Question

What will be the uncertainty in momentum if we know the position certainly?

Answer 1

It would be infinite... which is clearly greater than `h//4pi`.

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Heisenberg's Uncertainty Principle commonly applies to electrons, such that...

`DeltaxDeltap >= h//4pi`

where `Deltaq` is the uncertainty in `q`, `x` is position, and `p` is momentum. `h` is Planck's constant.

The inequality requires that if one uncertainty gets too low, the other uncertainty must increase until the inequality is satisfied.

Thus, if `Deltax -> 0`, the only way for the product to be greater than `h//4pi` is if `color(blue)(Deltap -> oo)`. What it physically means to have infinite momentum uncertainty is that the electron motion is spread out infinitely over all coordinates.

In fact, although we never have absolute certainty about an electron's position, we usually are more certain about the position of the electron rather than its momentum. The physical proof of that is:

As the wave function (the solution to the Schrodinger equation) must be bounded, that generates what we regard as orbital probability densities, seen above.