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4. Suppose you have a system of N-10 quantum harmonic oscillators described by the Boltzmann distribution....
(b) For a system of N independent harmonic oscillators at temperature T, all having a common...
(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C. 3. This problem involves a collection of N independent harmonic oscillators, all having a common angular frequency w (such as in an Einstein solid or in the...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 ..xM with an en...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 ..xM with an energy given by: the Ci's and xs are constants. The Ci's are all positive MkBT A) Prove that (E) M regardless of the value of the constants. This result is known as the equipartition theorem each classical quadratic degree of freedom contributes to the average thermal energy of an equilibrium system. kBT B) Use the equipartition theorem to derive the familiar...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 XM with an ener...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 XM with an energy given by: the Ci's and x0s are constants. The Ci's are all positive A) Prove that (E) regardless of the value of the constants. This result is known as the equipartition theorem - each classical quadratic degree of freedonm contributes-BT to the average thermal energy of an equilibrium system. MkBT 2 3nRT B) Use the equipartition theorem to derive the...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and =...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥ, = îl + Ĥ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... 1....
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and =...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
The multiplicity of a system of N quantum oscillators with the total internal energy U and...
The multiplicity of a system of N quantum oscillators with the total internal energy U and energy quantum a is Ω(,- ,N) = and q 1 and use Stirling's approximation, N(N), to derive U (T, N). You can replace N-1 with N in multiplicity equation to simplify writing given that N1 (N-1+9)!. Assume that both N » 1
1. Imagine you have two simple harmonic oscillators. Oscillator 1 is characterized by k1 10 kg...
1. Imagine you have two simple harmonic oscillators. Oscillator 1 is characterized by k1 10 kg s-2 and oscillator 2 by k25 kg s-2. The oscillators are displaced by the (a) If they each oscillate with a 1 kg mass, what is the period for the oscillators to (b) For some initial displacement d, where do the oscillators first come back into same amount in the same direction and are released together (in phase). come back into phase? phase?
Statistical physics. A system of a large number (N) of identical particles is described by Maxwell Boltzmann distribution function. There are only two possible energy levels, separated by an energ...
Statistical physics.
A system of a large number (N) of identical particles is described by Maxwell Boltzmann distribution function. There are only two possible energy levels, separated by an energy gap of 3 m e V. Degeneracy of each level is one. Let N be equal to number of hydrogen atoms in 1 gm of hydrogen. Calculate average energy of the particles at room temperature
A system of a large number (N) of identical particles is described by Maxwell Boltzmann...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The syste...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C. 3. This problem involves a collection of N independent harmonic oscillators, all having a common angular frequency w (such as in an Einstein solid or in the...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 ..xM with an energy given by: the Ci's and xs are constants. The Ci's are all positive MkBT A) Prove that (E) M regardless of the value of the constants. This result is known as the equipartition theorem each classical quadratic degree of freedom contributes to the average thermal energy of an equilibrium system. kBT B) Use the equipartition theorem to derive the familiar...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Suppose that you have a classical equilibrium system that is described by many continuous variables, x1 XM with an energy given by: the Ci's and x0s are constants. The Ci's are all positive A) Prove that (E) regardless of the value of the constants. This result is known as the equipartition theorem - each classical quadratic degree of freedonm contributes-BT to the average thermal energy of an equilibrium system. MkBT 2 3nRT B) Use the equipartition theorem to derive the...
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥ, = îl + Ĥ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... 1....
H2 Consider two harmonic oscillators described by the Hamiltonians łty = ħws (atât ta+2) and = ħwz (6+6 +) with â (h) and at (@t) being the annihilation and creation operators for the first (second) oscillator, respectively. The Hamiltonian of two non-interacting oscillators is given by Ĥg = îl + Ħ2. Its eigenstates are tensor products of the eigenstates of single-oscillator states: Ĥm, n) = En,m|n, m), where İn, m) = \n) |m) and n, m = 0,1,2, ... a)...
The multiplicity of a system of N quantum oscillators with the total internal energy U and energy quantum a is Ω(,- ,N) = and q 1 and use Stirling's approximation, N(N), to derive U (T, N). You can replace N-1 with N in multiplicity equation to simplify writing given that N1 (N-1+9)!. Assume that both N » 1
1. Imagine you have two simple harmonic oscillators. Oscillator 1 is characterized by k1 10 kg s-2 and oscillator 2 by k25 kg s-2. The oscillators are displaced by the (a) If they each oscillate with a 1 kg mass, what is the period for the oscillators to (b) For some initial displacement d, where do the oscillators first come back into same amount in the same direction and are released together (in phase). come back into phase? phase?
Statistical physics.
A system of a large number (N) of identical particles is described by Maxwell Boltzmann distribution function. There are only two possible energy levels, separated by an energy gap of 3 m e V. Degeneracy of each level is one. Let N be equal to number of hydrogen atoms in 1 gm of hydrogen. Calculate average energy of the particles at room temperature
A system of a large number (N) of identical particles is described by Maxwell Boltzmann...
4. (20 points). Consider a quantum harmonic oscillator with characteristic frequency w. The system is in thermal equilibrium at temperature T. The oscillator is described by the following density matrix: A exp kaT where H is the usual harmonic oscillator Hamiltonian and kB is Boltzmann's constant. Working in the Fock (photon number) basis: a. Find the diagonal elements of ρ b. Determine the normalization constant A. c. Calculate the expectation value of energy (E
4. (20 points). Consider a quantum...
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