..Solve the following questions

1. Replace the following classical mechanical expressions with their corresponding quantum mechanical operators. a. K.E. =- myz in three-dimensional space. b. p = mv, a three-dimensional cartesian vector. c. y-component of angular momentum: Ly = zPx - XPz- 2. Transform the following operators into the specified coordinates: ALTY- from cartesian to spherical polar coordinates. b. Lz = - from spherical polar to cartesian coordinates. 3. Match the eigenfunctions in column B to their operators in column A. What is the eigenvalue for each eigenfunction? Column A Column B i. (1-x2) d- 4x4 - 12x2 + 3 dx 2 ii, 12 5x4 dx 2 3x + e-3x iv. 12 dx 2 - 2x 7 x2 . 4x + 2 v. x d2 dy? + (1-x) 4x 3 - 3x 4. Show that the following operators are hermitian. 3 a. l'x b. Lx5. For the following basis of functions ( 2p_p 2pp and 2p. ), construct the matrix representation of the Ly operator (use the ladder operator representation of L,). Verify that the matrix is hermitian. Find the eigenvalues and corresponding eigenvectors. Normalize the eigenfunctions and verify that they are orthogonal. 2p-1 8 1/2 a re-z/2a Sin el 1 Z 2p. 1/2 2a re-2/2a Cos 5/2 2p1 8 1/2 a re-2/2a Sin el 6. Using the set of eigenstates (with corresponding eigenvalues) from the preceding problem, determine the probability for observing a z-component of angular momentum equal to 1h if the state is given by the Lx eigenstate with Oh Lx eigenvalue. 7. Use the following definitions of the angular momentum operators: Lex= y - . z - , Ly = z - . x - La = T X y - , and 12 = 13 + 13 + L; . and the relationships: [x. Px] = ih , [y. py] = ih , and [z, pa] = ih . to demonstrate the following operator identities: a. [Lx, Ly] = ih Lz. b. [Ly, Ly] = ih Lx. c. [LzLx] = in Ly. d. [L,, L3] = 0, e. [Ly, L ] = 0, f. [LL?] = 0. 8. In exercise 7 above you determined whether or not many of the angular momentum operators commute. Now, examine the operators below along with an appropriate given