Quantum mechanics electron spin

According to quantum mechanics, electrons in atoms occupy the allowed energy levels of atomic orbitals that are described by four quantum numbers the principal, the azimuthal, the magnetic, and the spin quantum numbers. The orbitals are usually expressed by the principal quantum numbers 1, 2, 3, —increasing from the lowest level, and the azimuthal quantum numbers conventionally eiqiressed by s (sharp), p (principal), d (diffuse), f (fundamental), — in order. For instance, the atom of oxygen with 8 electrons is described by (Is) (2s) (2p), where the superscript indicates the munber of electrons occupying the orbitals, as shown in Fig. 2-1. 

Hence, above a certain density, stellar matter manifests quite different properties which can only be described by quantum mechanics. Electrons in the medium begin to oppose gravity in a big way through their exaggerated individualism. In fact, elementary particles with half-integral spin, such as electrons, neutrons and protons, all obey the Pauli exclusion principle. This stipulates that a system cannot contain two elements presenting exactly the same set of quantum characteristics. It follows that two electrons with parallel spins cannot have the same velocity. 

M. Peric, B. Engels, S. D. Peyerimhoff, Theoretical Spectroscopy on Small Molecules Ab Initio Investigations of Vibronic Structure, Spin-Orbit Splittings and Magnetic Hyperfine Effects in the Electronic SPectra of Triatomic Molecules. In Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy,... 

Much as a pitched baseball is given a spin about its axis, so also does the electron spin about. However, the microscopic quantum mechanical electron behaves very differently from a baseball. Baseballs can be old, new, clean, dirty, and can differ from one another in myriad ways. On the other hand, all electrons are absolutely identical to one another. There is a subtler difference as well. Baseballs may spin rapidly, slowly, or in the case of a knuckleball, not at all. Every electron in the universe (about 1080 of them ) is spinning at exactly the same rate. The magnitude of electron spin is an intrinsic and immutable characteristic of the electron. Only the axis about which the electron spins can be changed. 

Spin The intrinsic angular momentum of an electron, nucleus, or other quantum-mechanical object. Spin-spin splitting Coupling, or the splitting of an NMR spectral peak by neighbors into more complex... 

According to quantum mechanics, electron is the fermion with 1/2 spin quantum number, which is the eigenvalue of spin angular momentum. In a system comprised of two electrons, the total spin quantum number S is 0 or 1. The spin wave function of the eigenstates S = 0 and S = 1 can be demonstrated by ... 

The resonance Raman spectrum of the class III ascorbate peroxidase isoenzyme II from tea leaves was consistent with an unusual five-coordinate quantum mechanically mixed-spin haem. Porphyrin-ring oxidation state marker bands were assigned for a range of Ru (Por)L2 complexes, where Por = TPP and substituted derivatives, L = PhNO. IR spectroscopy was used to probe the coupling of ground state molecular vibrations with low-energy electronic transitions of Ru(III, II) porphyrin dimer species. ... 

A central notion in quantum chemistry is a wave function. This is a function characterizing a state of the system. Therefore, it depends on the variables that are adequate for the given system. This means that the wave function has to depend, at least, on spatial coordinates describing motions of the particles in the investigated system. Moreover, the wave function depends on so-called spin variables (spin is an additional degree of freedom included a posteriori in nonrelativistic quantum mechanics). This spin dependency can be built into the wave function by introducing a spin function. For instance, for the electron with a label 1 its wave function depends on the spatial coordinates Xi,yi, and zi and is multiplied by the spin function a(oi) or /l(o i), where Oi is a spin variable. The spin functions must fiilfiU the following requirements ... 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The spin in quantum mechanics was introduced because experiments indicated that individual particles are not completely identified in terms of their three spatial coordinates [87]. Here we encounter, to some extent, a similar situation A system of items (i.e., distributions of electrons) in a given point in configuration space is usually described in terms of its set of eigenfunctions. This description is incomplete because the existence of conical intersections causes the electronic manifold to be multivalued. For example, in case of two (isolated) conical intersections we may encounter at a given point m configuration space four different sets of eigenfunctions (see Section Vni). 

The fact that an electron has an intrinsic spin comes out of a relativistic formulation of quantum mechanics. Even though the Schrodinger equation does not predict it, wave functions that are antisymmetric and have two electrons per orbital are used for nonreiativistic calculations. This is necessary in order to obtain results that are in any way reasonable. 

The most common description of relativistic quantum mechanics for Fermion systems, such as molecules, is the Dirac equation. The Dirac equation is a one-electron equation. In formulating this equation, the terms that arise are intrinsic electron spin, mass defect, spin couplings, and the Darwin term. The Darwin term can be viewed as the effect of an electron making a high-frequency oscillation around its mean position. 

HyperChem quantum mechanics calculations must start with the number of electrons (N) and how many of them have alpha spins (the remaining electrons have beta spins). HyperChem obtains this information from the charge and spin multiplicity that you specify in the Semi-empirical Options dialog box or Ab Initio Options dialog box. N is then computed by counting the electrons (valence electrons in semi-empirical methods and all electrons in fll) mitio method) associated with each (assumed neutral) atom and... 

In addition to total energy and gradient, HyperChem can use quantum mechanical methods to calculate several other properties. The properties include the dipole moment, total electron density, total spin density, electrostatic potential, heats of formation, orbital energy levels, vibrational normal modes and frequencies, infrared spectrum intensities, and ultraviolet-visible spectrum frequencies and intensities. The HyperChem log file includes energy, gradient, and dipole values, while HIN files store atomic charge values. 

For a quantum mechanical calculation, the single point calculation leads to a wave function for the molecular system and considerably more information than just the energy and gradient are available. In principle, any expectation value might be computed. You can get plots of the individual orbitals, the total (or spin) electron density and the electrostatic field around the molecule. You can see the orbital energies in the status line when you plot an orbital. Finally, the log file contains additional information including the dipole moment of the molecule. The level of detail may be controlled by the PrintLevel entry in the chem.ini file. 

In the classical picture of an electron orbiting round the nucleus it would not surprise us to discover that the electron and the nucleus could each spin on its own axis, just like the earth and the moon, and that each has an angular momentum associated with spinning. Unfortunately, although quantum mechanical treatment gives rise to two new angular momenta, one associated with the electron and one with the nucleus, this simple physical... 

From a quantum mechanical treatment the magnitude of the angular momentum due to the spin of one electron, whether it is in the hydrogen atom or any other atom, is given by... 

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. 

The electrons do not undergo spin inversion at the instant of excitation. Inversion is forbidden by quantum-mechanical selection rules, which require that there be conservation of spin during the excitation process. Although a subsequent spin-state change may occur, it is a separate step from excitation. 

The fourth quantum number is called the spin angular momentum quantum number for historical reasons. In relativistic (four-dimensional) quantum mechanics this quantum number is associated with the property of symmetry of the wave function and it can take on one of two values designated as -t-i and — j, or simply a and All electrons in atoms can be described by means of these four quantum numbers and, as first enumerated by W. Pauli in his Exclusion Principle (1926), each electron in an atom must have a unique set of the four quantum numbers. 

The Born interpretation of quantum mechanics tells us that s)dTds gives the chance of finding the electron in the spatial volume element dr and with spin coordinate between s and s + ds. Since probabilities have to sum to 1, we have... 

The Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

Quantum mechanical model, 138-139 Quantum number A number used to describe energy levels available to electrons in atoms there are four such numbers, 140-142,159q electron spin, 141 orbital, 141... 

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