Bound States of the Hydrogen Atom

In this section we discuss the bound states of the hydrogen atom. These are states where the electron stays with the nucleus. In contrast, an electron with lots of energy could simply speed past the nucleus without getting trapped. Such an unbound electron does not stop long enough form a coherent atom hence in our study of the atom, it makes sense to study only the bound states. 

At long last, it is time to appeal to the Schrodinger operator 

The Schrodinger operator can be used to make predictions about measurements of the energy of the electron in a hydrogen atom. For example, suppose (j) e satisfies the Schrodinger eigenvalue equation 

Proposition 8.14 Each negative eigenvalue E of the Schrodinger operator has a finite number of linearly independen t eigenfunctions. 

The proof depends on Proposition A.3 of Appendix A, which ensures that all solutions of the Schrodinger equation can be approximated by linear combinations of solutions where the radial and angular variables have been 

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We can put these representations (one for each eigenvalue of the Schrodinger operator) together to form a representation of su(2) on the vector space of bound states of the hydrogen atom. We will see in Section 8.6 that there is a physically natural representation of the larger Lie algebra 5o(4) = 5m(2) 5m(2) on the set of bound states of the hydrogen atom. 

Both Equations (64) and (65) have the same form and they can be interpreted as Schrodinger equations in circular-like coordinates for harmonic oscillators [33], as indicated by their respective kinetic energy and quadratic potential energy terms. The identification and interpretation are even more convincing if we parametrize the negative energy of the bound states of the hydrogen atom as... 

Again we introduce dimensionless variables r = jSp and E = — (l/n )(ft /2/,ij8 ), where n is a parameter characterizing the energy. Note that the energy has been chosen as a negative quantity this implies that the discussion will deal only with the bound states of the hydrogen atom. The zero of potential energy is at r oo, in which state the two particles move independently. The equation becomes... 

We now consider the bound states of the hydrogen atom, with < 0. In this case, the quantity in parentheses in (6.66) is positive. Since we want the wave functions to remain finite as r goes to infinity, we prefer the minus sign in (6.66), and in order to get a two-term recursion relation, we make the substitution... 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

Indeed, the first comment on the ground state of the hydrogen atom in a spherical enclosure contains the idea and the methodology, which have proven to be valid for all the bound states in the successive situations of confinement, including the removal of the degeneracy of their energy levels by far-away boundaries as well as the quantitative evaluation of the positions of each boundary for which the specific states have zero energy. 

A procedure that is commonly used in such cases is to construct trial functions for molecules from the exact eigenfunctions that apply to the atoms from which the molecules are formed. The eigenfunction for the Is state of the hydrogen atom is given by equation (1.27) with Z = 1, and we may therefore write the eigenfunction for the electron bound to proton A as... 

One last point must be mentioned. So far we have assumed that the complete sets have been discrete the expansions have been written in terms of functions ipi, each with a coefficient Ci, where i is an integer (or set of integers) enumerating the chosen functions and their coefficients. It is important to realize that not all complete sets are discrete and that not all infinite discrete sets are complete. The wavefunctions of the bound (negative-energy) states of the hydrogen atom form an infinite discrete set, but the set is not complete. The complete set includes the continuum of wavefunctions that describe a free electron (positive energy) scattered by the nucleus. If we wish to expand a function in terms... 

There are also unbound states for which the energy is positive. The unbound states are quite different from the bound ones, in that they are finite at infinity and at the origin. There is a continuous range of positive energies, and these correspond to ionization of the hydrogen atom. We will not need to consider the unbound states in this text. 

Suppose we take an electron out of one of the hydrogen atoms, leaving behind a H ion. The electron will be re-attracted to the ion by a potential - e /R. It will be bound in the ground state for this potential, which has a radius Uh = h /me. If, however, there are enough ionized electrons in the lattice, they will screen the electron from the positive ion core, according to a potential - (e /R) exp(- kR), where X is a screening parameter in the Thomas-Fermi modeP X... 

We can use the representation theory of the Lie algebra 50(4) along with the stunning fact that there is a representation of 5o(4) on the space of bound states of the Schrodinger operator with the Coulomb potential to make a satisfying prediction about the dimensions of the shells of the hydrogen atom and the energy levels of these shells. 

Until recently there was no firm theoretical evidence that a positron could bind to any atom other than positronium it had been rigorously proved by Armour (1978, 1982) that it cannot bind to atomic hydrogen, and the evidence that it cannot bind to helium is overwhelming. The most likely candidates were the highly polarizable alkali atoms, and states of the positron-atom system below the positron-atom scattering threshold do indeed exist. However, they were all believed to lie above the threshold for positronium scattering by the corresponding positive ion, and were therefore not true bound states. 

Abstract. Muonium is a hydrogen-like system which in many respects may be viewed as an ideal atom. Due to the close confinement of the bound state of the two pointlike leptons it can serve as a test object for Quantum Electrodynamics. The nature of the muon as a heavy copy of the electron can be verified. Furthermore, searches for additional, yet unknown interactions between leptons can be carried out. Recently completed experimental projects cover the ground state hyperfine structure, the ls-2s energy interval, a search for spontaneous conversion of muonium into antimuonium and a test of CPT and Lorentz invariance. Precision experiments allow the extraction of accurate values for the electromagnetic fine structure constant, the muon magnetic moment and the muon mass. Most stringent limits on speculative models beyond the standard theory have been set. 

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