I was reading a bit about Quantum Mechanics, Schrodinger's equation and its probabilistic interpretation (found this very insightful intro here https://plus.maths.org/content/schrodinger-1), my background is in mathematics so I was interested to know how this interpretation came about from a more mathematical point of view:

Schrodinger's equation that describes the motion of a particle in three dimensions is:

$\frac{ih}{2\pi}\frac{\partial\Psi}{\partial t} = - \frac{h^2}{8\pi^2m} ( \frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} + \frac{\partial^2 \Psi}{\partial z^2} ) + V \Psi$,

where $V$ is the potential energy of the particle, $m$ is its mass and $h$ is Planck's constant. The solution to this equation is $\Psi(x,y,z,t)$. A time-independent version of this equation for a function $\psi(x,y,z)$ is:

$\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} + \frac{8\pi^2m}{h^2} (E-V)\psi = 0$,

where $E$ is the total energy of the particle. The solution to the full equation is

$\Psi = \psi e^{-(2\pi i E/h)t}$.

Max Born's interpretation of the wave function $\Psi$ is that $\mid \Psi(x,y,z,t) \mid^2$ is a probability density function that determines the position of the particle in $\mathbb{R}^3$ at time $t$, how did he arrive at this formulation? I was interested in the more measure-theoretic mathematics behind this, anyone know?


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