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... 

Optical fibres behave as wave guides for light. The original ones were made of glass but now polymeric materials are mostly used as they are cheaper than glass and metal wires. 

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. 

Wave-particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, then we can apply the resnlts of classical electromagnetic theory to particles. By analogy, the probability is the sqnare of the amplitnde. Zero-point energy is a con-seqnence of the Heisenberg nncertainty relation all particles bound in potential wells have finite energy even at the absolnte zero of temperature. 

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... 

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... 

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... 

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. 

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. 

Ketterle, W., Nobel lecture When atoms behave as waves Bose-Einstein condensation and the atom laser. Rev. Mod Phys., 74, 1131, 2002. 

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. 

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