Electron behaving as waves

Wave mechanics is based on the fundamental principle that electrons behave as waves (e.g., they can be diffracted) and that consequently a wave equation can be written for them, in the same sense that light waves, soimd waves, and so on, can be described by wave equations. The equation that serves as a mathematical model for electrons is known as the Schrodinger equation, which for a one-electron system is... 

Within the atom, electrons behave as waves. Different shapes and sizes of these waves are possible around the nucleus. These are known as orbitals . The simplest orbital is spherical, but more complex orbital shapes are possible. Any orbital, irrespective of its size or shape, can hold a maximum of two electrons. 

Electrons Behave as Waves Standing Waves in One and Two Dimensions Standing Waves in Three Dimensions Atomic Orbitals Mixing Atomic Orbitals into Molecular Orbitals Bonding and Antibonding MOs of Hydrogen... 

Beyond predicting what types of bonds are present in a molecule, however, the Lewis structure tells us fairly little about the details of a chemical bond. To understand just how electrons are shared, we must reahze that electrons behave as waves and that when waves overlap, they interfere with each other. When the waves buildup they are said to interfere constructively and the chemical implication is the formation of a chemical bond. Overlap can be achieved in more than one way, so that we can distinguish types of bonds as either sigma or pi bonds. 

The electron behaves as a standing wave with an integral number of half wavelengths fitting into the one-dimensional box, with boundary conditions... 

If an electron behaves as a wave as it moves in a hydrogen atom, a stable orbit can result only when the circumference of a circular orbit contains a whole number of waves. In that way, the waves can join smoothly to produce a standing wave with the circumference being equal to an integral number of wavelengths. This equality can be represented as... 

This discussion extends to electrons, and small particles in general, what we have already found for photons, that is, the appearance of electrons behaving as classical particles or as a wave depends on the experiment being performed. On the other hand, it illustrates a basic feature of microsystems there is an unavoidable and uncontrollable interaction between the observer and the observed. No matter how cleverly one devises an experiment, there is always some disturbance involved in any measurement and such a disturbance is intrinsically indeterminate (see also Section 1.3). 

Max Planck noted that in certain situations, energy possessed particlelike properties. A French physicist, Louis deBroglie, hypothesized that the reverse could be true as well Electrons could, at times, behave as waves rather than particles. This is known today as deBroglie s wave-particle duality. 

It is contrary to our intuition that electrons might behave as waves. The repercussions of this notion are that the electron does not have a definite... 

Why Do Electrons Behave As Particles on Mondays hut As Waves on Tuesdays ... 

Jammer, when he refers to researches in modern physics, presumably means the philosophical difficulties created by quantum physics. Quantum theory was first introduced to explain a number of experimental laws concerning phenomena of thermal radiation and spectroscopy which are inexplicable in terms of classical radiation theory. Eventually it was modified and expanded into its present state. The standard interpretation of the experimental evidence for the quantum theory concludes that in certain circumstances some of the postulated elements such as electrons behave as particles, and in other circumstances they behave as waves. The details of the theory are unimportant to us except in respect of the Heisenburg uncertainty relations . One of these is the well known formula Ap Aq > hl4ir where p and q are the instantaneous co-ordinates of momentum and position of the particle, Ap and Aqi are the interval errors in the measurements of p and q, and h is the Universal Planck s constant. The interpretation of this formula is, therefore, that if one of these co-ordinates is measured with great precision, it is not possible to obtain simultaneously an arbitrarily precise value for the other co-ordinate. The equations of quantum theory cannot, therefore, establish a unique correspondence between precise positions and momenta at one time and at another time nevertheless the theory does enable a probability with which a particle has a specified momentum when it has a given position. Thus quantum theory is said to be not deterministic (i.e, not able to be precisely determined) in its structure but inherently statistical. Nagel [25] points out that this theory refers to micro-states and not macro-states. Thus although quantum... 

For improved magnification, one exploits the dual nature of matter, that particles also behave as waves. The wavelengths are obtained from the momentum as mV = TjihlX, where m is the mass of the particle, Vis its velocity, h is Planck s constant, and k is the resulting wavelength. For electrons, the wavelength works out to be about 0.005 nm, calculated resolution 0.003 nm, and the aetual resolution more like 2 nm. Since electric and magnetic fields can be used like lenses, unlike... 

Overlap is what is needed for constructive or destructive interference of electron waves. If electrons didn t behave as waves, they wouldn t interfere. So the wave behavior of electrons implies the importance of overlap. 

As stated by de Broglie s duality principle, aU particles, especially electrons, can behave as waves under appropriate circumstances. This principle states in its simplest form that a particle of mass m moving at a speed v has a wavelength given by... 

The technique of electron diffraction has been highly developed. In flie electron microscope the wave characteristics of electrons are used to obtain pictures of tiny objects. This microscope is an important tool for studying surface phenomena at very high magnifications. Figure 6.14 is a photograph of an electron microscope image, which demonstrates that tiny particles of matter can behave as waves. 

However, electrons do not have to be observed as particles. The quantum corral (Figure 2.11) shows that electrons can move across a copper surface but they are trapped by a ring of iron atoms, put in place (at low temperature) using an STM (Chapter 12). The electrons behave as standing waves and are observed as ripples. The agreement between the experimental results and the predictions based on these models is what constitutes scientific proof. 

This model assumes that the electron behaves as a standing wave (wave-particle duality) and is subject to boundary conditions similar to those applied to the tension waves of a violin string fixed at both ends. The standing waves have nodes (regions of no vibration or zero electron density) and antinodes (regions of maximum vibration and maximum electron density). 

Given that electrons behave like waves, we need to be able to reconcile the predictions of quantum mechanics with the existence of objects, such as biological cells and the organelles within them. 

The d orbitals are more complex in shape and arrangement in space. In 1925 Touis de Broglie suggested that electrons behaved like waves. This led to the idea of electron probability clouds. The electron probability cloud for one type of d orbital is very strange -it is like a modified p orbital with a ring around the middle (Figure 3.8). You will not need to know the d-orbital shapes at AS level, but you will for A level when studying the transition elements (see Chapter 24). 

Electrons behave as both waves and particles. The consequences of their wave and particle nature ate derived through the formalism of quantum mechanics. The requirement for conservation of energy and momentum forces the electrons to select specific states described by quantum numbers, analogous to resonant vibrations of a string on a musical instrument. The resonant states associated with each set of quantum numbers results in a set of wave functions which describe the probability of finding an electron around a given location at a given time. The wave functions of the resonant states are found as follows. 

The optimization of the Haitree-Fock wave function may be carried out using any of the standard techniques of numerical analysis. However, for many purposes, it is better to use an alternative scheme, which mote directly reflects the physical contents of the Hartree-Fock state. Thus, from the discussion in Section S.l, we recall that an antisymmetric product of spin orbitals represents a state where each electron behaves as an independent particle (but subject to Fermi correlation as discussed in Section S.2.8). This observation suggests that the optimal determinant - that is, the Hartree-Fock determinant in (5.4.3) - may be found by solving a set of effective one-electron Schrbdinger equations for the spin orbitals. Such an approach is indeed possible the effective one-electron Schrddinger equations are called the Hartree-Fock equations and the associated Hamiltonian is the Fock operator... 

A closed-shell Hartree-Fock state is represented by a variationally optimized Slater determinant. Such a wave fiiiKtion represents a state where each electron behaves as an independent particle (subject to Fermi correlation as discussed in Section 5.2.8). We should therefore be able... 

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