De Broglie’s relationship

Incidentally, the uncertainty principle associated with the name of Heisenberg, well known in quantum mechanics, follows from the expression given here when de Broglie s relationship connecting the momentum of a particle with its wavelength is included. 

By 1925 it was realized that the classical ideas that described matter didn t work at the atomic level. Some progress—Planck s quantum theory, Einstein s application of quantum theory to light, Bohr s theory of hydrogen, de Broglie s relationship—had been made, but it was all very specific and not generally applied to atoms and molecules. 

X rays. This result verified de Broglie s relationship, at least for electrons. Larger chunks of matter, such as balls, have such small wavelengths (see Sample Exercise 7.3) that they are impossible to verify experimentally. However, we believe that all matter obeys de Broglie s equation. 

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. 

Now, to associate this kinetic energy with a wave, we can use de Broglie s relationship (A = h/p) to get... 

The next step is to substitute de Broglie s relationship for the wavelength of a matter wave. 

Using de Broglie s relationship between the wavelength and the momentum further yields... 

How many values of k are there As many as the number of translations in the crystal or, alternatively, as many as there are microscopic unit cells in the macroscopic crystal. So let us say Avogadro s number, give or take a few. There is an energy level for each value of k (actually a degenerate pair of levels for each pair of positive and negative k values. There is an easily proved theorem that E(k) = E( — k). Most representations of E(k) do not give the redundant (- ), but plot ( k ) and label it as E(k)). Also the allowed values of k are equally spaced in the space of k, which is called reciprocal or momentum space. The relationship between k = 2x7 X and momentum derives from the de Broglie relationship X = hip. Remarkably, k is not only a symmetry label and a node counter, but it is also a wave vector, and so measures momentum. 

Electrons moving in circles around the nucleus, as in Bohr s theory, can be thought of as forming standing waves that can be described by the de Broglie equation. However, we no longer believe that it is possible to describe the motion of an electron in an atom so precisely. This is a consequence of another fundamental principle of modern physics, Heisenberg s uncertainty principle, which states that there is a relationship... 

Dalton s atomic theory, overview, 1 De Broglie equation, 23 Delocalization energy, definition, 174 Density functional theory chemical potential, 192 computational chemistry, 189-192 density function determination, 189 exchange-correlation potential and energy relationship, 191-192 Hohenberg-Kohn theorem, 189-190 Kohn-Sham equations, 191 Weizsacker correction, 191 Determinism, concept, 4 DFT, see Density functional theory Dipole moment, molecular symmetry, 212-213... 

Alternatively, Einstein s relationship relates the wavelength of light A to the linear momentum p according to X = h/p (where h = 6.626 x 10 [Ls] is the Planck constant). De Broglie applied this to the material wave of the mass m to obtain... 

We base our quantum version of radiation on one relationship, Planck s law (Eq. 1.2) Ephoton The quantum mechanics of matter also grow out of a single equation, one that ascribes a property to matter completely alien to the classical view a wavelength. At the tiny distances over which atoms and subatomic particles interact, matter exhibits the characteristics of waves, consistent with a de Broglie wavelength. 

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