Department of Physics Temple University Introduction to Quantum Mechanics, Physics 3701 - Solution set for homework # 6

Department of Physics Temple University Introduction to Quantum Mechanics, Physics 3701 - Solution set for homework # 6 Consider an arbitrary physical system whose four-dimensional state space i s spanned by a basis of four eigenvectors jj; mzi common to J^2 and Jz (j = 0 or 1; -j ≤ mz ≤ +j), of eigenvalues j(j + 1)¯h2 and mz¯ h, such that: • a) Express in terms of the kets jj; mz , the eigenstates common to J^2 and J^x to be denoted by jj; mx . We must first form the matrix of the operator J^x in the basis fjj; mz g. If we recall the following relation + J^-) (3) then we may use Eqs. ?? and ?? to write the matrix of the J^x operator in the given basis. We first calculate the individual matrix elements. First, the J^+ terms j0; 0 = 0 (7) then the J - terms Now we may write the operator J^x in matrix form, using Eq. ??. in the basis fj1; 1 ; j1; 0 and (12) To find the eigenvalues of this matrix, it is necessary to diagonalize it. The determinant is solved in the usual way, resulting in a characteristic equation given by This produces four roots, two of which are zero. This λ = 0 eigenvalue is thus two-fold degenerate. The other two eigenvalues are +¯h and -¯ h. Substitution of these eigenvalues into Eq. ?? allows us to solve for the eigenvectors. We are therefore able to write the eigenvectors in the new basis, jj; mx in terms of the old basis, jj; mz . The following are the eigenvalues that correspond to the eigenvectors in both bases, in the These are the eigenstates common to the operators J^2 and J^x • b) Consider a system in the normalized state: Note that this state is normalized. Therefore, we must have the following relation between the coefficients: i) What is the probability of finding 2¯h2 and ¯h if J^2 and J^x are measured simultaneously?