Consider the matrix representation of Lx, Ly and L₂ for the case l = 1 (see Matrix Representation of Operators class notes pp. 11-12). (a) Construct the matrix representation of L2 for l = 1. (b) What are the eigenvalues and corresponding eigenvectors of L²? (c) Are the eigenvectors of L2 the same as those of L₂? Explain. (d) Compute L² \x; +1), where [x;+1) is the eigenvector of La corresponding to eigenvalue +ħ.
Consider the matrix representation of Lx, Ly and L₂ for the case l = 1 (see Matrix Representation of Operators class notes pp. 11-12). (a) Construct the matrix representation of L2 for l = 1. (b) What are the eigenvalues and corresponding eigenvectors of L²? (c) Are the eigenvectors of L2 the same as those of L₂? Explain. (d) Compute L² \x; +1), where [x;+1) is the eigenvector of La corresponding to eigenvalue +ħ.
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Transcribed Image Text:Consider the matrix representation of \( L_x, L_y, \) and \( L_z \) for the case \( \ell = 1 \) (see Matrix Representation of Operators class notes pp. 11-12).
(a) Construct the matrix representation of \( L^2 \) for \( \ell = 1 \).
(b) What are the eigenvalues and corresponding eigenvectors of \( L^2 \)?
(c) Are the eigenvectors of \( L^2 \) the same as those of \( L_z \)? Explain.
(d) Compute \( L^2 \, |x; +1\rangle \), where \( |x; +1\rangle \) is the eigenvector of \( L_x \) corresponding to eigenvalue \( +\hbar \).
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