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Springer eBooks

Nonrelativistic Hydrogen Atom

James K. Freericks

Published Jan 1, 2026• 0 citations
Publisher

Abstract

In this chapter, we focus on three-dimensional problems that have rotational symmetry, with a concentrated effort on describing the properties of the hydrogen atom. This is an exactly solvable problem that is extremely important. To get there, we first have to describe how we work in spherical coordinate systems, and we need to develop the general rules for the factorization method. We develop the concept of a vector operator, of radial momentum, and the rules needed to solve problems by the Schrödinger factorization method. We first examine the isotropic harmonic oscillator, and then we establish the energy scales of hydrogen and solve first for the energies and eigenstates and then the wavefunctions in position and momentum space. We derive the Kramers–Pasternak relations for expectation values of powers of r and then focus on important experiments, which include the Pickering–Fowler lines, the discovery of deuterium, visualizing the atom in momentum space with e-2e spectroscopy, Rydberg atoms, the Lamb shift, and hyperfine structure and its applications to radio astronomy.

DOI
10.1007/978-3-031-87845-9_11

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In this chapter, we focus on three-dimensional problems that have rotational symmetry, with a concentrated effort on describing the properties of the hydrogen atom. This is an exactly solvable problem that is extremely important. To get there, we first have to describe how we work in spherical coordinate systems, and we need to develop the general rules for the factorization method. We develop the concept of a vector operator, of radial momentum, and the rules needed to solve problems by the Schrödinger factorization method. We first examine the isotropic harmonic oscillator, and then we establish the energy scales of hydrogen and solve first for the energies and eigenstates and then the wavefunctions in position and momentum space. We derive the Kramers–Pasternak relations for expectation values of powers of r and then focus on important experiments, which include the Pickering–Fowler lines, the discovery of deuterium, visualizing the atom in momentum space with e-2e spectroscopy, Rydberg atoms, the Lamb shift, and hyperfine structure and its applications to radio astronomy.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.
Abstract In Chapter 5 we considered the factorization of a radial Hamiltonian for a particle in a spherically symmetric potential which is independent of angular momentum. This factorization provides shift operators for energy and angular momentum in the eigenkets of the radial Hamiltonian for the oscillator and Coulomb potentials. These operators are linear functions of the momentum operator *p and they are also nonlinear functions of the position operator r: in the coordinate representation of wave mechanics they yield first-order differential operators. In this chapter we show that for the three-dimensional isotropic harmonic oscillator and the Coulomb potential, one can also construct shift operators for energy and angular momentum that are linear functions* of r and nonlinear functions of p. In the momentum representation of wave mechanics the shift operators yield first-order differential operators. Applications to the calculation of energy eigenvalues, matrix elements, and radial momentum-space wavefunctions are considered.
Abstract In this chapter, we go to three dimensions in space and look at the solution of the time independent Schrödinger equation for the hydrogen atom. The Hamiltonian is then the kinetic energy plus the potential energy due to the Coulomb coupling between the positively charged nucleus and the electron. We construct the angular momentum operator and find that the partial differential equation for the angular momentum eigenfunctions of the spherical coordinates θ,ϕ is the same as the angular part of the ∇2 operator in spherical coordinates. The angular momentum eigenfunctions are the spherical harmonics, with two quantum numbers, l and m, and the solution to the radial part of the Hamiltonian including the Coulomb potential are Laguerre polynomials with one quantum number, called the principle quantum number, n. The hydrogen wave function is the product of a Laguerre polynomial and a spherical harmonic with three quantum numbers. Since these are two- and three-dimensional functions for angular momentum and hydrogen respectively, they are best understood in a series of plots. The chapter concludes by giving the historical letter names to specific orbitals, since they continue to be used today.
The factorization method of Schrödinger shows us how to determine the energy eigenstates without needing to determine the wavefunctions in position or momentum space. A strategy to convert the energy eigenstates to wavefunctions is well known for the one-dimensional simple harmonic oscillator by employing the Rodrigues formula for the Hermite polynomials in position or momentum space. In this work, we illustrate how to generalize this approach in a representation-independent fashion to find the wavefunctions of other problems in quantum mechanics that can be solved by the factorization method. We examine three problems in detail: (i) the one-dimensional simple harmonic oscillator; (ii) the three-dimensional isotropic harmonic oscillator; and (iii) the three-dimensional Coulomb problem. This approach can be used in either undergraduate or graduate classes in quantum mechanics.