91Ó°ÊÓ

Mathematically, the uncertainty relation can be stated as $\mathit{Δ}\mathit{x}\mathit{Δ}\mathit{p}\mathbf{â\copyright ¾}\frac{\mathbf{h}}{\mathbf{2}}$.

Fourier established this theory, which connects the function spreading in space$\mathrm{Δ}x$ to the range of wavenumber$\mathrm{Δ}k$ , long before the Heisenberg uncertainty principle. The relation states that $\mathrm{Δ}x\mathrm{Δ}kâ\copyright ¾\frac{1}{2}$.

In other words, if we require a function with a very small spatial bandwidth, we must cover a wide variety of wavelengths or wavenumbers; $\left(\mathrm{λ}=\frac{2\mathrm{π}}{k}\right)$as a result, we must combine many sinusoidal functions.

If we only take note that the momentum is equal to , then this may be directly connected to the Heisenberg connection. Consequently, if we substitute that back in, we will reestablish our original relationship $\mathrm{Δ}x\mathrm{Δ}kâ\copyright ¾\frac{h}{2}$. So, the same conclusion is drawn: the more wavenumbers added together (greater uncertainty in momentum), the less uncertainty there is in location, and the opposite is also true.