*Srinivisa:* What is the math behind the statistical interpretation of Quantum Mechanics?

Physics is all about motion, start with a system $\rm S$ composed of a particle of mass $m$, moving along an axis, let it be subject to a known force, put some physical laws out there, mix well and BAM: $x(t)$ will determine for you the position of the particle at any time! Once you know that, you can find its velocity, its momentum... and a lot of other useful stuff. Of course, this is just under the framework of

*classical* mechanics.

Quantum Mechanics is very different since we can't have any function that determines with absolute certainty the position of a particle at a given time, and thus we can't determine the velocity of the particle without being uncertain. In fact, by Heisenberg's uncertainty principle, the more you know about a particle's position the less you know about its velocity, and the opposite is true. So instead of the well-defined $x(t)$, Quantum Mechanics uses what we call a

**wavefunction**.

As the double-slit experiment shows, light can behave like a particle or like a wave. In fact, even electrons display this same behavior with that famous experiment, which turns out to be very useful, since it will help solving the following problem. If we think of electrons as individual particles orbiting the nucleus of an atom, like the planets in our solar system orbiting our sun, then we get to a serious problem. Indeed, to a very serious one. Since the negatively charged electron is attracted by the positively charged nucleus by the electromagnetic force, then the electron will be continuously accelerating, and would thus radiate away its energy and fall into the nucleus.

That's why quantum mechanics came and proposed that we shouldn't think of the electron not just as a particle, but also as a wave. And this wave is basically described by the wavefunction, just like a 'classical' particle is described by the $x(t)$ function. We get the wavefunction of a particle (denoted as $\psi(x,t)$ or as $\Psi(x,t)$) by solving Schrödinger's equation: $$\underset{\textit{The time dependent Schrödinger equation.}}{\boxed{\displaystyle\,\,

i\hbar\dfrac{\partial \Psi}{\partial t} =

-\dfrac{\hbar^2}{2m}\dfrac{\partial^2\Psi}{\partial x^2}+\mathrm{V}\Psi.\,\,}}$$
The wavefunction is mathematically expressed as: $$\Psi(x,t)=\mathrm{A}e^{\displaystyle i(kx-\omega t)}$$ where $\mathrm{A}$ is the amplitude of the wave, $e$ is a constant which is approximately equal to $2.71$, $i$ is the square root of minus $1$, $k$ is the momentum divided by $\hbar$ which is the same as $h/2\pi$ where $h$ is Planck's constant, and finally $\omega$ denotes the frequency of the wave times $2\pi$.

But isn't a particle, by its nature, located at a single and unique point, whereas a wave is spread out in space? How can we make sense of such object? It's in an attempt to answer those questions that the Born interpretation (or the statistical interpretation) of the wave function was born. The latter propose that $|\Psi(x,t)|^2$ tell us the probability of finding the particle at point $x$, at time $t$. More precisely: $$\int_a^b |\Psi(x,t)|^2\,\mathrm dx=\left\{

\begin{matrix}

\text{probability of finding the particle} \\

\text{between $a$ and $b$, at time $t$.}\\

\end{matrix}

\right\}.$$
Note:

* The wavefunction itself is complex, but *$|\Psi|^2$

* is real and nonnegative. *
Here is a plot of wavefunctions of identical particles, have fun!

Best wishes, $\mathcal H$akim.