Relation de Broglie

The product of mass and speed is called the linear momentum, p, of a particle, and so this expression is more simply written as the de Broglie relation ... 

The kinetic energy of a particle of mass m is related to its speed, v, by F.K = rmr. We can relate this energy to the wavelength of the particle by noting that the linear momentum is p = mv and then using the de Broglie relation (Eq. 7) ... 

When expressed in three dimensions, the de Broglie relation is... 

In principle, the calculation of bonding in two or three dimensions follows the same scheme as outlined for the chain extended in one dimension. Instead of one lattice constant a, two or three lattice constants a, b and c have to be considered, and instead of one sequential number k, two or three numbers kx, ky and k- are needed. The triplet of numbers k = (kx, ky, kz) is called wave vector. This term expresses the relation with the momentum of the electron. The momentum has vectorial character, its direction coincides with the direction of k the magnitudes of both are related by the de Broglie relation [equation (10.5)]. In the directions a, b and c the components of k run from 0 to nja, njb and n/c, respectively. As the direction of motion and the momentum of an electron can be reversed, we also allow for negative values of kx, ky and kz, with values that run from 0 to —nja etc. However, for the calculation of the energy states the positive values are sufficient, since according to equation (10.4) the energy of a wave function is E(k) = E(—k). 

For a particle in a one dimensional box with its potential energy zero, the de Broglie relation from its energy expression, deduced ... 

The de Broglie relation, p = ftk, is valid for any particle. An beam of particles with mass M and kinetic energy E is associated with a wavelength... 

In what I broadly regard as structure (essentially quantum theory), the equation that epitomizes the transition from classical mechanics to quantum mechanics, is the de Broglie relation, k = hip, for it summarizes the central concept of duality. Stemming from duality is the aspect of reality that distinguishes quantum mechanics from classical mechanics, namely superposition y = y/A + y/R with its implication of the roles of constructive and destructive interference. Then of course, there is the means of calculating wavefixnctions, the Schrodinger equation. For simplicity I will write down its time-independent form, Hip = Eip, but it is just as important for a physical chemist to be familiar with its time-dependent form and its ramifications for spectroscopy and reaction. 

A second interpretation of the Aharonov-Bohm effect was devised by Boyer [65,66], who used matter waves associated to moving electrons. Waves coming from each slit interfere with a phase shift = 2jidistance between two slits. If P is the impulse of an electron in the beam, the de Broglie relation gives us P 2nh/X. This results in the fact that the phase... 

It is easy to see why the wavelike properties of particles had not been noticed. According to the de Broglie relation, a particle of mass 1 g traveling at 1 m-s-1 has a wavelength of... 

Strategy In each case, we use the de Broglie relation. The mass of a proton is given in Table B.l, and the speed of light is given inside the back cover. Expect the macroscopic object, the marble, to have a very short wavelength. Remember to express all quantities in kilograms, meters, and seconds, and use 1 J = 1 kg-m2-s-2. 

Now, using the de Broglie relation E = hv = me2, one can estimate the mass of the photon as related to the conductivity coefficient a. If one considers only plane waves in the Z direction, then... 

This is unphysical. But if we take the phase velocity in de Broglie relation, we get a physical solution, namely, the real nonzero rest mass of photon. Since the mid 1990s, there has been much interest in the vg / c solutions of Maxwell equations [40]. However, in our framework, taking the phase velocity in de Broglie relation (59), we get... 

To understand the resolution of electron microscopes and later diffraction techniques the wave approach is more instructive. The resolving power of electron microscopes is substantially better than that of a light microscope, since the wavelength of the electrons, given by the de Broglie relation... 

The neutron is a particle of about the mass of the hydrogen atom and its wavelength is given by the de Broglie relation... 

It can also be shown that Schrodinger s wave equation is none other than a form of the classical differential equation for a wave phenomenon in which the new feature is to be found in the application of it to electrons by means of the experimentally verified De Broglie relation which, in turn follows, as was seen, from a combination of the fundamental relations of Planck and Einstein in the form Av = me2 (p. 107). 

This is the Schrodinger equation for a particle moving in one dimension. The development provided here is not a derivation of this central equation of quantum mechanics rather, it is a plausibility argument based on the idea that the motions of particles can be described by a wave function with the wavelength of the particle being given by the de Broglie relation. 

Discnss the de Broglie relation and use it to calculate the wavelengths associated with particles in motion (Section 4.4, Problems 29-32). 

E8.4 Using the de Broglie relation given in Section 8.2, we have ... 

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