Question

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`.

Answer 1

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.

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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.