hw11sol - Homework 11 Solution Prob. 11.1 Consider two...

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Homework 11 Solution Prob. 11.1 Consider two canonical systems that are initially in respective equilibrium at different temperature T 1 and T 2 and µ 1 and µ 2 . They are brought to contact at t = 0. (a) How does the entropy of the total system change at t = 0? (This is actually one of the definition of time) (5 pts) With the interaction, the entropy will increase. When the system migrates towards T 1 = T 2 and µ 1 = µ 2 , the total entropy will reach a maximum (if the two systems are identical, then this is towards the center of the Pascal tree). Time can actually be defined as the direction of increasing entropy when the system has a constant energy (or no energy input). From here, you can derive the second law of Thermodynamics. (b) If T 1 > T 2 and µ 1 = µ 2 for t < 0, describe the exchange between the two systems for t 0. When will the exchange stop? (5 pts) Since µ 1 = µ 2 for all t , the exchange is to make T 1 goes towards T 2 for t > 0. The particle on system 1 with the higher T side has a higher kinetic energy, and hence higher probability of going towards system 2. To maintain µ 1 = µ 2 , some particle with lower kinetic energy will flow back to system 1. This phenomenon is often called thermal diffusion. (c) If T 1 = T 2 and µ 1 > µ 2 for t < 0, describe the exchange between the two systems for t 0. When will the exchange stop? (5 pts) Since T 1 = T 2 for all t , the exchange is to make µ 1 goes towards µ 2 for t > 0. The particle on system 1 has a higher concentration (higher µ with fixed V and T ), and hence higher probability of going towards system 2. This phenomenon is often called particle diffusion. Prob. 11.2 For a 3D particle in the 3D simple harmonic oscillator described by: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 z y x r m z y x m z y x z y x m z y x m z y x H o o , , 2 1 , , 2 , , 2 1 , , 2 , , ˆ 2 2 2 2 2 2 2 2 2 2 ψ ϖ + - = + + + - = The energy eigenvalues are: + = + + + = 2 3 2 3 0 0 n n n n E z y x n , where 0 2 3 is referred to as the “zero-point energy” (okay, I know this name was twisted in the movie of “The Incredibles”) (a) For n = 2, what are the possible eigenstates? How many states are there? (5 pts) ( n x , n y , n z ) = (2, 0, 0) have three possible eigenstates. There are another three in (1, 1, 0).
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This note was uploaded on 11/26/2010 for the course ECE 3060 at Cornell University (Engineering School).

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hw11sol - Homework 11 Solution Prob. 11.1 Consider two...

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