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?
[Physics] Max Born’s statistical interpretation of the wave function
probabilityquantum mechanicswavefunction
Related Solutions
In position-space (that is, when your functions are functions of x), the function $\int|\Psi|^2$ gives the probability of finding the particle in a given range. The expectation value of x is where you'd expect to find the particle. It is often essentially the weighted average of all the positions where the probability density, $|\Psi|^2$, is the weighting function (that's not exactly what it is, but it's a useful analogy). Similarly, you can find the expectation value for any measurable quantity. In this space, the difference between the two is that the expectation value is a number that represents the expected average position of the particle over many measurements whereas the probability is a number that gives you the probability for finding the particle within the limits of integration.
However, you can use any different basis. For example, you could choose momentum-space, $\left|\Psi\right>$ is $\Psi(p)$ (quantum physicists please don't kill me for that affront to notation). In momentum space, the integral $\int|\Psi|^2$ is now the probability of the particle having a given range of momenta. However, the expectation value of x is still the average measurement of x. What, you ask, is the point? The expectation value is a number that can be found in any basis that represents the "on-average" value of a measurement. The probability found by $\int|\Psi|^2$ is the probability that a particle will be found existing within a specified range of values for the basis you are using.
$\int_{x_1}^{x_2}|\Psi|^2dx$ is "there is #% chance that the particle will be found between $x_1$ and $x_2$"
$\left<\Psi\right|x\left|\Psi\right>$ is "the expected average position of the particle over a large number of sample measurements is at $x$=#"
$|\Psi|^2(x)$ is a function "the probability per unit length of finding the particle at this position is #%"
For a quantum system with one degree of freedom on the closed interval $I$, the Hilbert space is $L^2(I)$. In this case the $I$ is the range for the spatial coordinate $x$, so that normalisation applies with respect to the Lebesgue measure on $I$. Now suppose that you have a dynamics described by the Hamiltonian $H$ on such Hilbert space, and that $\psi$ is an eigenstate of $H$ with eigenvalue $E$. The time evolution of $\psi$ is
$$\psi\mapsto e^{iEt}\psi.$$
If we naively try to integrate this function, we get to
$$\int_{\mathbb R}|e^{iEt}\psi(x)|^2\text d t = \int_{\mathbb R}|\psi(x)|^2\text d t = |\psi(x)|^2\int_{\mathbb R}\text dt,$$
which is infinite for every $x\in I$ for which $\psi(x)\neq 0$ or zero otherwise. We then have a problem trying to attach the meaning of probability to such integral. You can interpret this result as saying that a particle will pass through $x$ infinitely often provided $\psi(x)\neq 0$ if you wait indefinitely, but this information is already deductible from $\psi$ itself, there is no need to do such an integral.
Best Answer
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.