Three Dimensional Problem - Advanced Quantum Chemistry and Spectroscopy - Problem, Exercises of Chemistry

Download Three Dimensional Problem - Advanced Quantum Chemistry and Spectroscopy - Problem and more Exercises Chemistry in PDF only on Docsity! Chemistry Problem Set 3 1.) Given: ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − = 310 012 001 ~ 054 203 121 ~ BA Find BA ~~ and AB~~ . Do the two matrices commute? 2.) Evaluate the following determinant: 1000 1100 1110 1111 − − −− 3.) Given the matrix: ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ −= 001 00 11 ~ i i A a) Is A~ Hermitian? b) Find the eigenvalues of A~ 4.) Consider a three-dimensional problem. In a given orthonormal basis the Hamiltonian is represented by the matrix: ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − =+= c c c HHH 00 00 00 200 030 001 ~~~ )1()0( c is a constant << 1. a) Find the exact eigenvalues of H~ b) Determine the eigenvalues by non-degenerate perturbation to second order. c) Compare the results of part a) and b) Note: you will find the binomial expansion useful for part c): (1+x)n ~ 1+nx; x < 1. docsity.com