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Quantum Theory of Two Coupled Particles

Z. Ficek

Published Mar 3, 2017• 0 citations
Publisher

Abstract

The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.

DOI
10.1201/9781315364384-17

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The hydrogen atom can be considered a two-particle system, an electron and a proton, with the Coulomb potential acting between them. This chapter shows how to solve the Schrodinger equation for the wave function of the two-particle system. It introduces the coordinates of the center of mass, which will allow us to split the Schrodinger equation into two independent equations, one for the center of mass motion and another for the relative motion of the electron and proton. The left- hand side of the Schrodinger equation depends solely on the spatial variables, whereas the right-hand side depends solely on time. The wave function can be written as a product of the spatial and time-dependent parts. It is very similar in form to the Schrodinger equation for the hydrogen atom except it involves two Laplacians. Since the Laplacians act on separate variables, the chapter helps to solve the Schrodinger equation using the method of separate variables.
The time-dependent Schrödinger equation (TDSE) contains two variables: position and time. Therefore, the solutions are functions of both space and time (x and t, if we limit ourselves to particles moving in one dimension). It turns out that the easiest way to "solve" such an equation of two variables is to perform a separation of variables. We show that if the potential does not change with time, then certain special solutions (eigenfunctions) can be written as a product of a space function and a time function: Ψ(x,t) = ψ(x) f(t). By assuming a "separable" solution of this form, and plugging into the TDSE, the TDSE itself can be separated into two equations: the time-independent Schrödinger equation (TISE) with solution ψ(x), and the space-independent Schrödinger equation (SISE) with solution f(t). Furthermore, f(t) always equals e -iωt , while ψ(x) depends on the potential. So, for potentials that do not vary with time, we need only solve the much simpler TISE in order to investigate the energy, momentum, and so on of the particle. This is why, for the remainder of this book, we only use the TISE.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge −e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the center of mass coordinate and the relative coordinates. In the hydrogen atom the attractive force exerted by the proton on electron prevents it from escape. The potential energy V of the system is −e2/r where r is the distance between the electron and the proton. Suppose the nucleus is held fixed at the origin of coordinates. V (r) is independent of θ and φ, that is, insensitive to the direction of r and depends only on the scalar magnitude r. The potential energy remains invariant under rotation and is said to be spherically symmetric. Figure 12.1 shows V (r) as a function of r. In this chapter the hydrogen atom problem in threedimension is solved and momentum eigenfunction is obtained. The problem in D-dimension where D may be noninteger is analysed. The magnetic field produced by a hydrogen atom is worked out. Finally, the system in parabolic coordinates is discussed.
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.
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, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
The problem of the motion of a particle attracted by two fixed centers of force according to the Coulomb force law can be treated by classical mechanics and has been used in quantum theory by Pauli and Messen for a theory of the hydrogen molecule ion.1 In the quantum mechanics, where the energy levels are determined as the “eigenwerte” of Schrodinger’s equation, the variables are separable and the boundary value problem is easily set up. But thus far a satisfactory treatment of the differential equations involved is lacking. Burrau2 has recently carried out a numerical integration of the problem for the lowest energy level of an electron moving under the influence of two fixed centers of Coulomb attraction as a function of the distance apart of these centers. In this paper, Burrau’s data are used to give a semi-quantitative discussion of the neutral hydrogen molecule. His values are: Nuclear separation 1.0 1.3 1.6 1.8 2.0 2.2 2.4 2.95 Electronic energy 2.896 2.648 2.436 2.309 2.204 2.109 2.025 1.836 The unit of separation is the Bohr l1 orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen.