Abstract “Identical particles and the helium atom” introduces bosons and fermions. Fermion states are expressed in terms of Slater determinants and the Pauli Principle. Helium is presented in such a way as to show what properties are and are not due to electron identity. Quantum states are described according as the space wave function is symmetric or antisymmetric under interchange of labels attached to the electrons. These in turn form singlet and triplet spin states when the electrons’ fermion identity is taken into account. Helium is an example of LS coupling, but a rather stunted example.
Abstract “Identical particles and the helium atom” introduces bosons and fermions. Fermion states are expressed in terms of Slater determinants and the Pauli Principle. Helium is presented in such a way as to show what properties are and are not due to electron identity. Quantum states are described according as the space wave function is symmetric or antisymmetric under interchange of labels attached to the electrons. These in turn form singlet and triplet spin states when the electrons’ fermion identity is taken into account. Helium is an example of LS coupling, but a rather stunted example.
This chapter discusses the properties of systems of identical particles. It presents the wave functions of a system composed of N identical and independent particles. From the properties of the wave functions, the chapter shows that identical particles can be characterized by symmetric as well as antisymmetric wave functions. As a result, it distinguishes two kinds of particles: bosons and fermions. Fermions have half integer spin, whereas bosons have integer spin. Since an arbitrary number of bosons can occupy the same state, atoms with integer spin can abruptly condensate into a single ground state when the temperature of the atoms goes below a certain critical value. In atoms, a limited number of electrons can occupy the same energy level. The Pauli principle prevents the energy states being occupied by an arbitrary number of electrons. Thus, the Pauli principle prevents all atoms having the same size and the same energy.
Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
This chapter addresses the quantum mechanics of identical particles, emphasizing the role of wavefunction symmetry in determining statistical behavior. Symmetric and antisymmetric wavefunctions are defined, and the implications of particle indistinguishability are examined, particularly regarding how it influences measurable outcomes and the concept of energy exchange degeneracy. The distinguishability of identical particles is discussed and shown to affect specific calculations. For systems where particle interactions can be neglected in the zero-order approximation, the total energy eigenfunction is expressed as a product of single-particle eigenfunctions. The Pauli exclusion principle is derived as a consequence of antisymmetrization. The relationship between symmetry type (bosonic or fermionic) and quantum statistics is elaborated. As an illustration, the ground state and the first excited s-state of the helium atom are analyzed under a simplified approximation. Metadata/Keywords: identical particles, symmetric wavefunction, antisymmetric wavefunction, Pauli exclusion principle, quantum statistics, energy exchange degeneracy, indistinguishability, helium atom, symmetry and statistics
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.
Bosons have symmetric wave function under particle exchange, while fermions have antisymmetric wave functions. This leads to differences in behavior of the two kinds of particles even in the gaseous state. Strong repulsive forces between atoms at close range are due to the Pauli principle that forbids two electrons to be in the same energy or position state. A repulsive Pauli pressure operates similarly in white dwarf stars. Bosons undergo Bose–Einstein condensation because of Bose statistics. The wave function symmetry of particles with spin can affect the magnetic states and the behavior of transport properties, for example, viscosity and thermal conductivity. The symmetry properties of bosons at a beam splitter are demonstrated in the Hong–Ou–Mandel effect or in a Michelson interferometer via the Hanbury Brown-Twiss effect. Researchers see boson bunching and fermion antibunching in experiments considered.
Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.