Excercise sheet 10

Matrices

1. Define matrices
	A=(({{1, 2}, {3, 4}})), B=(({{0, 1}, {1, 1}})),
and the vectors 
	X=(({{1}, {2}})), Y=(({{3}, {4}})).
Do the following
(a) Calculate the expressions A B,  B A ,  A X,  
      X·Y and A B - B A
(b) Find the determinant of A, det A.
(c) Solve the equation A Z = Xfor Z. Check the result by substituting Y back into the equation
(d) Find the inverse (A^(-1)) and calculate (A^(-1)) A.
(e) Find the transposed matrix (A^T).
(f) Find the eigenvalues and eigenvectors of A.
(g) Using Mathematicas help system, can you find a command that calculates the cross-product? Use it to calculate X×Y.

2. Vector calculus Mathematica commands can be found in the package Calculus`VectorAnalysis`. Can you find a way to calculate the gradient ∇f(x, y, z), divergence \(TraditionalForm\`∇\(\(·\)\(\(G\&_\)(x, y, z)\)\)\) and curl \(TraditionalForm\`∇\(\(×\)\(\(G\&_\)(x, y, z)\)\)\) for the scalar function f(x, y, z) = 1/(Sqrt[(x^2) + (y^2) + (z^2)]) and vector function \(TraditionalForm\`\(G\&_\)(x, y, z) = \((x\^2 + y, y + z, z\ x)\)\)?

Schrödinger equation in matrix form

In the lecture notes, the Schrödinger equation was solved for the harmonic potential. Modify the potential to include the term ϵ (r^4), that is, use V(r) = (r^2) + ϵ (r^4)  Set ϵ = 0.1. How do the energies change compared to the original, harmonic potential? How about the eigenfunctions?