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(Solved): Quantum Mechanics Vol 2 Cohen-Tannoudj Chapter XI Complement HXI Exercise 3. Can Anybody help me wit ...



Quantum Mechanics Vol 2 Cohen-Tannoudj Chapter XI Complement HXI Exercise 3.

Can Anybody help me with a solution to this problem please?

3. A particle of mass m, constrained to move in the Oy plane, has a Hamiltonian:
1
I
Y
Ho
=
p² P²
= +
2m 2m
+ mw² (X² + y²)
(

 

3. A particle of mass m, constrained to move in the Oy plane, has a Hamiltonian: 1 I Y Ho = p² P² = + 2m 2m + mw² (X² + y²) (a two-dimensional harmonic oscillator, of angular frequency w). We want to study the effect on this particle of a perturbation W given by: W = X?W? + A?W? where X? and ? are constants, and the expressions for W? and W? are: W? = mw²XY W? = ?w C (L? is the component along Oz of the orbital angular momentum of the particle). In the perturbation calculations, consider only the corrections to first order for the energies and to zeroth order for the state vectors. a. Indicate without calculations the eigenvalues of Ho, their degrees of degeneracy and the associated eigenvectors. In what follows, consider only the second excited state of Ho, of energy 3?w and which is three-fold degenerate. b. Calculate the matrices representing the restrictions of W? and W? to the eigensub- space of the eigenvalue 3?w of Ho. c. Assume X? = 0 and X? < 1. Calculate, using perturbation theory, the effect of the term X?W? on the second excited state of Ho. e. Assume X? << ? < 1. Considering the results of question c to be a new unper- turbed situation, calculate the effect of the term X2 W2. f. Now assume that ?? = 0 and X2 < 1. Using perturbation theory, find the effect of the term X?W? on the second excited state of Ho. h. Finally, assume that ?? < ?? < 1. Considering the results of question f to be a new unperturbed situation, calculate the effect of the term A? W?.


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