Excercise sheet 10 -- solutions

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 Z 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 ∇·Overscript[G, _](x,y,z) and curl ∇×Overscript[G, _](x,y,z) for the scalar function f(x,y,z)=1/(x^2 + y^2 + z^2)^(1/2) and vector function Overscript[G, _](x,y,z)=(x^2+y,y+z,z x)?

Solutions

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?

Solution

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