension of the Law of Total Probability, and the hidden-measurements interpretation and pilot-wave theory both claim to be able to derive the Born rule from other axioms), but these use a different set of axioms than mainstream quantum-mechanics.
A: Wave Mechanics was heavily influenced by classical electromagnetic waves, and in classical EM, the intensity of a wave is proportional to the square of the field's strength.  Luckily, this turned out to be true in QM as well.
