PX156 - C6 - the uncertainty principle

• $1927$, werner heisenberg: if the x-component the momentum of a particle is measured with an uncertainty of $\mathrm{\Delta }{p}_x$, then its x-position cannot, at the same time, be measured more accurately than $\mathrm{\Delta }x=\frac{\hslash }{2\mathrm{\Delta }{p}_x}:$

- with uncertainties defined in RMS sense, ie: $\mathrm{\Delta }x={\sqrt{⟨x^2⟩-⟨x⟩}}^2$, where $⟨A⟩=$ average value of $A$

• alternatively:

• from wave packets and top-hat distribution:

  • if $a(k)$ is sharply peaked, then the wave pulse is relatively broader
  • the width of $a(k)$ curve is inversely proportional to the probability distribution function, $\psi$
  • for a narrow range of $k$, corresponding to a narrow range of momentum, $p$, and a small uncertainty, $\mathrm{\Delta }p$, is only possible for a large number of $x$, and therefore larger uncertainty, $\mathrm{\Delta }x$

• eg: consider a ball of mass, $\sim 200g$. measure its position, $\sim \pm 10\mu m$
  $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$
  $\mathrm{\Delta }p\sim \frac{\hslash }{\mathrm{\Delta }x}\sim \frac{{10}^{-34}}{{10}^{-5}}\sim {10}^{-29}kgm{s}^{-2}$
  $\mathrm{\Delta }v=\frac{\mathrm{\Delta }p}{m}\sim \frac{{10}^{-25}}{0.2}\sim 5\times {10}^{-29}m{s}^{-1}$

• eg: consider a hydrogen atom

  • using bohr's model: $r_n=n^2{a}_0$
  • taking $n=1:$
  $$a_0=\frac{{ϵ}_0{h}^2}{\pi m{e}^2}\approx 0.54\AA$$
  • electron is localized at the scale of $2a_0:$
  $$\mathrm{\Delta }p\ge \frac{\hslash }{2a_0}\sim \frac{\hslash }{a_0}$$
  • angular momentum:

• eg: showing that it can be shown that an electron is bound in an atom
  • in a 1D region $\sim 0.5\AA$
  • $\mathrm{\Delta }p\sim p\sim \frac{\hslash }{a_0}\sim \frac{{10}^{-34}}{{10}^{-10}}\sim {10}^{-24}kgm{s}^{-1}$
  $$KE=\frac{p^2}{2m}\sim \frac{({10}^{-24}{)}^2}{2\times 9.11\times {10}^{-31}}\sim 5\times {10}^{-19}J\approx 3eV$$
  • from here: rydberg constant $=13.6eV$