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Pauli paramagnetism Consider an ideal spin-1/2 Fermi gas in the presence of an external magnetic field...
Consider a canonical ensemble of ? paramagnetic particles of magnetic moment ? in an external magnetic...
Consider a canonical ensemble of ? paramagnetic particles of
magnetic moment ? in an external magnetic field B and find
a. An expression for the average magnetic moment per
particle
b. The total magnetization.
c. Demonstrate the Curie’s law of paramagnetism that predicts
that at high temperature the magnetization of the material is
proportional to the ratio between the external field and the
temperature
Consider a canonical ensemble of N paramagnetic particles of magnetic moment μ in an external magnetic...
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D sp...
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef as a function in temperature. Hint: vou will need to know that Ep =
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef...
Statistical_Mechanics 2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D....
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
In in the study of thermodynamic systems and magnetic gases, consider a monoatomic ideal (Boltzmann) gas...
In in the study of thermodynamic systems and magnetic gases, consider a monoatomic ideal (Boltzmann) gas in which the spin(-1/2) atoms have magnetic moment μ. The gas is placed in magnetic field B pointing up. At a given temperature T, (c) Find the heat capacity at constant volume and magnetic field, cva, per atom of the gas. (Use the equipartition theorem for the translational motion of the atoms).
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic fi...
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B),...
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
3. A simple model of a Neutron star is an ideal gas of neutrons (each with spin 1/2 in units of h...
3. A simple model of a Neutron star is an ideal gas of neutrons (each with spin 1/2 in units of h). Aside from the kinetic energy of the neutrons, one must consider the gravitational energy, which for a homogeneous star of mass M and radius R, is 3GM2 5R where G 6.67 x 10-11m3kg-'s-2 is the universal gravitational constant (i) We suppose in this problem that the Fermi temperature is large enough for T0 What general condition determines the...
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic...
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic field in the r-direction. The spin is initially (at time t 0) in the z-direction. (a) Write down the Hamiltonian for the electron spin. (Do you remember the elementary magnetic moment of electron, the Bohr magneton μΒ, in terms of electron mass me and the elementary charge e?) b) Write down the Schrödinger equation for the spin (c) Describe the motion of spin direction
To first approximation, a neutron star may be treated as a non-interacting gas of neutrons (spin-...
To first approximation, a neutron star may be treated as a non-interacting gas of neutrons (spin-1/2 fermions). ( a) Find an expression for the degeneracy (zero point) energy of the star as a function its mass M and radius R. (Don’t worry that our calculation for degenerate Fermi systems was done for particles in a cubical box, the same results apply to a sphere of the same volume.) (b) Write down an expression for the gravitational potential energy of the...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
Consider a canonical ensemble of ? paramagnetic particles of
magnetic moment ? in an external magnetic field B and find
a. An expression for the average magnetic moment per
particle
b. The total magnetization.
c. Demonstrate the Curie’s law of paramagnetism that predicts
that at high temperature the magnetization of the material is
proportional to the ratio between the external field and the
temperature
Consider a canonical ensemble of N paramagnetic particles of magnetic moment μ in an external magnetic...
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef as a function in temperature. Hint: vou will need to know that Ep =
2. Consider we have put an ideal Fermi gas with (N) average particles of spin and mass m in 2D space of area A at finite temperature T. Derive the fermi energy ef...
Statistical_Mechanics
2 20 points) 2D ideal Fermi gas 24 Consider an ideal Fermi gas in 2D. It is contained in an area of dimensions L x L. The particle mass is m. (a) Find the density of states D(e) N/L2 (b) Find the Fermi energy as a function of the particle density n = (c) Find the total energy as a function of the Fermi energy ef. (d) Find the chemical potential u as a function of n and T....
In in the study of thermodynamic systems and magnetic gases, consider a monoatomic ideal (Boltzmann) gas in which the spin(-1/2) atoms have magnetic moment μ. The gas is placed in magnetic field B pointing up. At a given temperature T, (c) Find the heat capacity at constant volume and magnetic field, cva, per atom of the gas. (Use the equipartition theorem for the translational motion of the atoms).
(4) The Pauli spin matrices are a set of 3 complex 2 x 2 matrices that are used in quantum mechanics to take into account the interaction of the spin of a particle with an external electromagnetic field. σ2 10), (a) Find the eigenvalues and corresponding eigenvectors for all three Pauli spin matrices. Show all of vour work (b) In Linear Algebra, two matrices A and B are said to commute if AB BA and their commutator defined as [A,...
Problem 4 A monoatomic Boltzmann ideal gas if spin / atoms in uniform magnetic field (B), has in additional to its usual kinetic translational energy, a magnetic energy of tuB per atom, where is u the magnetic moment. It is assumed that the gas is so diluted that the interaction of magnetic moments can be neglected. a) What is the partition function for a canonical ensemble of N such atoms. b) Calculate C, from the partition function. c) Draw the...
3. A simple model of a Neutron star is an ideal gas of neutrons (each with spin 1/2 in units of h). Aside from the kinetic energy of the neutrons, one must consider the gravitational energy, which for a homogeneous star of mass M and radius R, is 3GM2 5R where G 6.67 x 10-11m3kg-'s-2 is the universal gravitational constant (i) We suppose in this problem that the Fermi temperature is large enough for T0 What general condition determines the...
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic field in the r-direction. The spin is initially (at time t 0) in the z-direction. (a) Write down the Hamiltonian for the electron spin. (Do you remember the elementary magnetic moment of electron, the Bohr magneton μΒ, in terms of electron mass me and the elementary charge e?) b) Write down the Schrödinger equation for the spin (c) Describe the motion of spin direction
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
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