Wave Functions for One and Two Atom Systems.

Keith Hannabuss

Abstract One often wants to study systems, such as the helium atom of Sections 12.3 and 14.2, that contain more than one particle. We describe two-particle systems by a wave function, ψ (r1, r2), where r1 is the position of the first particle, and r2 that of the second. If we interchange the two particles, then the wave function becomes ψ (r2, r1). (16.1)

Jens C. Zorn, E. A. Zorn

Wei-Can Syu, Da-Shin Lee, Chi-Yong Lin

We theoretically study binary Bose-Einstein condensates trapped in a single-well harmonic potential to probe the dynamics of collective atomic motion. The idea is to choose tunable scattering lengths through Feshbach resonances such that the ground-state wave function for two types of the condensates are spatially immiscible where one of the condensates, located at the center of the potential trap, can be effectively treated as a potential barrier between bilateral condensates of the second type of atoms. In the case of small wave-function overlap between bilateral condensates, one can parametrize their spatial part of the wave functions in the two-mode approximation together with the time-dependent population imbalance $z$ and the phase difference $\ensuremath{\phi}$ between two wave functions. The condensate in the middle can be approximated by a Gaussian wave function with the displacement of the condensate center $\ensuremath{\xi}$. As driven by the time-dependent displacement of the central condensate, we find the Josephson oscillations of the collective atomic motion between bilateral condensates as well as their anharmonic generalization of macroscopic self-trapping effects. In addition, with the increase in the wave-function overlap of bilateral condensates by properly choosing tunable atomic scattering lengths, the chaotic oscillations are found if the system departs from the state of a fixed point. The Melnikov approach with a homoclinic solution of the derived $z, \ensuremath{\phi}$, and $\ensuremath{\xi}$ equations can successfully justify the existence of chaos. All results are consistent with the numerical solutions of the full time-dependent Gross-Pitaevskii equations.

T. M. Kalotas, Matt Lloyd, L. M. Delves

Nevill Francis Mott

Abstract Section I.—The exclusion principle of Pauli was introduced into the old quantum theory as an empirical fact that had been brought to light in the ordering of spectra. The new mechanics has provided some sort of explanation of the principle, for in a closed system with many electrons the mathematically possible stationary states can be separated into a number of groups, with the property that transitions between stationary states in different groups cannot occur. One of these group is made up of the stationary states corresponding occur. One of the these groups is made up of the stationary states corresponding to wave functions that are antisymmetrical in the co-ordinates of the electrons. Apart from accidental degeneracies, the stationary states in different groups have different energy values. The exclusion principle then states that only energy values belonging to the antisymmertical group are found in nature. The exclusion principle has had great success not only in explaining the spectra of helium and of more complicated atoms, but also, under the form of the Fermi-Dirac statistics, in accounting for metallic conduction and ferromagnetism. In all these phenomena we are dealing with systems is stationary states, possessing energy values which are discrete, although they may lie very close together. Now, as was first emphasised by Oppenheimer, we must also use antisymmetrical wave-functions to describe aperiodic phenomena, such as the collision between an electron and an atom. If we do not, we obtain probabilities for the formation of atoms whose wave-funtions are not antisymmetrical, as we shall show in section 4, where we consider the collision between an electron and a helium atom. A helium atom described by a symmetrical wave-function would show a singlet series in palce of the observed triplets and triplet series in place of the observed singlets. The wave-functions of open systems are essentially degenerate; the symmetical and antisymmetrical solution are not separated from one another by a finite energy difference; but for any arbitrary value of the energy (and of the other integrals of the motion) we can form a symmetrical and an antisymmetrical solution. This is somewhat fundamental difference between open and closed systems. For closed systems containing two electrons there exist only the symmetrical and the antisymmetical solution; but for open systems we might take any combination of the two. In fact, to describe an observable phenomenon such as a collision, the wave-function that it would first occur to us to use is a combination of the two. To fix our ideas we shall discuss the collision between two electrons. Our arguments could equally well be applied to the collision between an electron and a hydrogen atom, the problem originally discussed by Born; but the former is the simpler case, and perhaps illustrates our theory better.

Charles Schwartz