5. Multiple-Particle Systems

AbstractSo far, only wave functions for single particles have been discussed. This chapter explains how the ideas generalize to more particles. The basic idea is simple: you just keep adding more and more arguments to your wave function.

That simple idea will immediately be used to derive a solution for the hydrogen molecule. The chemical bond that keeps the molecule together is a two-electron one. It involves sharing the two electrons in a very weird way that can only be described in quantum terms.

Now it turns out that usually chemical bonds involve the sharing of two electrons like in the hydrogen molecule, not just one as in the hydrogen molecular ion. To understand the reason, simple approximate systems will be examined that have no more than two different states. It will then be seen that sharing lowers the energy due to

twilightterms. These are usually more effective for two-electron bonds than for single electron-ones.Before systems with more than two electrons can be discussed, a different issue must be addressed first. Electrons, as well as most other quantum particles, have intrinsic angular momentum called

spin. It is quantized much like orbital angular momentum. Electrons can either have spin angular momentumor in a given direction. It is said that the electron has spin . Photons can have angular momentum, 0, orin a given direction and have spin 1. Particles with half-integer spin like electrons are called fermions. Particles with integer spin like photons are called bosons. For quantum mechanics there are two consequences. First, it means that spin must be added to the wave function as an uncertain quantity in addition to position. That can be done in various equivalent ways. Second, it turns out that there are requirements on the wave function depending on whether particles are bosons or fermions. In particular, wave functions must stay the same if two identical bosons, say two photons, are interchanged. Wave functions must change sign when any two electrons, or any other two identical fermions, are interchanged.

This so-called antisymmetrization requirement is usually not such a big deal for two electron systems. Two electrons can satisfy the requirement by assuming a suitable combined spin state. However, for more than two electrons, the effects of the antisymmetrization requirement are dramatic. They determine the very nature of the chemical elements beyond helium. Without the antisymmetrization requirements on the electrons, chemistry would be something completely different. And therefore, so would all of nature be. Before that can be properly understood, first a better look is needed at the ways in which the symmetrization requirements can be satisfied. It is then seen that the requirement for fermions can be formulated as the so-called Pauli exclusion principle. The principle says that any number

of identical fermions must occupy different quantum states. Fermions are excluded from entering the same quantum state. At that point, the atoms heavier than hydrogen can be properly discussed. It can also be explained why atoms prevent each other from coming too close. Finally, the derived quantum properties of the atoms are used to describe the various types of chemical bonds.

- 5.1 Wave Function for Multiple Particles
- 5.2 The Hydrogen Molecule
- 5.2.1 The Hamiltonian
- 5.2.2 Initial approximation to the lowest energy state
- 5.2.3 The probability density
- 5.2.4 States that share the electrons
- 5.2.5 Variational approximation of the ground state
- 5.2.6 Comparison with the exact ground state

- 5.3 Two-State Systems
- 5.4 Spin
- 5.5 Multiple-Particle Systems Including Spin
- 5.5.1 Wave function for a single particle with spin
- 5.5.2 Inner products including spin
- 5.5.3 Commutators including spin
- 5.5.4 Wave function for multiple particles with spin
- 5.5.5 Example: the hydrogen molecule
- 5.5.6 Triplet and singlet states

- 5.6 Identical Particles
- 5.7 Ways to Symmetrize the Wave Function
- 5.8 Matrix Formulation
- 5.9 Heavier Atoms
- 5.9.1 The Hamiltonian eigenvalue problem
- 5.9.2 Approximate solution using separation of variables
- 5.9.3 Hydrogen and helium
- 5.9.4 Lithium to neon
- 5.9.5 Sodium to argon
- 5.9.6 Potassium to krypton
- 5.9.7 Full periodic table

- 5.10 Pauli Repulsion
- 5.11 Chemical Bonds