Question

Rutherford's scattering experiments gave the first indications that an atom consists of a small, dense, positively charged nucleus surrounded by negatively charged electrons. His experiments also allowed for a rough determination of the size of the nucleus. In this problem, you will use the uncertainty principle to get a rough idea of the kinetic energy of a particle inside the nucleus.
Consider a nucleus with a diameter of roughly 5.0×10−15 meters.

Part A

Consider a particle inside the nucleus. The uncertainty Δx in its position is equal to the diameter of the nucleus. What is the uncertainty Δp of its momentum? To find this, use ΔxΔp≥ℏ2where ℏ=h2π.

Express your answer in kilogram-meters per second to two significant figures.

You did not open hints for this part.

ANSWER:

Δp = kg⋅m/s

Answer 1

The uncertainty on the momentum is $1.1\cdot 10^{-20} kg m/s$

Explanation:

We can solve the problem by using the uncertainty principle, which states that:

$\Delta x \Delta p \geq \frac{h}{4\pi}$

where

$\Delta x$ is the uncertainty on the position

$\Delta p$ is the uncertainty on the momentum

$h=6.63\cdot 10^{-34} Js$ is the Planck constant

For the nucleus in this problem, the uncertainty on the position is equal to the size of the nucleus, therefore

$\Delta x = 5.0\cdot 10^{-15} m$

Therefore we can substitute into the equation and solve for $\Delta p$ to find the uncertainty on the momentum:

$\Delta p \geq \frac{h}{4\pi \Delta x}=\frac{6.63\cdot 10^{-34}}{4\pi (5.0\cdot 10^{-15})}=1.1\cdot 10^{-20} kg m/s$

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