A bound state is a quantum state in which a particle is confined within a potential well and unable to escape, due to the attractive forces of the potential. This results in a discrete energy level for the particle.
A scattering matrix is a mathematical tool used to describe the probability of a particle scattering off of a potential, and the resulting changes in its properties (such as energy or momentum).
To find the bound state of a Hamiltonian, one must solve the Schrödinger equation for the system. This involves finding the eigenvalues and eigenvectors of the Hamiltonian matrix, which represent the energy levels and corresponding wave functions of the bound states.
The scattering matrix of a Hamiltonian can be found by solving the Schrödinger equation for the system and then using the resulting wave functions to calculate the scattering matrix elements. These elements represent the probability amplitudes for the particle to scatter into different states.
Understanding the bound state and scattering matrix of a Hamiltonian is crucial in many areas of physics, including quantum mechanics, nuclear physics, and solid state physics. It allows us to predict and analyze the behavior of particles and systems, and provides valuable insights into the underlying physical processes at play.