Quantum Mechanics With Application

This chapter is devoted to a short review of classical mechanics with the aim of defining the classical Hamiltonian, and establishing the relation between space symmetry and conservation laws.

We devote this chapter to the concept of the superposition of waves to construct wave packets that satisfy the uncertainty principle and the Schrodinger equation.

In this chapter we turn to a solution of the Schrodinger equation for a simple one dimensional potential as a tool for understanding the meaning of the wave function with an introduction to symmetry and conservation laws.

This chapter is devoted to the application of the Schrodinger equation to one dimensional models of interesting phenomena. In this way we can get an understanding of quantum effects, and the role of symmetry, without the complexity of the mathematics encountered in three dimensional problems.

Here we turn to some applications of quantum mechanics to systems in atoms, nuclei and solids that can be modeled in terms of one dimensional quantum systems.

In the present chapter we turn to the one-dimensional harmonic oscillator potentials for which the Schrodinger equation admits analytic solutions. This will allow us to examine the vibrational spectrum of quantum systems, and in particular, diatomic molecules.

In this chapter we commence with the solution of the Schrodinger equation in three dimension and consider the case when the potential has spherical symmetry. This requires the introduction of spherical polar coordinates and the reduction of the three dimensional partial differential equation to a set of ordinary differential equations. The solution of these equations for the angular dependence will allow us to examine the rotational spectrum of diatomic molecules.

This chapter is devoted to the solution of the radial Schrodinger equation for three different spherically symmetric potentials whose solutions are used to discuss the approximation methods in atomic, molecular and nuclear problems in Chapter 13.

With the central role played by scattering in any measurement on a quantum system, in this chapter we introduce the basic formulation of scattering theory for a central potential. As a simple application we consider scattering by a square-well, as well as the scattering by a Coulomb potential.

Having solved the Schrodinger equation for several potentials, we turn our attention in this chapter to the matrix formulation of quantum mechanics and the different representations we can use to simplify the solution of problems for both bound and scattering problems.

With symmetry playing a central role in quantum mechanics, in this chapter we commence with the simplest of symmetries of translation in space and time. Here we consider the group structure of the symmetry operators. This is followed by a rotational symmetry that plays a central role in atomic and nuclear physics and is a major component of this chapter. Finally we examine time and space reversion and isospin that are central to the study of nuclear and particle physics.

With a limit on the small number of problems that admit exact solutions, we turn in this chapter to approximation methods that can be used for bound state problems. The most commonly used method is the perturbation theory for which one use a small parameter to form the basis of a systematic expansion, or a variational method in which one hopes to set the best limit on the energy of a system. We will consider both of these approximation methods with applications to problems in atomic and molecular physics.

In this chapter we return to scattering theory and set the foundation for approximation methods in calculating the scattering amplitude. We also address the problem of solving the Schrodinger equation for scattering in other than the coordinate representation. We convert the Schrodinger equation into an integral equation for the scattering amplitude and discuss ways of solving such integral equations. In the process we apply the formalism to electronatom scattering and nucleon-nucleon scattering.

Full text available

Full text available