If the uncertainty in the position is equal to the wavelength of an electron, how certain can we be about the velocity? Determine an expression for `v_x//Deltav_x`.

Well, referring to the Heisenberg Uncertainty Principle, there is one formulation of it that is fairly easy to use in calculations:

`bb(DeltaxDeltap_x >= ℏ)`,

or

`bb(DeltaxDeltap_x >= h/(2pi))`,

where `ℏ = h/(2pi)` is the reduced Planck's constant, and `h = 6.626xx10^(-34) "J"cdot"s"`. You may also see `ℏ/2`, or `h/(4pi)`, but that's beside the point. The point is, it's on the order of `ℏ`.

The de Broglie wavelength is:

`lambda = h/(mv)`

If the uncertainty in the position becomes numerically equal to `lambda`, then we can plug in `lambda = h/(mv_x) = Deltax` to get:

`cancel(h)/(mv_x)Deltap_x >= cancel(h)/(2pi)`

Since `p = mv`, `Deltap = mDeltav` for a particle experiencing little relativistic effects on its mass:

`1/(cancel(m)v_x)cancel(m)Deltav_x >= 1/(2pi)`

`(Deltav_x)/v_x >= 1/(2pi)`

Flipping both sides, we get:

`color(blue)(v_x/(Deltav_x) <= 2pi)`

Since `lambda` is very small (on the order of `nm` for electrons), we expect that the uncertainty in the velocity is very large so that the inequality `DeltaxDeltap_x >= h/(2pi)` is maintained.

That should make sense because if `Deltav_x` is very large, only then would `v_x/(Deltav_x) <= 2pi` hold true, since `v_x` for an electron is generally quite large as well.

EXAMPLE

For instance, the `1s` electron in hydrogen atom travels at around `1/137` times the speed of light, or `v_x = 2.998xx10^8 xx 1/137 = 2.188xx10^6 "m/s"`. That means:

`(2.188xx10^6)/(Deltav_x) <= 2pi`

`color(red)(Deltav_x >= 3.483 xx 10^5)` `color(red)("m/s")`

i.e. the uncertainty in the velocity of a `1s` electron is AT LEAST `~~15.92%` of the velocity of the electron (not at most... at least) when you are sure of the position of the electron to within `"nm"` of horizontal distance.

It physically means that if we were to try to predict its velocity, we are extremely unsure of which way it's going and at what actual velocity.

In real life, if this were to be the case, then if you shined a laser through a slit of a few `"nm"` of width, you would see a bunch of constructive and destructive interference on the far walls, indicating the high uncertainty in the velocity along the `x` axis:

Of course, the de Broglie relation is for electrons, as photons have no mass, but both behave as waves, and so, the slit experiment applies to both.