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)=21kx2) with k=1 : V(x)=29029x=3x=0x=3 (A) (2 points) 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 "t". (B) (2 points) Calculate, by hand, the characteristic polynomial of the matrix in (A) by calculating det(HI)=0. Write down this characteristic polynomial but do not solve it. (C) (2 points) 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