Question 1: Write a matrix that approximates (1) the second derivative of \(\psi\), and (2) approximates multiplication by \(V(x)\). The second matrix should be a diagonal matrix whose entries are \(V\left(x_{j}\right)\) for \(-N \leq j \leq N\). Hence write down the matrix that represents the left hand side of the Schrödinger equation.

Question 2: Find the smallest 10 eigenvalues of this matrix when \(N=100\) and \(h=0.05\) with (a) \(V(x)=\) \(\frac{1}{2} x^{2}\), (b) \(V(x)=\frac{1}{10} x^{4}-x^{2}\), (c) \(V(x)=\frac{1}{10} x^{4}-x^{2}+\frac{1}{10} x\). For (a) compare with the true answer.

Question 3: Plot the four eigenfunctions corresponding to the smallest four eigenvalues for (a), (b) and (c).

Question 4: Approximate \(X\) and \(D\) by \(D_{N}\) and \(X_{N}\), the top left \(N \times N\) submatrices. Show that \(D_{N}\) is skew symmetric, and \(X_{N}\) is symmetric. Show that \(-D_{N}^{2}\) and \(X_{N}^{2}\) are positive semi-definite.

Question 5: For \(N=100\), show that the ten smallest eigenvalues of \(-0.5 D_{N}^{2}+0.1 X_{N}^{4}-X_{N}^{2}+0.1 X\) are about the same as the eigenvalues from Question 3.

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