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1. Transform (using the coordinate system provided below) the following functions accordingly: Z Y X a. from cartesian to spherical polar coordinates 3x + y - 4z = 12 b. from cartesian to cylindrical coordinates y2 + z2 =9 c. from spherical polar to cartesian coordinates r =2 Sin Cos 2. Perform a separation of variables and indicate the general solution for the following expressions: a. 9x + 16y- =0 X b. 2y + +6=0 X 3. Find the eigenvalues and corresponding eigenvectors of the following matrices:NCONN DONNE b. NNC 4. For the hermitian matrix in review exercise 3a show that the eigenfunctions can be normalized and that they are orthogonal. 5. For the hermitian matrix in review exercise 3b show that the pair of degenerate eigenvalues can be made to have orthonormal eigenfunctions. 6. Solve the following second order linear differential equation subject to the specified "boundary conditions": d2x dt2 + k2x(t) =0, where x(t=0) = L, and OX(1 0) =0. dt Exercises 1. Replace the following classical mechanical expressions with their corresponding quantum mechanical operators. a. K.E. = - mv- - 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: a. Lx = Ty - Z 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 d2 5x4 dx 2 iii. x - e3x + e-3x ive d2 - 2x d x2 - 4x + 2 dx 2 * dx V. X- d2 4x3 - 3x dx 2 + (1-x)