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Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with fre...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with frequency w in the microcanonical ensemble. Determine the phase space volume and the corresponding entropy. Please be sure to include the quantum correction h3N. Please note that the Hamiltonian for the system isN mwqi You may also want to note that the volume of a d-dimensional which can be simplified by using r sphere of radius R is given by
1: Consider a system of N classical distinguishable...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
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...
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...
Question 1 A system of N identical non-interacting magnetic ions of spin Y%, has energy u...
Question 1 A system of N identical non-interacting magnetic ions of spin Y%, has energy u tHo for each spin. μο is the magnetic moment in a crystal at absolute temperature T in a magnetic field B. For this system calculate: a) The partition function, Z. b) Free energy, F. c) The entropy. S d) The average energy, U e) The average magnetic moment, M
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the follo...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
3. Consider the generalized one-dimensional Ising model with link-dependent interactions J. In other words, we have N spins s, in a chain and energy N-1 SiSi+1 We recover a standard Ising model (with...
3. Consider the generalized one-dimensional Ising model with link-dependent interactions J. In other words, we have N spins s, in a chain and energy N-1 SiSi+1 We recover a standard Ising model (with B 0 and open boundary condi tions) if we set JiJN-1-J a) Show that the canonical partition function is Hint: You can do this by induction and carefully comparing the states of the N-spin chain to those of the (N - 1)-spin chain.] b) Show that the...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
1: Consider a system of N classical distinguishable one-dimensional harmonic oscillators with frequency w in the microcanonical ensemble. Determine the phase space volume and the corresponding entropy. Please be sure to include the quantum correction h3N. Please note that the Hamiltonian for the system isN mwqi You may also want to note that the volume of a d-dimensional which can be simplified by using r sphere of radius R is given by
1: Consider a system of N classical distinguishable...
1 The Gibbs Paradox Consider N particles, each of mass m, in a 3-dimensional volume V at temperature T. Each particle i has momentum pi. Assume that the particles are non-interacting (ideal gas) and distinguishable. a) (2P) Calculate the canonical partition function N P for the N-particle system. Make sure to work out the integral. b) (2P) Calculate the free energy F--kBTlnZ from the partition function Z. Is F an extensive quantity? c) (2P) Calculate the entropy S F/oT from...
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...
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...
Question 1 A system of N identical non-interacting magnetic ions of spin Y%, has energy u tHo for each spin. μο is the magnetic moment in a crystal at absolute temperature T in a magnetic field B. For this system calculate: a) The partition function, Z. b) Free energy, F. c) The entropy. S d) The average energy, U e) The average magnetic moment, M
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
3. Consider the generalized one-dimensional Ising model with link-dependent interactions J. In other words, we have N spins s, in a chain and energy N-1 SiSi+1 We recover a standard Ising model (with B 0 and open boundary condi tions) if we set JiJN-1-J a) Show that the canonical partition function is Hint: You can do this by induction and carefully comparing the states of the N-spin chain to those of the (N - 1)-spin chain.] b) Show that the...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
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