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2. A gas that consists of N atoms, each of which and be in any of...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
2.A single particle has energy levels 0, ?,-?, 2 ?, and-2? a) Write the single particle...
2.A single particle has energy levels 0, ?,-?, 2 ?, and-2? a) Write the single particle partition function, z. b) Write the canonical partition function, Z, for an c) Write the probability P, for the particle to have d) At T-0.1 &/kB, calculate P, for the particle to have e) Calculate the average energy of an assembly of N assembly of N such particles. energy 0, e,-?,2?, or-2e. energy of 0 and e. particles at very high temperatures
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you su...
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2...
Problem 6.15. Mean energy of a toy model of an ideal Bose gas (a) Calculate the...
Problem 6.15. Mean energy of a toy model of an ideal Bose gas (a) Calculate the mean energy of an ideal gas of N 2 identical bosons in equilibrium with a heat (b) Calculate the mean energy for N 2 distinguishable particles assuming that each particle can (c) If E1 is the mean energy for one particle and Eg is the mean energy for the two-particle system, bath at temperature T, assuming that each particle can be in one of...
please explain Q Search I guv a Ev O O Av +20 to 12. Two infinite...
please explain
Q Search I guv a Ev O O Av +20 to 12. Two infinite surface charge densities are arranged as shown. Sensors to measure the electric field are placed at points A, B, and C. What is the correct ranking of the magnitude of the electric field at these three points, E4, Eb, and Ec? a. EA = Ep = EC b. Eg > EA = EC c. Ex = Ec > EB d. EA > EB >...
5. Consider a quantum mechanical system made of N identical particles. There are total M possible...
5. Consider a quantum mechanical system made of N identical particles. There are total M possible energy levels that each of these particles can occupy. (a) According to statistical thermodynamics, the probability that a particle occu- pies ith energy level with energy e; is proportional to e-Bes where B = r and T is the temperature. k is a universal constant called Boltzmann constant. What is the probability for a given particle to occupyith energy level? (b) On average, how...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-partic...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
Q.7) Consider a systems of N>>1 identical, distinguishable and independent particles that can be placed in three energy levels of energies 0, E and 2€, respectively. Only the level of energy sis degenerate, of degeneracy g=2. This system is in equilibrium with a heat reservoir at temperature T. a) Obtain the partition function of the system. b) What is the probability of finding each particle in each energy level? c) Calculate the average energy <B>, the specific heat at constant...
1-r' Problem 16.12 (30 pts) This chapter examines the two-state system but consider instead the infinite-state system consisting of N non-interacting particles. Each particle i can be in one of an infinite number of states designated by an integer, n; = 0,1,2, .... The energy of particle i is given by a = en; where e is a constant. Note: you may need the series sum Li-ori = a) If the particles are distinguishable, compute QIT,N) and A(T,N) for this...
2. Two noninteracting particles, each of mass m, are in the 1-D harmonic oscillator potential Describe their ground state and the next two excited states, that is, their corresponding (i) wave functions, (i) energy eigenvalues, and (ii) degeneracies if two particles are (a) distinguishable particles, (b) the identical bosons, and (c) the identical fermions. 3. Suppose one particle is in the ground state, and the other is in the first excited state for tw particles described in Prob. 2. Calculate...
2.A single particle has energy levels 0, ?,-?, 2 ?, and-2? a) Write the single particle partition function, z. b) Write the canonical partition function, Z, for an c) Write the probability P, for the particle to have d) At T-0.1 &/kB, calculate P, for the particle to have e) Calculate the average energy of an assembly of N assembly of N such particles. energy 0, e,-?,2?, or-2e. energy of 0 and e. particles at very high temperatures
Answer a total of any THREE out of the four questions. Put the solution to each problem in a separate blue book and put the number of the problem and your name on the front of each book. If you submit solutions to more than three problems, only the first three problems as listed on the exam will be graded. Some possibly useful information: Sterling's asymptotic series: N In N-N + 1 ln(2nN] In N! N → oo, as 2...
Problem 6.15. Mean energy of a toy model of an ideal Bose gas (a) Calculate the mean energy of an ideal gas of N 2 identical bosons in equilibrium with a heat (b) Calculate the mean energy for N 2 distinguishable particles assuming that each particle can (c) If E1 is the mean energy for one particle and Eg is the mean energy for the two-particle system, bath at temperature T, assuming that each particle can be in one of...
please explain
Q Search I guv a Ev O O Av +20 to 12. Two infinite surface charge densities are arranged as shown. Sensors to measure the electric field are placed at points A, B, and C. What is the correct ranking of the magnitude of the electric field at these three points, E4, Eb, and Ec? a. EA = Ep = EC b. Eg > EA = EC c. Ex = Ec > EB d. EA > EB >...
5. Consider a quantum mechanical system made of N identical particles. There are total M possible energy levels that each of these particles can occupy. (a) According to statistical thermodynamics, the probability that a particle occu- pies ith energy level with energy e; is proportional to e-Bes where B = r and T is the temperature. k is a universal constant called Boltzmann constant. What is the probability for a given particle to occupyith energy level? (b) On average, how...
1. Consider a quantum system comprising three indistinguishable particles which can occupy only three individual-particle energy levels, with energies ε,-0, ε,-2e and ε,-3. The system is in thermal equilibrium at temperature T. Suppose the particles are bosons with integer spin. i) How many states do you expect this system to have? Justify your answer [2 marks] (ii) Make a table showing, for each state of this system, the energy of the state, the number of particles (M, M,, N) with...
Problem 3: (40 points) One-dimensional relativistic gas: Here we consider a non-interacting gas of N relativistic particles in one dimension. The gas is confined in a container of length L, i.e., the coordinate of each particle is limited to 0 <q < L. The energy of the ith particle is given by ε = c (a) Calculate the single particle partition function Z(T,L) for given energy E and particle number N. [12 points] (b) Calculate the average energy E and...
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