PX262 - A7 - the uncertainty principle

PX156 - C6 - the uncertainty principle

• wave-particle duality limits precision on how well we can measure the momentum and the position of a particle
• when measuring the position of a particle with a microscope, by illuminating particles with light of wavelength, $\lambda$, and a microscope with an aperture, $\alpha :$
  • in optics, the uncertainty in the position:$$\begin{array}{}\text{(1)}& \mathrm{\Delta }x=\frac{\lambda }{\sin \alpha }\end{array}$$
  • as photons behave as particles, they can affect the momentum:$$\begin{array}{}\text{(2)}& \mathrm{\Delta }p\simeq p\sin \alpha =\frac{2\pi \hslash \sin \alpha }{\lambda }\end{array}$$
• combining $(1)$ and $(2):$$$\mathrm{\Delta }x\mathrm{\Delta }p=2\pi \hslash$$
• there is a fundamental limit in quantum mechanics when trying to measure the position and the momentum of a particle simultaneously, called the heisenberg's uncertainty principle:$$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{1}{2}\hslash$$