Applications of integral transformations in mechanical and quantummechanical problems

Abstract

There are many mathematical techniques of which it is unclear how they find their application. We exemplify the diverse application of inte- gral transformations by solving three mechanical and quantummechanical problems in the solution of which integral transformations play a cen- tral role. After introducing the general formalism of integral operators we use the Abel operator to determine an exact parametrisation of the isochrone curve. Then we use Fourier transformations to obtain an exact solution for the ground state energy of the one-dimentional Heisenberg model of E0/JL = −0.44, which we compare to a numerical result of E0/JL = −0.43. Lastly we use the Laplace transformation to find the form of the effective Liouvillian Leff which acts on the reduced density S matrix ρS of a single spin- 1 particle coupled to a large reservoir in Laplace 2 space. Lastly, we reflect on these results and conclude that the ability to use integral transformations is essential for any physicist.