The power of the Finite Difference Approximation is that it can be used for potentials that are not analytically solvable! Consider the following potential as a crude approximation to the harmonic oscillator potential (V(x)=1/2kx?) with k = 1: = -3 V (x) = 1 = 3 (A) Using the N=3 Finite Difference Approximation, write down the 3x3 matrix representation of the Hamiltonian, H. for the above potential. The final matrix should include the "t" value in the lecture notes - no need to plug anything else in - just keep it as "(". (B) Calculate, by hand, the characteristic polynomial of the matrix in (A) by calculating det(H - 20) =0. Write down this characteristic polynomial but do not solve it. (C) It's pretty crazy how complicated this expression already is for such a simple system. You can try to find the roots by hand, or you can use Mathematica's Solve function (Solve[f(x)=0.x]) function to find the three roots of this equation. These are the three eigenvalues of the Hamiltonian. (D) Now, set t=1.0 in your H matrix and use Mathematica's Eigensystem routine to calculate the eigenvalues and their associated eigenvectors, and make sure that the eigenvectors are normalized. If you have trouble writing a matrix in Mathematica, go to "Palettes"-"Basic Math Assistant" and click on the "Advanced" tab. This will have picture of an empty two-by-two matrix on the right that you can click on to set up a matrix. Additional rows and columns can be added using the "Row+" and "Col+" buttons. (E) Compute the expectation value of the position operator for the lowest energy eigenfunction you determined in (D) () using matrix vector multiplication (by hand). Here, 'T' is the lowest energy eigenfunction you calculated in (D). Remember from the lecture that the position operator is a diagonal matrix in the basis of position kets, with the values of the position at each location on the diagonal. (F) Extend your basis to five total states (N = 5) by adding basis states at x = -6 and x = 6. Using this new matrix, use the Eigensystem routine to calculate the Eigenvalues and Eigenvectors of this system. (G) Draw the wavefunctions corresponding to three lowest energy eigenvalues you got from (F) using the corresponding eigenfunctions you calculated in (F).EXAMPLE V= D Lets solve the ID Particle in a box " numerically" using a basis of 3 position states 1X0> 1Xi> 1X2> - Step 1: Build the matrix 2 + + v (xo ) I -t = Matrix will be 3x3 -t - 2+ + V( x ) 2+ + V(x. ) - t 9 0 2 .t 2t + V(x, ) - t 0 - t 2+ + v(x 2 ) 1 2 -t 2 N - -t V ( XD ) = V ( x1 ) = V(x2 ) = 0 2 2+ + v( x2) because ID PIB