Probability Amplitudes

In Quantum Mechanics, we understand this wave-particle duality using (complex) probability amplitudes which satisfy a wave equation.

\begin{displaymath}\bgroup\color{black}\psi(\vec{x},t)=e^{i(\vec{k}\cdot\vec{x}-\omega t)}=e^{i(\vec{p}\cdot\vec{x}-Et)/\hbar}\egroup\end{displaymath}

The probability to find a particle at a position \bgroup\color{black}$\vec{x}$\egroup at some time \bgroup\color{black}$t$\egroup is the absolute square of the probability amplitude \bgroup\color{black}$\psi(\vec{x},t)$\egroup.


To compute the probability to find an electron at our thought experiment detector, we add the probability amplitude to get to the detector through slit 1 to the amplitude to get to the detector through slit 2 and take the absolute square.


Quantum Mechanics completely changes our view of the world. Instead of a deterministic world, we now have only probabilities. We cannot even measure both the position and momentum of a particle (accurately) at the same time. Quantum Mechanics will require us to use the mathematics of operators, Fourier Transforms, vector spaces, and much more.

Jim Branson 2013-04-22