Question #b701d

The idea here is that the Heisenberg Uncertainty Principle tells us that we cannot measure both the position and the momentum of a particle with arbitrarily high precision.

In other words, we will always have a very high uncertainty in measuring the position of a particle if we have a very precise measurement of its momentum.

Similarly, we will always have a very high uncertainty in measuring the momentum of a particle if we have a very precise measurement of its position.

In fact, the uncertainty in position and the uncertainty in momentum must always satisfy the 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, equal to `6.626 * 10^(-34)"kg m"^2"s"^(-1)`

Now, the uncertainty in momentum will depend on the mass of the particle, `m`, and the uncertainty in velocity, which we'll call `Deltav`

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

In your case, you have `m` as the mass of an electron, listed as

`m ~~ 9.10938 * 10^(-31)"kg"`

Plug this back into the inequality to find

`Deltax * m * Deltav >= h/(4pi)`

Rearrange to solve for `Deltav`

`Deltav >= h/(4pi) * 1/(Deltax * m)`

Now, notice that you have a very small uncertainty in position

`497 color(red)(cancel(color(black)("pm"))) * "1 m"/(10^(12)color(red)(cancel(color(black)("pm")))) = 4.97 * 10^(-10)` `"m"`

which means that in order for the Heisenberg Uncertainty Principle to hold, you must have a very high uncertainty in velocity.

Plug in your values to get the uncertainty in velocity

`Deltav = (6.626 * 10^(-34)color(red)(cancel(color(black)("kg"))) "m"^color(red)(cancel(color(black)(2)))"s"^(-1))/(4 * pi) * 1/(479 * 10^(-12)color(red)(cancel(color(black)("m"))) * 9.10938 * 10^(-31)color(red)(cancel(color(black)("kg"))))`

`color(darkgreen)(ul(color(black)(Deltav = 1.21 * 10^(5)color(white)(.)"m s"^(-1))))`

The answer is rounded to three sig figs.

As predicted, you have a very high uncertainty in velocity.