Uncertainty matter waves

For matter waves hk is the particle momentum and the uncertainty relation AxAp > h/2, known as the Heisenberg uncertainty principle. 

For our present purposes we shall take as our additional postulate the supposition that the mechanical behaviour of matter on the atomic scale is in accordance with the Schrodinger wave equation. Its numerical solution, for appropriate conditions, expresses the observable properties without contravening the principle that it is impossible to make an exact and simultaneous specification of position and velocities. It may be remarked that it is because of this principle of uncertainty that wave mechanics seek to describe the state of a system by means of the function whose purpose is to describe probabilities and not certainties. 

De Broglie s relationship suggests that electrons are matter waves and thus should display wavelike properties. A consequence of this wave-particle duality is the limited precision in determining an electron s position and momentum imposed hy the Heisenberg uncertainty principle. How then are we to view electrons in atoms To answer this question, we must begin by identifying two types of waves. 

Briefly describe each of the following ideas or phenomena (a) atomic (line) spectrum (b) photoelectric effect (c) matter wave (d) Heisenberg uncertainty principle (e) electron spin (f) Pauli exclusion principle (g) Hund s rule (h) orbital diagram (i) electron charge density (j) radial electron density. 

In classical mechanics both the position of a particle and its velocity at any given instant can be determined with as much accuracy as the experimental procedure allows. However, in 1927 Heisenberg introduced the idea that the wave nature of matter sets limits to the accuracy with which these properties can be measured simultaneously for a very small particle such as an electron. He showed that Ax, the product of the uncertainty in the measurement of the position x, and Ap, the uncertainty in the measurement of the momentum p, can never be smaller than M2tt ... 

Combining the wave nature of matter and the probability concept of the Uncertainty Principle, M. Born proposed that the electronic wavefunction is no longer an amplitude function. Rather, it is a measure of the probability of an event when the function has a large (absolute) value, the probability for the event is large. An example of such an event is given below. 

The realization that both matter and radiation interact as waves led Werner Heisenberg to the conclusion in 1927 that the act of observation and measurement requires the interaction of one wave with another, resulting in an inherent uncertainty in the location and momentum of particles. This inability to measure phenomena at the subatomic level is known as the Heisenberg uncertainty principle, and it applies to the location and momentum of electrons in an atom. A discussion of the principle and Heisenberg s other contributions to quantum theory is located here http //www.aip.org/historv/heisenberg/. 

I ve been using marbles and atom-size insects as an analogy for electrons, but I don t want to leave you with the misconception that electrons can only be thought of as solid objects. In the introduction to this book and in the first chemistry book, I discussed how we can think of electrons (and all particles, for that matter) as collections of waves. It is this wave nature of electrons that is the basis for quantum mechanics, which is the math we use to come up with the uncertainty principle. So, while it is often convenient to consider electrons to be tiny, solid objects, you should always be aware of the model of electrons as waves. 

The story of the evolution of physics in the twentieth century is the story of the elaboration and acceptance of a wave-mechanical conception of the primary nature of matter. No model of matter can fail to take into account that contemporary physics has recaptured a Pythagorean intuition too long forgotten by the followers of the commonsense physics of Newton. Common sense is gone from physics Planck banished it when he discovered the discrete nature of radiation, and Heisenberg s Principle of Uncertainty made a return to the notion of simple location forever impossible. Our own theory is thoroughly kymatic, or wavelike. 

The Heisenberg Uncertainly Principle arises from the dual nature (wave-particle) of matter. It states that there exists an inherent uncertainty in the product of the position of a particle and its momentum, and that this uncertainty is on the order of Planck s constant. 

We believe that such reconciliation Is not adequate. Moreover, we believe that the thermodynamic behavior of matter is due to quantum uncertainties of the same nature but broader than those associated with wave functions and invoked in the uncertainty principle. 

Wave-particle duality means that matter has wavelike properties and energy has particle-like properties. These properties become observable only at the atomic scale. Because of wave-particle duality, we can never know the exact position and momentum of an electron simultaneously [uncertainty principle). 

Acceptance of the dual nature of matter and energy and of the uncertainty principle culminated in the fi eld of quantum mechanics, which examines the wave nature of objects on the atomic scale. In 1926, Erwin Schrddinger derived an equation that is the basis for the quantum-mechanical model of the hydrogen atom. The model describes an atom that has certain allowed quantities of energy due to the allowed frequencies of an electron whose behavior is wavelike and whose exact location is impossible to know. 

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