This chapter discusses the classic circular motion as a basis for the circular motion of a particle. The derivation of the bidimensional Laplace operator in polar coordinate system is presented next in order to solve the Schrödinger equation of the particle in a ring (whose wave function is a part of the solution of the hydrogen atom). Subsequently, there appears the derivation of the wave equation for the particle in a ring which leads to an energy equation and its comparison with the classic equation of the angular momentum leads to the corresponding quantum angular momentum equation. Afterwards, there appears the derivation of the angular momentum in spherical polar coordinate which is part of the tridimensional Laplace operator in spherical polar coordinate which is obtained in the next section. Then, the tridimensional Laplace operator in spherical polar coordinate is used in Schrödinger equation of the particle in a sphere (a product of angular equation and azimuthal equation) where the azimuthal equation was already solved (from the particle in a ring given in a previous section) and then this section is devoted to derive the angular or polar equation and the corresponding quantum angular momentum equation. (195 words).
This chapter discusses the classic circular motion as a basis for the circular motion of a particle. The derivation of the bidimensional Laplace operator in polar coordinate system is presented next in order to solve the Schrödinger equation of the particle in a ring (whose wave function is a part of the solution of the hydrogen atom). Subsequently, there appears the derivation of the wave equation for the particle in a ring which leads to an energy equation and its comparison with the classic equation of the angular momentum leads to the corresponding quantum angular momentum equation. Afterwards, there appears the derivation of the angular momentum in spherical polar coordinate which is part of the tridimensional Laplace operator in spherical polar coordinate which is obtained in the next section. Then, the tridimensional Laplace operator in spherical polar coordinate is used in Schrödinger equation of the particle in a sphere (a product of angular equation and azimuthal equation) where the azimuthal equation was already solved (from the particle in a ring given in a previous section) and then this section is devoted to derive the angular or polar equation and the corresponding quantum angular momentum equation. (195 words).
This chapter considers rotational motion as the second class of motion of an object around a fixed point and angular momentum, which is one of the most important topics in quantum mechanics. It discusses the rotational motion and angular momentum in terms of solutions of the Schrödinger equation and see how its properties emerge from the operators for angular momentum and their commutation relations. It also examines the quantum mechanical description of a particle travelling on a circular ring, which applies to the motion of a bead on a circle of wire and any body rotating in a plane. The chapter describes the moment of inertia the property, which determines the characteristics of the rotational motion of a body. It cites a particle of mass m travelling on a circle of radius r in the xy-plane and its potential energy that is constant and taken to be zero.
This chapter physical properties of the hydrogen atom and its Hamiltonian function as a particle in a spherically symmetric potential having a wave function as a product of the radial function and spherical harmonic function. Hereafter, it comes to two independent wave equations: the radial equation and the angular equation. After some derivation, it is shown that the angular equation is a product of the azimuthal angle equation and polar angle equation (as discussed in the previous chapter). The azimuthal angle equation is similar to the particle in a ring problem from the previous chapter and after some derivation the polar angle equation becomes an associated Legendre equation with well-known polynomials as solutions. Next, the solution to the radial equation is presented which becomes after some derivation an associated Laguerre equation with well-known polynomials as solutions and from this result is obtained the energy equation for the hydrogen-like atom. Subsequently, it is derived the normalization constant of the radial function. Afterwards, the numerical analysis of the radial equation of the hydrogen atom along with its corresponding Fortran code is presented. Finally, the hydrogen atomic orbitals equations and plots are depicted followed by the fine structure of the hydrogen (198 words).
Abstract In this chapter we introduce the concept of orbital angular momentum into the mathematical language of quantum mechanics. We discuss the commutation relations of the corresponding operators and their roles as generators of rotations in the context of the theory of Lie groups and Lie algebras, focusing on the three-dimensional rotation group SO(3). We further examine how scalar and vector-valued operators transform under spatial rotations. We then turn to the derivation of the eigenfunctions and eigenvalues of the angular-momentum operators, for which we follow a dual approach in analogy to that employed for the harmonic oscillator in Chapter 5. Using both the algebraic ladder-operator method and an analytical method using spherical polar coordinates we discover the relevant azimuthal and magnetic quantum numbers and their relation to the spherical harmonics and the Legendre polynomials
In quantum mechanics, there is a profound distinction between orbital angular momentum and spin angular momentum in which the former can be associated with the motion of a physical object in space but the latter cannot. The difference leads to a radical deviation in the formulation of their corresponding dynamics in which an orbital angular momentum can be described by using a coordinate system but a spin angular momentum cannot. In this work, we show that it is possible to treat spin angular momentum in the same manner as orbital angular momentum by formulating spin dynamics using Schrödinger equation in an intrinsic coordinate system. As an illustration, we apply the formulation to the dynamics of a hydrogen atom and show that the intrinsic spin angular momentum of the electron can take half-integral values and, in particular, the intrinsic mass of the electron can take negative values. We also consider a further extension by generalising the formulation so that it can be used to describe other intrinsic dynamics that may associate with a quantum particle, for example, when a hydrogen atom radiates a photon, the photon associated with the electron may also possess an intrinsic dynamics that can be described by an intrinsic wave equation that has a similar form to that for the electron.
Chapter 12 uses the results presented in the previous chapter to deal with systems consisting of two particles interacting via a central potential. It therefore begins by transforming the Hamiltonian of the two-particle system into one that separates the center-of-mass motion from the relative motion. When the potential is spherically symmetric, the relative motion is conveniently described in spherical coordinates to take advantage of the conservation of angular momentum and the solution of the corresponding eigenvalue problem. The chapter contains a detailed discussion of the rigid rotor and the free particle, as well as the selection rules for angular momentum, in preparation for the study of the most important problem of atomic physics, the hydrogen atom. The H spectrum is discussed in detail, and the introduction of minimal coupling to the electromagnetic field allows for a discussion of the normal Zeeman effect as well as the Aharonov-Bohm effect.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.