Question #277d6

Heisenberg Uncertainty Principle states that the position and the velocity of a particle cannot be measured with accuracy and this accuracy decreases as the size of the particle decreases

A formal definition of the Heisenberg Uncertainty Principle states that the product of the position and momentum uncertainties `sigma_vecx` and `sigma_(vecp_x)` is no less than `ℏ//2`, where `h` is Planck's constant, `6.626xx10^(-34) "J"cdot"s"`, and `ℏ = h//2pi` is the reduced Planck's constant.

That is,

`\mathbf(sigma_(vecx)sigma_(vecp_x) >= ℏ//2)`

(By the way, this happens to hold for any direction, not just the `x` direction, so this extends to our three dimensions.)

Mathematically, this is expressed using the following inequality

`color(blue)(ul(color(black)(Deltax * Deltap >= h/(4pi))))`

Here

• `Deltax` is the uncertainty in position
• `Deltap` is the uncertainty in momentum
• `h` is Planck's constant

The uncertainty in momentum will depend on the mass of the particle, `m`, and on its uncertainty in velocity, `Deltav`

`color(blue)(ul(color(black)(Deltap = m * Deltav)))`

But first we'll calculate

`p = mv`

`m = 9.11*10^(-31)kg`
`v = 2.2xx10^6m"/"s`

`p = 9.11*10^(-31)kg xx 2.2xx10^6m"/"s`

`p = 2.0042xx10^-24`

But since we have been given that the electron velocity is known to be within 10% of 2.2 X 10^6 m/s

`Deltap = 10/100p`

`Deltap = 2.0042xx10^-24 * 0.1 = 2.0042xx10^-25`

`Deltax>=( 6.626*10^(-34)"kg" *" m"^2"/"s)/((4pi)*2.0042xx10^-25(kg*m)/s)`

`Deltax >= 2.63xx10^(-10)"m"`

Answer B

`2.63xx10^(-10)m >= "Diameter of H"`

The uncertainty is higher as the speed of the electron is less in a hydrogen nucleus than the give situation.

The proton also plays a major role in decreasing the uncertainty and this uncertainty is equal to the diameter of the hydrogen