Problem 3.42. Consider the probability density function p(v) = (a/n)3/2 e-av² for the velocity v of a particle. Each of the three velocity components can range from -∞o to +∞o and a is a constant. (a) What is the probability that a particle has a velocity between ur and ur + dur, Vy and vy + duy, and v₂ and v₂ +du₂? (b) Show that p(v) is normalized to unity. Use the fact that So √ (3.97) Note that this calculation involves doing three similar integrals that can be evaluated separately. (c) What is the probability that u 20, vy ≥ 0, vz 20 simultaneously? au² du
Problem 3.42. Consider the probability density function p(v) = (a/n)3/2 e-av² for the velocity v of a particle. Each of the three velocity components can range from -∞o to +∞o and a is a constant. (a) What is the probability that a particle has a velocity between ur and ur + dur, Vy and vy + duy, and v₂ and v₂ +du₂? (b) Show that p(v) is normalized to unity. Use the fact that So √ (3.97) Note that this calculation involves doing three similar integrals that can be evaluated separately. (c) What is the probability that u 20, vy ≥ 0, vz 20 simultaneously? au² du
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Statistical Physics
P3.42
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Transcribed Image Text:Problem 3.42. Consider the probability density function p(v) = (a/)³/2 e-av² for the velocity
v of a particle. Each of the three velocity components can range from -∞ to +∞o and a is a
constant. (a) What is the probability that a particle has a velocity between ₁ and Ur + dur, Vy
and vy + duy, and v₂ and v₂ + dv₂? (b) Show that p(v) is normalized to unity. Use the fact that
(3.97)
Note that this calculation involves doing three similar integrals that can be evaluated separately.
(c) What is the probability that v, 20, vy ≥ 0, vz 20 simultaneously?
fe
е
-au²
du
1
√
2
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