In a three-dimensional problem, consider a particle of mass m and of potential energy: mw V(X,...

 

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In a three-dimensional problem, consider a particle of mass m and of potential energy: mw V(X, Y, Z): + (1-) 4X (1+) (x +Y?) 2 3 3 where w and X are constants which satisfy: 0 <1 < 4 a. What are the eigenstates of the Hamiltonian and the corresponding energies? b. Calculate and discuss, as functions of A, the variation of the energy, the parity and the degree of degeneracy of the ground state and the first two excited states. Use the following guide in answering part (b) of this problem: (b-i) As a function of 2, determine the ground state energy Eo, eigenvector(s), degeneracy, and parity of eigenvector(s). Sketch Eo(). (b-ii) At à = 0, determine the energies, eigenvectors, parity, and degeneracy of the first two excited levels. (b-iii) Suppose << 1. Determine how the energy levels from (b-i)-(b-ii) evolve. Sketch your result, indicating eigenvectors, parity, and degeneracies. (b-iv) What happens to the Hamiltonian when 2 = 3/4? Determine the eigenvalues, eigenvectors, and comment on their degeneracy and parity. For 2< 3/4, describe how these energy levels evolve. In a three-dimensional problem, consider a particle of mass m and of potential energy: mw V(X, Y, Z): + (1-) 4X (1+) (x +Y?) 2 3 3 where w and X are constants which satisfy: 0 <1 < 4 a. What are the eigenstates of the Hamiltonian and the corresponding energies? b. Calculate and discuss, as functions of A, the variation of the energy, the parity and the degree of degeneracy of the ground state and the first two excited states. Use the following guide in answering part (b) of this problem: (b-i) As a function of 2, determine the ground state energy Eo, eigenvector(s), degeneracy, and parity of eigenvector(s). Sketch Eo(). (b-ii) At à = 0, determine the energies, eigenvectors, parity, and degeneracy of the first two excited levels. (b-iii) Suppose << 1. Determine how the energy levels from (b-i)-(b-ii) evolve. Sketch your result, indicating eigenvectors, parity, and degeneracies. (b-iv) What happens to the Hamiltonian when 2 = 3/4? Determine the eigenvalues, eigenvectors, and comment on their degeneracy and parity. For 2< 3/4, describe how these energy levels evolve.