I'm reading a quantum mechanics book, and it has the following equation:

$$ \Delta x \approx \frac{\lambda}{\sin\alpha} \sim \frac{h}{mc\sin\alpha} $$

What is the difference between $\approx$ and $\sim$?

  • $\begingroup$ What's the context? It looks like they're using de Broglie's relation $\lambda=h/p$ and then taking $p=mc$ (which could make sense under the right conditions). That kind of detail is important when interpreting notation in a physics text v. a math text. $\endgroup$ – Semiclassical Aug 29 '14 at 0:17
  • $\begingroup$ OK. After looking at it for awhile I know what they are doing... $\Delta x \approx \frac{\lambda}{\sin \alpha}$ is referring to the positional uncertainty of a particle measured by a microscope with angular aperture $2\alpha$. However, due to the Crompton effect, the wavelength will increase proportional to $\frac{h}{mc}$ when a photon hits a particle. $\endgroup$ – daviewales Aug 29 '14 at 2:57

$\approx$ is used as the mathematical "approximately" symbol $-$ this means $ \Delta x $ has approximately the same value as $ \frac {\lambda}{\sin \alpha} $. However, $\sim$ is used a proportionality symbol, so there is some factor tacked onto $\frac{h}{mc\sin \alpha}$, which may be $1000$ or $2$ or something else.

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