Many-particle operator eigenfunction

In conventional quantum mechanics, a wavefunction d ribing the ground or excited states of a many-particle system must be a simultaneous eigenfunction of the set of operators that commute with the Hamiltonian. Thus, for example, for an adequate description of an atom, one must introduce the angular momentum and spin operators L, S, L, and the parity operator H, in addition to the Hamiltonian operator. 

Operators, Eigenfunctions and Eigenvalues. Inasmuch as all matter is (from a chemist s point of view) a combination of atomic nuclei and electrons, it is clear that the motions, or rather the distributions, of these particles must explain all observable properties, and there are many of these for example the energy of a molecule, its dipole moment, and its shape. In the end the observable property must be a number (with the relevant units) and these values are called eigenvalues . How can one obtain them ... 

The many-body perturbation theory is developed in terms of some set of single particle states, Pi which are eigenfunctions of some single-particle operator, /,... 

In a many-electron system, one must combine the spin functions of the individual electrons to generate eigenfunctions of the total Sz =Li Sz(i) ( expressions for Sx = j Sx(i) and Sy = j Sy(i) also follow from the fact that the total angular momentum of a collection of particles is the sum of the angular momenta, component-by-component, of the individual angular momenta) and total S2 operators because only these operators commute with the full Hamiltonian, H, and with the permutation operators Pjj. No longer are the individual S2(i) and Sz(i) good quantum numbers these operators do not commute with Pjj. 

Electrons (and many other particles) have associated with them an intrinsic angular momentum that has come to be called spin . One of the greatest successes of relativistic quantum mechanics is that spin is seen to arise naturally within the relativistic formalism, and does not need to be added post facto as it is in non-relativistic treatments. As with orbital angular momentum, spin angular momentum has x, y, and z components, and the operators 5, Sy, and S, together with orthonormal eigenfunctions a and fi of electron spin, are defined from ... 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

The approach is rather different from that adopted in most books on quantum chemistry in that the Schrbdinger wave equation is introduced at a fairly late stage, after students have become familiar with the application of de Broglie-type wavefunctions to free particles and particles in a box. Likewise, the Hamiltonian operator and the concept of eigenfunctions and eigenvalues are not introduced until the last two chapters of the book, where approximate solutions to the wave equation for many-electron atoms and molecules are discussed. In this way, students receive a gradual introduction to the basic concepts of quantum mechanics. 

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