Q3: Perturbation of the Three-Dimensional Harmonic Oscillator 25.0 points (graded) The spectrum of the three-dimensional

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Q3 Perturbation Of The Three Dimensional Harmonic Oscillator 25 0 Points Graded The Spectrum Of The Three Dimensional 1 (93.84 KiB) Viewed 31 times

Q3: Perturbation of the Three-Dimensional Harmonic Oscillator 25.0 points (graded) The spectrum of the three-dimensional isotropic harmonic oscillator is degenerate. In this problem, we see how a certain perturbation reduces the degeneracy. (This problem does not require the tools of degenerate perturbation theory.) Consider a system described by the Hamiltonian H = H (0) + 8H . The unperturbed Hamiltonian is given by 1 + H(0) = where x = ($1,22,23). P = (P1P2, P3). The perturbation &H is given by 8H = 'WL2. where L. = 23P1 - 13 is the component of angular momentum in the y direction, and ) is unit free. PARTA 4.0 points (graded) Set 1 = 0, so that H = H). Use creation and annihilation operators for "oscillator quanta" in the 1, 2 and 3 directions, with number operators Ni.N2 N3, respectively. Denote eigenstates of these number operators by their eigenvalues, as ni, n2, ns) What is the energy Ennains of the state ni,n2, ng) ? Write your answers in terms of n.72 and nz. w and h. How many linearly independent states are there with energy E= ?W? Degeneracy = PART B 8.0 points (graded) Express SH in terms of creation and annihilation operators. Write your answers in terms of às âu and âs . a . a and át, 1 and w. (Recall that you can use for the Imaginary unit.) H = Find the matrix representation of 8H in the E= hw degenerate subspace spanned by 11) = (1,0,0), 12) = 10,1,0,13) = 10,0,1). Write your answers in terms of X, w and h. Note that this is a matrix input. 8H =