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Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unper...
Please solve the problem as soon as possible. Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn...
Please solve the problem as soon as possible.
Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0.
Problem 1: Consider a two level system with Hamiltonian: Using the first...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction...
1. Sudden switched-on perturbation calculate ca(t) and c,(t) to second order for a two-level syst...
1. Sudden switched-on perturbation calculate ca(t) and c,(t) to second order for a two-level system with Hab, Hba independent of time and Haa H-O. Use or a two-level system WI initial conditions ca(0) 1, c(0) - 0. Sketch the probability Pa-b(t) (b) This problem can be solved exactly. Find the exact solution and compare it to your solution in part a
1. Sudden switched-on perturbation calculate ca(t) and c,(t) to second order for a two-level system with Hab, Hba independent...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v,...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as...
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for ...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for a two-level system with a (13) Use the ansatz ψ-ee(t)e-iwt/21e) + cg(t)ewt/21g) (14) for the state a) Derive the (exact) differential equations for ce(t),cg(t) b) Use a Fourier-series ansatz, ce- en einwptan ,eg Ση einWptbn. Show that the equations hold. Find m (consider the case separately) wWp c) Find an iterative procedure to solve these equations to higher and higher accuracy. Calcu- late the leading order...
Quantum Mechanics Problem 1. (25) Consider an infinite potential well with the following shape: 0 a/4...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Statistical_Mechanics(1) . (15 points) Fermions in a two-level or three-level system with degeneracy. Consider a system...
Statistical_Mechanics(1)
. (15 points) Fermions in a two-level or three-level system with degeneracy. Consider a system of(N independent fermions. Assume that single-particle Hamiltonian have only two energy levels, with energy co = 0 and ej = e. However, the two levels have degeneracies no and ni, which are O integers. Hint: Note that 1 1 = 1 (4) (ebe-ys e 1 e- 1 (a) For the case of N = 1 = no = n\. Find the chemical potential//i, as...
Please solve the problem as soon as possible.
Problem 1: Consider a two level system with Hamiltonian: Using the first order time-dependent perturbation theory, obtain the probability coefficients cn (t) if the perturbation is applied at t >0 and the system is originally in the ground state. Hint: When solving the problem, first you may need to find the energies and wave functions of the unperturbed Hamiltonian A0.
Problem 1: Consider a two level system with Hamiltonian: Using the first...
Consider a quantum mechanical system with 4 states and an unperturbed Hamiltonian given by 1 0 0 0 Ho E0 0 2 0 a small perturbation is added to this Hamiltonian 0 0 1 0 where e is much smaller than E a) [10pts] What are the energy eigenvalues of the unperturbed system of the following states? 1 o 2o 0 and which energy levels are degenerate? b) [10pts Find a good basis for degenerate perturbation theory instead of c)...
Q1) Consider 2.dimensional infinite "well" with the potential otherwise The stationary states are ny = (a) sin ( x) sin (y,) The corresponding energies are n) , 123 Note that the ground state, ?11 is nondegenerate with the energy E00)-E1)-' r' Now introduce the perturbation, given by the shaded region in the figure ma AH,-{Vo, if 0<x otherwise y<a/2 (a) What is the energy of the 1.st excited state of the unperturbed system? What is its degree of degeneracy,v? (b)...
Exercise 4: Fine structure of hydrogenic atoms a) Consider a Hamiltonian H-Ho + λΗ. with Mr a small perturbation. Show that in (non-degenerate) perturbation theory the first order correction to the unperturbed, discrete energy level E(Holis given by and the second order by b) Apply this to evaluate the first order corrections to the energy levels (the so-called fine structure) of a hydrogenic atom, that arise due to relativistic corrections. Confirm that the answer for the total first order correction...
1. Sudden switched-on perturbation calculate ca(t) and c,(t) to second order for a two-level system with Hab, Hba independent of time and Haa H-O. Use or a two-level system WI initial conditions ca(0) 1, c(0) - 0. Sketch the probability Pa-b(t) (b) This problem can be solved exactly. Find the exact solution and compare it to your solution in part a
1. Sudden switched-on perturbation calculate ca(t) and c,(t) to second order for a two-level system with Hab, Hba independent...
3. Consider a system whose Hamiltonian H, admits two eigenstates y, (with eigenvalues F) and v, (with eigenvalues E,). Assume E, E, and they are () orthogonal, (ifi) normalized and (ii) non-degenerate. After the perturbation is on the diagonal matrix elements become zero ie, <4, l H'l Ψ)-(4, I H'ly,)-0, while the off diagonal equals to a constant value ie. (v, l H'l%)-(wil H'ly)-c Using the 2nd order perturbation theory evaluate the energy of the perturbed system.
Problem 3: Time-Independent Perturbation Theory Consider the particle in a 1D box of size L, as in Fig. 3. A perturbation of the form. V,δ ((x-2)2-a2) with a < L is applied to the unperturbed Hamiltonian of the 1D particle in a box (solutions on the equation sheet). Here V is a constant with units of energy. Remember the following propertics of the Dirac delta function m,f(x)6(x-a)dx f(a) 6(az) が(z) = = ds( dz E, or Ψ(x)-En 10 0.0 0.2...
2 Two-level system Consider the time-dependent tion ihub = Hub Hamiltonian Schrödinger equa- for a two-level system with a (13) Use the ansatz ψ-ee(t)e-iwt/21e) + cg(t)ewt/21g) (14) for the state a) Derive the (exact) differential equations for ce(t),cg(t) b) Use a Fourier-series ansatz, ce- en einwptan ,eg Ση einWptbn. Show that the equations hold. Find m (consider the case separately) wWp c) Find an iterative procedure to solve these equations to higher and higher accuracy. Calcu- late the leading order...
Quantum Mechanics Problem
1. (25) Consider an infinite potential well with the following shape: 0 a/4 3al4 a h2 where 4 Using the ground state wavefunction of the original infinite potential well as a trial function, 2πχ trial = 1-sin- find the approximation of the ground state energy for this system with the variational method. (Note, this question is simplified by considering the two components of the Hamiltonian, and V, on their own) b) If we had used the 1st...
Statistical_Mechanics(1)
. (15 points) Fermions in a two-level or three-level system with degeneracy. Consider a system of(N independent fermions. Assume that single-particle Hamiltonian have only two energy levels, with energy co = 0 and ej = e. However, the two levels have degeneracies no and ni, which are O integers. Hint: Note that 1 1 = 1 (4) (ebe-ys e 1 e- 1 (a) For the case of N = 1 = no = n\. Find the chemical potential//i, as...
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