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Question

# Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of vx = 100 m/s. The rifle’s barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Solution

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### Knowns & Concept

The mass of the particles is $m=1.00\, \text{g}=1.00\times 10^{-3}\, \text{kg}$, and their speed is $v=100\, \dfrac{\text{m}}{\text{s}}$. The diameter of the hole is $d=2.00\, \text{mm}=2.00\times 10^{-3}\, \text{m}$. We need to find the distance $x$ (see the attached picture). From $\textbf{the Heisenberg uncertainty principle}$, we can write:

$d\Delta p_y \geq \dfrac{\hbar}{2}$

and we choose the case when $d\Delta p_y$ has the smallest value $\dfrac{\hbar}{2}$, where $\hbar=\dfrac{h}{2\pi}$ is the reduced Planck's constant $h=6.626 \times 10^{-34}\, \text{Js}$. Also, from the similarity of the blue and the green triangles at the picture, we can write:

$\dfrac{1\, \text{cm}}{x}=\dfrac{\Delta p_y}{p_x}$

where $p_x=mv$ is the momentum of the particles along $x$ axis. Now, we will express $\Delta p_y$ as:

$\Delta p_y=\dfrac{p_x\times 1\, \text{cm}}{x}$

and substitute $p_y$ at the Heisenberg principle:

$d\times \dfrac{p_x\times 1\, \text{cm}}{x}= \dfrac{\hbar}{2}$

Rearrange:

$x=\dfrac{2dp_x \times 1\, \text{cm}}{\hbar}=\dfrac{4\pi dp_x \times 1\, \text{cm}}{h}$

Substitute numerical values:

$x=\dfrac{4\times 3.14\times 2.00\times 10^{-3}\, \text{m}\times 1.00\times 10^{-3}\, \text{kg} \times 100\, \dfrac{\text{m}}{\text{s}}\times 1\times 10^{-2}\, \text{m}}{6.626 \times 10^{-34}\, \text{Js}}$

$\boxed{x=3.79\times 10^{28}\, \text{m}}$

Clearly, the required distance is larger than the diameter of the visible Universe $D=2\times 10^{26}\, \text{m}$.

$\dfrac{x}{D}=\dfrac{3.79\times 10^{28}}{2\times 10^{26}}=190$

The distance x is 190 times larger than the observable Universe!

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