Q1. Obtain the eigenvalues and eigenvectors of a 10 by 10 random symmetric matrix. (Hint: generate a random 10 by 10 random general (possibly asymmetric) matrix and symmetrise it by a number of approaches e.g. add it to its transpose and divide by 2). Check the eigenvectors for orthonormality.
Q2. Find the Laplace Transforms of 
, 
, 
and 
.
Q3. Write a Mathematica function that yield the available energy
levels below some maximum energy for a system of s quantum harmonic
oscillators with different energy gaps 
.
Q4. Given a reversible chemical reaction system
:
(i) Solve this system in general for the time dependence of concentrations using Mathematica via matrix methods.
(ii) Solve the same problem by direct integration.
(iii) Give the specific solution for an initial concentration of A of 1 unit and that of B and C is zero.
Q5. Write a Mathematica notebook to solve the 1D Schrödinger equation for an arbitrary polynomial in x over the domain 0<x<L (assume infinite potential walls outside this domain.