(a)Write down the classical and the quantum partition functions for two non-interacting atoms confined to in...
(a)Write down the classical and the quantum partition functions for two non-interacting atoms confined to in their motion along a line of length L.
(b)Obtain the internal energy of the system and the entropy by using the partition functions
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(a)Write down the classical and the quantum partition functions
for two non-interacting atoms confined to in...
(a)Write down the classical and the quantum partition functions for two non-interacting diatomic molecules in a...
(a)Write down the classical and the quantum partition functions for two non-interacting diatomic molecules in a volume V, where each molecule is treated as a simple harmonic oscillator. (b)Obtain the internal energy of the system and the entropy by using the partition functions
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. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for t...
1. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is...
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
Two non - interacting particles, with the same mass, are in a one - dimensional potential...
Two non - interacting particles, with the same mass, are in a
one - dimensional potential which is zero along a length
2a, and infinite elsewhere. What are the
values of the four lowest energies of the system? What are the
degeneracies of these energies if the two particles are: a)
identical, with spin ; b) identical, with spin 1.
a) Write down the Lagrangian L(x1, x2, 81, 82) for two particles of equal masses, m1...
a) Write down the Lagrangian L(x1, x2, 81, 82) for two particles of equal masses, m1 = m2 = m, confined to the x axis and connected by a spring with potential energy U = kx2 . [Here x is the extension of the spring, x = x1 - x2-1, where l is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.) (b) Rewrite L in terms of the...
PROBLEM 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic...
PROBLEM 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic oscillator is of the form d dp e Δ ΔΊΔ ) is the regulating spatial and momentum resolution cutoffs, which are often Chosen to be at the scale of the atoms (and n) and are important for making entropy dimensionless but they drop out in parts (b) and (c). Moreover, Z factorizes as Z ZzZp with Z. 3 Calculate the partition function and the...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
Classical Mechanics problem: Consider the two coupled pendulums shown in the figure below. Each of the...
Classical Mechanics problem:
Consider the two coupled pendulums shown in the figure below. Each of the pen- dulums has a length L and the spring constant is k. The pendulums' position can be specified by the angles ¢\ and ø2. The relaxed length of the spring is such that the equi librium position of the pendulums is at ¢2 = 0 with the two pendulums vertical a.) Find the lagrangian L of this system. You can assume the angular deflections...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
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. Show that for a classical ideal gas, Q1 alnQ1 NK Hint: Start with the partition function for the classical ideal gas ( Q1) and use above equation to find the value of right-hand side and compare with the value of r we derive in the class. (Recall entropy you derived for classical gas) NK Making use of the fact that the Helmholtz free energy A (N, V, T) of a thermodynamic system is an extensive property of the system....
Consider a classical particle confined on a segment with length 2L. If a harmonic potential is introduced on this segment, then the Hamiltinian becomes and application of the equipartition theorem predicts the average values of the kinetic and potential energies, e.g., <V>-kT/2. On the hand, making L sufficiently small, one can ensure that V(x)=ma,2x2/2ckT/2 for all x between-L and L. We conclude that the average value is larger than any allowed value of V! Please explain this paradox (assume that...
Two non - interacting particles, with the same mass, are in a
one - dimensional potential which is zero along a length
2a, and infinite elsewhere. What are the
values of the four lowest energies of the system? What are the
degeneracies of these energies if the two particles are: a)
identical, with spin ; b) identical, with spin 1.
a) Write down the Lagrangian L(x1, x2, 81, 82) for two particles of equal masses, m1 = m2 = m, confined to the x axis and connected by a spring with potential energy U = kx2 . [Here x is the extension of the spring, x = x1 - x2-1, where l is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.) (b) Rewrite L in terms of the...
PROBLEM 1 5 points] In classical statistical mechanics, the canonical partition function for a single harmonic oscillator is of the form d dp e Δ ΔΊΔ ) is the regulating spatial and momentum resolution cutoffs, which are often Chosen to be at the scale of the atoms (and n) and are important for making entropy dimensionless but they drop out in parts (b) and (c). Moreover, Z factorizes as Z ZzZp with Z. 3 Calculate the partition function and the...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
Classical Mechanics problem:
Consider the two coupled pendulums shown in the figure below. Each of the pen- dulums has a length L and the spring constant is k. The pendulums' position can be specified by the angles ¢\ and ø2. The relaxed length of the spring is such that the equi librium position of the pendulums is at ¢2 = 0 with the two pendulums vertical a.) Find the lagrangian L of this system. You can assume the angular deflections...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
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