Symmetry Properties of Wave Functions in Magnetic Crystals

1962 ◽  
Vol 127 (2) ◽  
pp. 391-404 ◽  
Author(s):  
J. O. Dimmock ◽  
R. G. Wheeler
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

The molecular orbital theory of chemical valency. XII. The excited states of diatomic molecules

1953 ◽  
Vol 216 (1124) ◽  
pp. 1-10 ◽  
Keyword(s):  
Wave Equation ◽  
Excited States ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Wave Functions ◽  
Triplet States ◽  
Orbital Theory ◽  
Lennard Jones ◽  
Singlet States ◽  
Symmetry Properties

A method is developed for the solution of the wave equation for two electrons in the presence of two centres. The work of Lennard-Jones & Pople (1951) on the ground state of such a system is generalized so as to apply to all the excited states. Full advantage is taken of the symmetry properties of the wave functions, both in three-dimensional and six-dimensional space, to reduce the wave equation to a number of component parts, each of a particular symmetry type. This leads to sets of equations with characteristic symmetry properties appropriate to singlet states and triplet states, whether even or odd, positive or negative in the standard notation ( 1 ∑ - g ).

scholarly journals Calculation of One-Electron Wave Functions and Energy Levels of N-Butane Molecule on the Basis of Slater Atomic Orbitals

2021 ◽  
Vol 75 (3) ◽  
pp. 229-233
Author(s):  
Faig Pashaev ◽  
Arzuman Gasanov ◽  
Musaver Musaev ◽  
Ibrahim Abbasov
Keyword(s):  
Group Theory ◽  
Energy Levels ◽  
Wave Functions ◽  
Electron Wave ◽  
Hamilton Operator ◽  
Atomic Orbitals ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Space Symmetry ◽  
Polyatomic Systems

Abstract It is known that the application of the group theory greatly simplifies the problems of polyatomic systems possessing to any space symmetry. The symmetry properties of such systems are their most important characteristics. In such systems, the Hamilton operator is invariant under unitary symmetry transformations and rearrangements of identical particles in the coordinate system. This allows to obtain information about the character of one-electron wave functions — molecular orbitals — the considered system, i.e. to symmetrise the original wave functions without solving the Schrödinger equation.

Electronic levels in simple conjugated systems, I. Configuration interaction in cyclobutadiene

1950 ◽  
Vol 202 (1071) ◽  
pp. 498-506 ◽  
Keyword(s):  
Molecular Orbital ◽  
Configuration Interaction ◽  
Energy Levels ◽  
Valence Bond ◽  
Wave Functions ◽  
The Other ◽  
Orbital Theory ◽  
Valence Bond Theory ◽  
Symmetry Properties ◽  
Bond Theory

In the simplest cyclic system of π-electrons, cyclobutadiene, a non-empirical calculation has been made of the effects of configuration interaction within a complete basis of antisymmetric molecular orbital configurations. The molecular orbitals are made up from atomic wave functions and all the interelectron repulsion integrals which arise are included, although those of them which are three- and four-centre integrals are only known approximately. In this system configuration interaction is a large effect with a strongly differential action between states of different symmetry properties. Thus the 1 A 1g state is several electron-volts lower than the lowest configuration of that symmetry, whereas for 1 B 1g the comparable figure is about one-tenth of an electron-volt. The other two states examined, 1 B 2g and 3 A 2g are affected by intermediate amounts. The result is a drastic change in the energy-level scheme compared with that based on configuration wave functions. Neither the valence-bond theory nor the molecular orbital theory (in which the four states have the same energy) gives a satisfactory account of the energy levels according to these results. One conclusion from the valence-bond theory which is, however, confirmed, is the somewhat unexpected one that the non-totally symmetrical 1 B 2g state is more stable than the totally symmetrical 1 A 1g . On the other hand, it is clear that the valence-bond theory, with the usual value for its exchange integral, grossly exaggerates the resonance splitting of the states, giving separations between them several times too great. Thus the valence-bond theory leads to large values of the resonance energy (larger, per π-electron, than in benzene) and so associates with the molecule a considerable π-electron stabilization. This expectation has no support in the present more detailed and non-empirical calculations.

scholarly journals The collision between two electrons

1930 ◽  
Vol 126 (801) ◽  
pp. 259-267 ◽  
Keyword(s):  
Wave Equation ◽  
Classical Theory ◽  
Wave Functions ◽  
The Other ◽  
Exclusion Principle ◽  
Wave Mechanics ◽  
Inverse Square Law ◽  
Symmetry Properties ◽  
Α Particles

It is well-known that the problem of the collision between two particles interacting according to the inverse square law is exactly soluble on the wave mechanics, and that the solution yields the same scattering laws as the classical theory. If, however, the two particles are identical, e.g. , two electrons or two α-particles, this is not necessarily the case; for the wave functions used must be antisymmetrical or symmetrical in the co-ordinates of the two particles; and this may affect the scattering laws. In this paper we shall discuss the collision between two particles possessing spin, such as electrons, and also between two particles without spin, such as α-particles. Assuming an inverse square law force between the particles, and neglecting the actual spin forces, we shall deduce from the symmetry properties of the wave functions a scatter­ing law differing considerably from the classical. We shall also mention the various methods by which the effect could be observed, and give some experi­mental evidence in its favour. The application of the exclusion principle to collision problems has been discussed by the author in a previous paper. Suppose we wish to describe the motion of two particles interacting in any field of force. We obtain a solution w (r 1 r 2 ) of the wave equation, where r 1 refers to the position of the first particle, and r 2 to that of the second. If we did not use antisymmetrical wave functions, we should argue that the probability that the first particle should be at r 1 and the second at r 2 would be | w (r 1 r 2 )| 2 , and therefore the probability that one particle should be at r 1 and the other at r 2 would be | w (r 1 r 2 )| 2 + | w (r 2 r 1 )|

Theory of Brillouin Zones and Symmetry Properties of Wave Functions in Crystals

1936 ◽  
Vol 50 (1) ◽  
pp. 58-67 ◽  
Author(s):  
L. P. Bouckaert ◽  
R. Smoluchowski ◽  
E. Wigner
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

Symmetry properties of wave functions in crystals

1949 ◽  
Vol 29 (3) ◽  
Author(s):  
T. Venkatarayudu ◽  
T. S. G. Krishnamurty
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

Symmetry properties of wave functions in crystals

1949 ◽  
Vol 29 (3) ◽  
Author(s):  
T. Venkatarayudu ◽  
T. S. G. Krishnamurty
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

NOTES ON THE SYMMETRY PROPERTIES OF NUCLEAR WAVE FUNCTIONS

Claude Bloch ◽  
1975 ◽  
pp. 57-89
Author(s):  
Claude BLOCH
Keyword(s):  
Wave Functions ◽  
Symmetry Properties

Symmetry properties of n-pion wave functions

1964 ◽  
Vol 26 (3) ◽  
pp. 418-441 ◽  
Author(s):  
M Grynberg ◽  
Z Koba
Keyword(s):  
Wave Functions ◽  
Symmetry Properties
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