Quantum mechanics helium atom

Liquid Helium-4. Quantum mechanics defines two fundamentally different types of particles bosons, which have no unpaired quantum spins, and fermions, which do have unpaired spins. Bosons are governed by Bose-Einstein statistics which, at sufficiently low temperatures, allow the particles to coUect into a low energy quantum level, the so-called Bose-Einstein condensation. Fermions, which include electrons, protons, and neutrons, are governed by Fermi-DHac statistics which forbid any two particles to occupy exactly the same quantum state and thus forbid any analogue of Bose-Einstein condensation. Atoms may be thought of as assembHes of fermions only, but can behave as either fermions or bosons. If the total number of electrons, protons, and neutrons is odd, the atom is a fermion if it is even, the atom is a boson. 

In the application of quantum mechanics to the helium atom, the following integral / arises and needs to he evaluated... 

H. A. Bethe and E. E. Salpeter (1957) Quantum Mechanics of One- and Two-Electron Atoms (Springer, Berlin Academic Press, New York reprinted by Plenum, New York, 1977). A comprehensive non-relativistic and relativistic treatment of the hydrogen and helium atoms with and without external fields. 

Helium has a very large quantum mechanical zero-point energy owing to its small atomic mass (E0 h1 /2mV2 2 where V is the atomic volume and m atomic mass). 

In previous sections, the basis for applying quantum mechanical principles has been illustrated. Although it is possible to solve exactly several types of problems, it should not be inferred that this is always the case. For example, it is easy to formulate wave equations for numerous systems, but generally they cannot be solved exactly. Consider the case of the helium atom, which is illustrated in Figure 2.7 to show the coordinates of the parts of the system. 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

The density of He I at the boiling point at 1 atm is 125 kg m 3 and the viscosity is 3 x 10 6 Pa s. As we would anticipate, cooling increases the viscosity until He II is formed. Cooling this form reduces the viscosity so that close to 0 K a liquid with zero viscosity is produced. The vibrational motion of the helium atoms is about the same or a little larger than the mean interatomic spacing and the flow properties cannot be considered in classical terms. Only a quantum mechanical description is satisfactory. We can consider this condition to give the limit of De-+ 0 because we have difficulty in defining a relaxation when we have the positional uncertainty for the structural components. 

The concept of resonance was introduced into quantum mechanics by Heisenberg16 in connection with the discussion of the quantum states of the helium atom. He pointed out that a quantum-mechanical treatment somewhat analogous to the classical treatment of a system of resonating coupled harmonic oscillators can be applied to many systems. The resonance phenomenon of classical mechanics is observed, for example, for a system of two tuning forks with the same characteristic frequency of oscillation and attached to a common base, hich... 

The last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

Fig. 32 shows on the left a conventional, localized domain model of the electronic environment of an atom that satisfies the Octet Rule. Each domain is occupied by two electrons. It is well known, however, that the assumption of two electrons per orbital is unnecessarily restrictive 27>122>. Better energies are obtained in quantum mechanical calculations if different orbitals are used for electrons of different spins, a fact first demonstrated in quantitative calculations on helium by Hylleraas 123> and Eckart 124>. Later, this "split-orbital method was applied to 71-electron systems 27,125) Its general application to chemical systems has been developed by Linnett 126>. 

How do these various atomic orbitals relate to the spatial distribution of electrons in molecules A molecule contains more than one atom (except for molecules like helium or neon), and certain electrons can move between the atoms —this interatomic motion is crucial for holding the molecule together. Fortunately, the spatial localization of electrons in molecules can be described using suitable linear combinations of the spatial distributions of electrons in various atomic orbitals centered about the nuclei involved. In fact, molecular orbital theory is concerned with giving the correct quantum-mechanical, or wave-mechanical, description... 

Why go to the trouble The fascination with hydrogen-like atoms starts with the same fascination physicists have with hydrogen itself. The reason is simphcity— just two particles are involved. For such a simple system, physicists can apply basic physical theories such as quantum mechanics, relativity, and quantum electrodynamics (QED) with minimal assumptions compromising the outcome. Often, for two-particle s) tems, physicists can solve the mathematical equations that arise exactly. This is not the case with the next simplest atom, helium—a three-particle system— and a relatively simple atom such as carbon confronts physical theory with formidable problems. The fascination is extended... 

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