Consider a Hilbert space spanned by the three functions phi1

Consider a Hilbert space spanned by the three functions phi_1 = x_1, phi_2 = x_2, phi_3 = x_3, and scalar product defined by (x_v| x_mu) = delta_v-mu Form the 3 times 3 matrix of each of the following operators: A_1 = sigma_i=1^3 x_i (partial differential/partial differential x_i), A_2 = x_1 (partial differential/partial differential x_2) - x_2 (partial differential/partial differential x_1). Form the column vector representing psi = x_1 - 2x_2 + 3x_3. Form the matrix equation corresponding to x = (A_1 - A_2) psi and verify that the matrix equation reproduces the result obtained by direct application of A_1 - A_2 to psi.

Solution

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