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Systems with variable numbers of particles1
y Imagine a solid with sites on which particles can...
1. Imagine that a particle can exist in one of the three states: |0 > , |1 >, |2> . We now consider 2 such part...
1. Imagine that a particle can exist in one of the three states: |0 > , |1 >, |2> . We now consider 2 such particles. How many distinguishable states are possible if the particles are (a) distinguishable (b) indistinguishable, classic (c) bosons (d) fermions. Write down the wave function of the system for all cases.
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
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...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
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...
Imagine a particle that can exist in only 3 states or levels. These energies levels are...
Imagine a particle that can exist in only 3 states or levels. These energies levels are –0.1 eV, 0.0 eV, and +0.1 eV. This particle is in eq with a reservoir at T = 300 K. A) Find the partition function for this particle. I want a number and a unit. Comment on what you would expect, and what the partition function tells you! B) Find the relative probability (%) for the particle to be in each of these...
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...
Dimoglobin as a case study in cooperativity. Note that each binding site can be occupied, ,...
Dimoglobin as a case study in cooperativity.
Note that each binding site can be occupied,
, or unoccupied,
, and that a parameter J describes the cooperativity between the O2
molecules when both states are occupied (i.e., the energy is not
just the sum of the individual binding energies).
a) Write down the weights (Gibbs factors) for each of the
different states for the dimoglobin system shown in Figure 2.
b) What is the formula that describes the energy of...
Part B All this are multiple questions Part C Part D Part E Question 3 (MCQ...
Part B
All this are multiple questions
Part C
Part D
Part E
Question 3 (MCQ QUESTION) [8 Marks) A hypothetical quantum particle in 10 has a normalised wave function given by y(x) = a.x-1.b, where o and bare real constants and i = V-1. Answer the following: a) What is the most likely x-position for the particle to be found at? Possible answers forder may change in SAKAI 14] a - b + ib a 0 Question 3 (MCQ...
(12%) Consider a system of non-interacting fermions in equilibrium with a heat bath at temperature T and a particle reservoir at chemical potential fl. Assume that we can neglect different spin orientations of the fermions. Each particle can be in one of three single-partiele states with energies 0, A and 2A. (a) Find the grand partition function of the system. (b) Find the mean number of particles and mean energy of the system. (C) Find the most probable microstate of...
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...
[1 44= 9 marks ] Question 5 Consider two identical particles in 1D which exist in single-particle (normalised) (x), and are in such close proximity they can be considered as indistinguishable. wave functions /a(x) and (a) Write down the symmetrised two-particle wave function for the case where the particles are bosons (VB) and the case where the particles are fermions (Vp). (b) Show that the expectation value (xjr2)B,F is given by: (T122) в,F — (а:)a (х)ь + dx x y:(")...
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...
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...
Dimoglobin as a case study in cooperativity.
Note that each binding site can be occupied,
, or unoccupied,
, and that a parameter J describes the cooperativity between the O2
molecules when both states are occupied (i.e., the energy is not
just the sum of the individual binding energies).
a) Write down the weights (Gibbs factors) for each of the
different states for the dimoglobin system shown in Figure 2.
b) What is the formula that describes the energy of...
Part B
All this are multiple questions
Part C
Part D
Part E
Question 3 (MCQ QUESTION) [8 Marks) A hypothetical quantum particle in 10 has a normalised wave function given by y(x) = a.x-1.b, where o and bare real constants and i = V-1. Answer the following: a) What is the most likely x-position for the particle to be found at? Possible answers forder may change in SAKAI 14] a - b + ib a 0 Question 3 (MCQ...
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