The formulation of quantum-mechanical problems in momentum space, which is largely ignored in traditional courses but is used in research, is discussed. In particular, the formulation of scattering problems, for which the momentum representation is quite natural, is neither generally well known nor obvious. This is illustrated by solving a simple problem, the one-dimensional delta function potential, through which the bound state and continuum wave function are found.
The formulation of quantum-mechanical problems in momentum space, which is largely ignored in traditional courses but is used in research, is discussed. In particular, the formulation of scattering problems, for which the momentum representation is quite natural, is neither generally well known nor obvious. This is illustrated by solving a simple problem, the one-dimensional delta function potential, through which the bound state and continuum wave function are found.
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
The momentum representation is seldom used in quantum mechanics courses. Some students are thence surprised by the change in viewpoint when, in doing advanced work, they have to use the momentum rather than the coordinate representation. In this work, we give an introduction to quantum mechanics in momentum space, where the Schrödinger equation becomes an integral equation. To this end we discuss standard problems, namely, the free particle, the quantum motion under a constant potential, a particle interacting with a potential step, and the motion of a particle under a harmonic potential. What is not so standard is that they are all conceived from momentum space and hence they, with the exception of the free particle, are not equivalent to the coordinate space ones with the same names. All the problems are solved within the momentum representation making no reference to the systems they correspond to in the coordinate representation.
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
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
Chapter Preview: We begin at the beginning with Plank's black-body radiation spectrum applied to the cosmic background. We move on to bound states; first finding them semi-analytically with a trial-and-error search, and then finding them numerically as an eigenvalue problem. We include relativistic effects in hydrogen bound states by solving the Klein-Gordon equation. We then examine the numerical solution of the time- independent Schrödinger equation, and various types of bound state wave functions. We move on to wave packet solutions of the time- dependent Schrödinger equation. Then we study scattering from: a 3-D square well, the Coulomb potential, three stationary disks, and on a billiard table (see also classical billiards in § 3.8 ). We next look at quantum mechanics in momentum space: integral equations, bound states, the delta-shell potential, hyperfine structure of hydrogen, SU3 symmetry of quarks. Next to last we examine several examples of coherent states. We end the chapter with the Feynman path integral formulation of quantum mechanics, and its solution via the Metropolis algorithm. If not enough, Chapter 8 discussion of quantum entanglement fits in well here too.
In the present paper the quantum mechanical representation in phase space introduced by Torres-Vega and Frederick (1993 J. Chem. Phys. 98 3103) has been successfully applied to the phase space description of the quantum potential scattering. 3D potential scattering is formulated from the view-points of the wave function and density operator in this new quantum phase-space representation. It is found that in the region where the potential acts predominantly, the scattered part of the wave function or density in quantum phase space depends closely not only on the potential, but also on a non-locality factor arising from the use of the eigenfunctions of the coordinate and momentum operators in the phase-space representation. Transition from the present formulation to the classical scattering is made.