Antisymmetrized wave function helium

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

Particles with antisymmetric wave function are called fermions - they have to obey the Pauli exclusion principle. Apart from the familiar electron, proton and neutron, these include the neutrinos, the quarks (from which protons and neutrons are made), as well as some atoms like helium-3. All fermions possess "half-integer spin", meaning that they possess an intrinsic angular momentum whose value is hbar = li/2 pi (Planck s constant divided by 27i) times a half-integer (1/2, 3/2, 5/2, etc ). In the theory of quantum mechanics, fermions are described by "antisymmetric states", which are explained in greater detail in the article on identical particles. 

Fia. 29-2.—Levels for configurations U, 1 2 , and 1 2p of the helium atom. spin-symmetric wave functions o. spin-antisymmetric wave functions... 

The analogy is even closer when the situation in oxygen is compared with that in excited configurations of the helium atom summarized in Equations (7.28) and (7.29). According to the Pauli principle for electrons the total wave function must be antisymmetric to electron exchange. 

Equation (7.23) expresses the total electronic wave function as the product of the orbital and spin parts. Since J/g must be antisymmetric to electron exchange the Ig and Ag orbital wave functions of oxygen combine only with the antisymmetric (singlet) spin wave function which is the same as that in Equation (7.24) for helium. Similarly, the Ig orbital wave function combines only with the three symmetric (triplet) spin wave functions which are the same as those in Equation (7.25) for helium. 

The first combination is symmetric, and the second is antisymmetric with respect to permutation. In the ground state of the helium atom both electrons occupy the same spatial orbital (Is), so that they must have the antisymmetric spin function for the total wave function to be antisymmetric they therefore form a singlet spin state. In order to find a helium atom with a triplet spin state (so-called spins parallel), the spatial part of the wave function must be antisymmetric with respect to interchange. 

Whilst we are discussing the Pauli principle, it is worthwhile to introduce a further method of expressing the antisymmetric nature of the electronic wave function, namely, the Slater determinant [1], Since both electrons in the ground state of the helium atom occupy the same space orbital, the wave function may be written in the form... 

The question as to which types of wave functions actually occur in nature can at present be answered only by recourse to experiment. So far all observations which have been made on helium atoms have shown them to be in antisymmetric states. We accordingly make the additional postulate that the wave function re-presenting an actual state of a system containing two or more electrons must be completely antisymmetric in the coordinates of the electrons that is, on interchanging the coordinates of any two electrons it must change its sign. This is the statement of the Pauli exclusion principle in wave-mechanical language. 

Those for the latter, called triplet states, are obtained by multiplying the antisymmetric orbital wave functions by the three symmetric spin functions.1 The spin-orbit interactions which we have neglected cause some of the triplet levels to be split into three adjacent levels. Transitions from a triplet to a singlet level can result only from a perturbation involving the electron spins, and since interaction of electron spins is small for light atoms, these transitions are infrequent no spectral line resulting from such a transition has been observed for helium. 

So far we have not taken into consideration the spins of the electrons. On doing this we find, exactly as for the helium atom, that in order to make the complete wave functions antisymmetric in the electrons, as required by Pauli s principle, the orbital wave functions must be multiplied by suitably chosen spin functions, becoming... 

By an argument identical with that given in Section 295 for the electrons in the helium atom we know that a system containing two identical protons can be represented either by wave functions which are symmetric in the protons or by wave functions which are antisymmetric in the protons. Let us assume... 

In the preceding sections we have discussed systems containing two nuclei, each with one stable orbital wave function (a Is function), and one, two, three, or four electrons. We have found that in each case an antisymmetric variation function of the determinantal type constructed from atomic orbitals and spin functions leads to repulsion rather than to attraction and the formation of a stable molecule. For the four-electron system only one such wave function can be constructed, so that two normal helium atoms, with completed K shells, interact with one another in this way. For the other systems, on the other hand, more than one function of this type can be set up (the two corresponding to the structures H- H+ and H+ H for the hydrogen molecule-ion, for example) and it is found on solution of the secular equation that the correct approximate wave functions are the sum and difference of these, and that in each case one of the corresponding energy curves leads to attraction of the atoms and the formation of a stable bond. We call the bonds involving two orbitals (one for each nucleus) and one, two, and three electrons the one-electron bond, the electron-pair bond, and the three-electron bond, respectively. 

The two-electron spin eigenfunctions consist of the symmetric functions a(l)o (2), /3(l)/3(2), and [a(l)/3(2) -I- j8(l)a(2)]/ V and the antisymmetric function [a(l)/3(2) —/3(l)a(2)]/ V. For the helium atom, each stationary state wave function is the product of a symmetric spatial function and an antisymmetric spin function or an antisymmetric spatial function and a symmetric spin function. Some approximate helium-atom wave functions are Eqs. (10.26) to (10.30). 

We begin with the ground state of He2. The separated helium atoms each have the ground-state configuration Is. This closed-subshell configuration does not have any unpaired electrons to form valence bonds, and the VB wave function is simply the antisymmetrized product of the atomic-orbital functions ... 

John C. Slater (1901-1976), American physicist, for 30 years a professor and dean at the Physics Department of the Massachusetts Institute of Technology, then at the University of Florida, Gainesville, where he participated in the Quantum Theory Project. His youth was in the stormy period of the intense development of quantum mechanics, and he participated vividly in it. For example, in 1926-1932, he published articles on the ground state of the helium atom, the screening constants (Slater orbitals), the antisymmetrization of the wave function (Slater determinant), and the algorithm for calculating the integrals (the Slater-Condon rules). 

In the ground state of the helium atom the spatial part of the wave-function, i/, j(I)v/jj.(2), is symmetric with respect to interchange of electrons, and therefore the total wavefunction will be antisymmetric only if the spin part of the wavefunction is also antisymmetric. Thus, the overall wavefunction must be ... 

Returning to the helium atom, suppose that electrons I and 2 occupy the two states 0 and b, respectively. The wave function in Equation (4.14) would be unacceptable because the two electrons are distinguishable upon interchange. Taking linear combinations of the product of the two states, however, provides two acceptable wave functions because now the electrons become indistinguishable upon interchange, as shown in Equations (4.15) and (4.16). The former of these equations is symmetric with respect to electron exchange (it yields the same mathematical expression), while the latter is antisymmetric. 

We now include spin in the He zeroth-order ground-state wave function. The function 15(1)15(2) is symmetric with respect to exchange. The overall electronic wave function including spin must be antisymmetric. Hence we must multiply the symmetric space function 15 (1) 15 (2) by an antisymmetric spin function. There is only one antisymmetric two-electron spin function, so the ground-state zeroth-order wave function for the helium atom including spin is... 

Now consider the excited states of helium. We found the lowest excited state to have the zeroth-order spatial wave function 2 / [l5 (l)25 (2) - 2s(l)ls(2)] [Eq. (9.103)]. Since this spatial function is antisymmetric, we must multiply it by a symmetric spin function. We can use any one of the three symmetric two-electron spin functions, so instead of the nondegenerate level previously found, we have a triply degenerate level with the three zeroth-order wave functions... 

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