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| ### Statement |
| ### Statement |
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| $5.8.5.$ [Insert the problem statement] |
| $5.8.5.$ The trajectory of an atom elastically reflected from the walls of a cube with |
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| dimensions $a$×$a$×$a$ is a square. The velocity of the atom is $v$. |
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| {a.} What is the average speed at which the impact site will move along each |
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| wall if the angle of incidence in the plane of the square is changed by ∆ ≪ 1? |
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| At what values of ∆ will the trajectory of the atom be closed? not closed? |
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| Determine the distance between adjacent parallel sections of the trajectories |
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| in the first and second cases. |
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| {b.} Why can we assume that the trajectory of an atom is usually not closed? |
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| What isthe probability of detecting an atom in the square of the area S located |
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| in the plane along which the atom moves in the case of an open trajectory? |
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| {c.} How will an atom move if you change the angle of its incidence perpendic |
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| ular to the plane of the square by ∆ ≪ 1? What is the probability of detecting |
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| such an atom in a region inside the cube whose volume is equal to V ? |
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| ### Solution |
| ### Solution |
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| {a.} In the figure, the motion along the trajectory is unfolded by mirror reflections into motion between two parallel straight lines. The corresponding points of the trajectories are marked with the same letters. From this figure it follows: | | {a.} In the figure, the motion along the trajectory is unfolded by mirror reflections into motion between two parallel straight lines. The corresponding points of the trajectories are marked with the same letters. From this figure it follows: |
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| \[ | | \[ |
| v' \approx \frac{x}{2A'B'} v \approx v \Delta \sqrt{2}; \quad \Delta \approx \frac{1}{2} \left[ \tan \left( \frac{\pi}{4} + \Delta \right) - 1 \right] = \frac{k}{2n}, | | v' \approx \frac{x}{2A'B'} v \approx v \Delta \sqrt{2}; \quad \Delta \approx \frac{1}{2} \left[ \tan \left( \frac{\pi}{4} + \Delta \right) - 1 \right] = \frac{k}{2n}, |
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| where \( k \) and \( n \) are integers with no common divisor, | | where \( k \) and \( n \) are integers with no common divisor, |
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| \tan(\pi/4 + \Delta) - 1 = k/n; \quad h_1 \approx 2a \Delta/k, \quad h_2 = 0. | | \tan(\pi/4 + \Delta) - 1 = k/n; \quad h_1 \approx 2a \Delta/k, \quad h_2 = 0. |
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| f{b.} It is improbable that \( \tan(\pi/4 + \Delta) - 1 \) is exactly equal to a simple fraction, for example 0.03, since near this number there can be arbitrarily many other numbers, for instance numbers of the form \( 0.03 + \sqrt{2}/n \), where \( n \) is an integer, which differ from 0.03 by an arbitrarily small amount. These numbers are called irrational, and in mathematics it is proven that the set of these numbers is more powerful than the set of simple fractions. If the number is irrational, then the trajectory is not closed. | | f{b.} It is improbable that \( \tan(\pi/4 + \Delta) - 1 \) is exactly equal to a simple fraction, for example 0.03, since near this number there can be arbitrarily many other numbers, for instance numbers of the form \( 0.03 + \sqrt{2}/n \), where \( n \) is an integer, which differ from 0.03 by an arbitrarily small amount. These numbers are called irrational, and in mathematics it is proven that the set of these numbers is more powerful than the set of simple fractions. If the number is irrational, then the trajectory is not closed. |
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| {c.} $P=V/a^3$, since trajectory is not closed. | | {c.} $P=V/a^3$, since trajectory is not closed. |
