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4. (a) Using &n = 2ML2 (n² + n3 + n²), show that the partition function of a massive particle (M) in a 3-dimensional box (V = L³) is given by Z, = n,V, where the quantum concentration is ną = (Mt/2nh²)³/2 ; you may need 'e-a²x² dx = \T/2a. 00 (b) Now the partition function for non-interacting N such particles is given by ZN = ZN /N!. Explain why. (f) Find the inter-particle spacing Ap at n = n, and argue that Ap~ħ/p, i. e. Ap is on the order of the thermal de Broglie wavelength, where p is the thermal average momentum.

4. (a) Using &n = 2ML2 (n² + n3 + n²), show that the partition function of a massive particle (M) in a 3-dimensional box (V = L³) is given by Z, = n,V, where the quantum concentration is ną = (Mt/2nh²)³/2 ; you may need 'e-a²x² dx = \T/2a. 00 (b) Now the partition function for non-interacting N such particles is given by ZN = ZN /N!. Explain why. (f) Find the inter-particle spacing Ap at n = n, and argue that Ap~ħ/p, i. e. Ap is on the order of the thermal de Broglie wavelength, where p is the thermal average momentum.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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4. (a) Using ɛn = 2ML2 (n² + n3 + n²), show that the partition function of a massive particle (M) in a 3-dimensional box (V = L³) is given by Z, = ngV, where the quantum concentration is no = (Mt/2nh²)³/2 ; you may need e-a?x² dx = \T/2a. (b) Now the partition function for non-interacting N such particles is given by ZN = ZN/N!. Explain why. (f) Find the inter-particle spacing Ap at n = ng and argue that Ap~ħ/p, i. e. 2p is on the order of the thermal de Broglie wavelength, where p is the thermal average momentum.
Transcribed Image Text:4. (a) Using ɛn = 2ML2 (n² + n3 + n²), show that the partition function of a massive particle (M) in a 3-dimensional box (V = L³) is given by Z, = ngV, where the quantum concentration is no = (Mt/2nh²)³/2 ; you may need e-a?x² dx = \T/2a. (b) Now the partition function for non-interacting N such particles is given by ZN = ZN/N!. Explain why. (f) Find the inter-particle spacing Ap at n = ng and argue that Ap~ħ/p, i. e. 2p is on the order of the thermal de Broglie wavelength, where p is the thermal average momentum.
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