A one-dimensional harmonic oscillator is composed of a particle of mass m charge q and potential...

 

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A one-dimensional harmonic oscillator is composed of a particle of mass m charge q and potential energy V(X) = mo?X?. We assume in this exercise that the particle is placed in an electric field 6(1) parallel to Ox and time-dependent, so that to V(x) must be added the potential energy: W (1) = - g8(t)X a. Write the Hamiltonian H (t) of the particle in terms of the operators a and a'. Calculate the commutators of a and a' with H(t). b. Let a(t) be the number defined by: (1) = ((1)la|4(1)> where (t)> is the normalized state vector of the particle under study. Show from he results of the preceding question that a(t) satisfies the differential equation: a(t) = - io a(t) + i(t) dt position and momentum of the particle? where 2(1) is defined by: 6(1) 2mho Jntegrate this differential equation, At time 1, what are the mean values of the c. The ket 9(1)) is defined by: 19(1)>= [a – a(1)] |V(1) > %3D where a(t) has the value calculated in b. Using the results of questions a and b, show that the evolution of (t)> is given by: ih - () > = [H() + ho]| o(t)> How does the norm of 9(1)> vary with time? d. Assuming that (0)> is an eigenvector of a with the eigenvalue a(0), show that (t)> is also an eigenvector of a, and calculate its eigenvalue. Find at time t the mean value of the unperturbed Hamiltonian Ho = H(t) – W (t) %3D as a function of a(0). Give the root-mean-square deviations 4X, AP and AH; how do they vary with time? e. Assume that at / = 0, the oscillator is in the ground state |4. >. The electric field acts between times 0 and T and then falls to zero. When i> T, what is the evolution of the mean values (X>(1) and (P>(1)? Application : assume that between 0 and T, the field &(1) is given by &(t) = &, cos (o't); discuss the phenomena observed (resonance) in terms of Aw what results can be found, and with what probabilities? %3D %3D o' - w. If, at 1> T, the energy is measured, A one-dimensional harmonic oscillator is composed of a particle of mass m charge q and potential energy V(X) = mo?X?. We assume in this exercise that the particle is placed in an electric field 6(1) parallel to Ox and time-dependent, so that to V(x) must be added the potential energy: W (1) = - g8(t)X a. Write the Hamiltonian H (t) of the particle in terms of the operators a and a'. Calculate the commutators of a and a' with H(t). b. Let a(t) be the number defined by: (1) = ((1)la|4(1)> where (t)> is the normalized state vector of the particle under study. Show from he results of the preceding question that a(t) satisfies the differential equation: a(t) = - io a(t) + i(t) dt position and momentum of the particle? where 2(1) is defined by: 6(1) 2mho Jntegrate this differential equation, At time 1, what are the mean values of the c. The ket 9(1)) is defined by: 19(1)>= [a – a(1)] |V(1) > %3D where a(t) has the value calculated in b. Using the results of questions a and b, show that the evolution of (t)> is given by: ih - () > = [H() + ho]| o(t)> How does the norm of 9(1)> vary with time? d. Assuming that (0)> is an eigenvector of a with the eigenvalue a(0), show that (t)> is also an eigenvector of a, and calculate its eigenvalue. Find at time t the mean value of the unperturbed Hamiltonian Ho = H(t) – W (t) %3D as a function of a(0). Give the root-mean-square deviations 4X, AP and AH; how do they vary with time? e. Assume that at / = 0, the oscillator is in the ground state |4. >. The electric field acts between times 0 and T and then falls to zero. When i> T, what is the evolution of the mean values (X>(1) and (P>(1)? Application : assume that between 0 and T, the field &(1) is given by &(t) = &, cos (o't); discuss the phenomena observed (resonance) in terms of Aw what results can be found, and with what probabilities? %3D %3D o' - w. If, at 1> T, the energy is measured,