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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...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unper...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unperturbed energies and E() and corresponding eigenfunctions ф ) and 4°, respectively. Assume E ) > E 0. such that the Bohr angular frequency wi(EEis positive. A time-independent perturbation V is switched on at time t 0 a) Write down the coupled set of equations for the coefficients ca(t) and c(t) of the wave function of the system: Note that we...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a pert...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
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...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harm...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
Exercise 8: Time dependence of a two-level system Consider a two-level system with stationary states a and b with unperturbed energies and E() and corresponding eigenfunctions ф ) and 4°, respectively. Assume E ) > E 0. such that the Bohr angular frequency wi(EEis positive. A time-independent perturbation V is switched on at time t 0 a) Write down the coupled set of equations for the coefficients ca(t) and c(t) of the wave function of the system: Note that we...
3. (a) Consider a 1-dim harmonic oscillator in its ground state (0) of the unperturbed Hamiltonian at t--0o. Let a perturbation Hi(t)--eEXe t2 (e, E and rare constants) be applied between - and too. What is the probability that the oscillator will be in the state n) (of the unperturbed oscillator) as t-> oo?(15%) (b) The bottom of an infinite well is changed to have the shape V(x)-ε sin® for 0Sxa. Calculate the energy shifts for all the excited states...
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...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
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