*Problem 4.34 (a) Apply S_ to |10) (Equation 4.177), and confirm that you get /2h|1–1). (b) Apply S+ to 100) (Equation 4.178), and confirm that you get zero, (c) Show that |1 1) and |1-1) (Equation 4.177) are eigenstates of S2, with the appropriate eigenvalue.
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- a2 Laplacian operator 72 = ax? ay? T əz2 in spherical polar coordinates is given by az? p² = () 1 a 1 1 a2 r2 sin e ae sin 0-) is an eigenfunction of the Laplacian operator and find the +- r2 sin 0 a0 r2 ar ar. r2 sin? 0 a20 sin 0 sin o Show that function r2 corresponding eigenvalue.Prove that ||A + B|| ≤ ||A|| + ||B||. This is called the triangle inequality; in twoor three dimensions, it simply says that the length of one side of a triangle ≤sum of the lengths of the other 2 sides. Hint: To prove it in n-dimensional space, write the square of the desired inequality using (10.2) and also use the Schwarz inequality (10.4). Generalize the theorem to complex Euclidean space by using (10.7) and (10.9).Do the problem in 1st quandrant. Explain how you determined the sign of each derivative. Draw pictures to illustrate your reasoning.
- please provide detailed solutions for a to d. thank you!Construct the 3D rotation ma- trices for rotations about the x-axis, y-axis, and z-axis by an angle o as described in I section 1.1.5. You should end up with three 3x3 matrices, one for each rotation axis. Demonstrate that your matrices do what they are supposed to do by considering >= 90 deg and applying them to an arbitrary vector A. Draw pictures to illustrate your reasoning.