The de Broglie Formula

In physics the concept is known as the property of all pairs of conjugate variables, such as position and momentum, mathematically related by Fourier transformation. The de Broglie formula that relates the momentum of matter waves to wavelength... 

For slow electrons the wavelength A and the accelerating voltage V of the primary beam are related by the de Broglie formula connecting Planck s constant h with electron momentum p... 

In 1924, Louis de Broglie argued on theoretical grounds that particles should have a wavelength associated with them. The de Broglie formula for the wavelength is... 

The time-independent Schrddinger equation Hilf = Eijf has been derived from the wave equation and the de Broglie formula. Solving this equation results in the stationary states and their energies. This is the basic equation of quantum chemistry. The prevailing weight of research in this domain is concentrated on solving this equation for... 

We recovered here the usual WKB formula for tunneling probability, which exhibits an exponentially decaying behavior. On the other hand, from Eq. (2.10), we observed immediately that resonances occur when the thickness of the barrier equals integer multiples of one half of the de Broglie wavelength in the barrier region. 

The de Broglie relationship states that any beam of moving particles will display wave properties according to the formula... 

In the special case of a particle at rest p = 0, we obtain Einstein s famous mass-energy equation E = mc. The alternative root E = — mc is now understood to pertain to the corresponding antiparticle. For a particle with zero rest mass, such as the photon, we obtain p = E/c. RecaUing that kv = c, this last four-vector relation is consistent with both the Planck and de Broglie formulas E = hv and p = h/X. 

Quite a different situation occurs in the micro-world. Here, because of small particle masses, the de Broglie wavelength is commensurate with the interatomic distances in crystals therefore, the corresponding experiment can be realized a crystal can serve as a diffraction grating (refer to Section 6.3.5). In fact, according to formula (7.1.1) and the... 

The neutron mass can be found from de Broglie formula (7.7.1) and rms ther-molyzed neutron velocity (eq. (3.3.7"0)... 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

This final formula, resulting from the coherent infinite sum of solutions, representing the undulatory part of a finite wave of de Broglie, is also a solution of the master nonlinear equation. 

In any case, just for the sake of exemplification, we shall derive here the formula for the uncertainty in momentum, following de Broglie s demonstration [32] almost step by step. In order to clarify the process, in Fig. 29 shows the detection region of the two types microscopes facing each other, for the case of the an horizontal incidence of light. (The same derivation could be obtained, of course, for any other incidence angle.) We must keep in mind that the reasoning... 

I wave-like properties, and allow de Broglie s formula (eqn 2.21) to be verified quantitatively. No-one who has used an LEED apparatus can be left in any doubt that this theory is correct. Simply turning a knob on the control panel changes the accelerating voltage for electrons, and hence their kinetic energy and momentum. The diffraction spots then move in exactly the way predicted by eqn 2.21, in conjunction with simple diffraction theory (see Problem 2 below). 

Comparison of de Broglie s equation (see Eq. 4.20) with Bohr s equation (see Eq. 4.22) shows that the wavelength of the standing wave is related to the linear momentum, f, of the electron by the following simple formula ... 

Louis de Broglie, then a doctoral student at Paris University, knew that whole-number behavior in physics was commonly associated with periodicity in a system. Perhaps a periodic nature had to be a property of electrons in atoms. Since waves are the quintessential example of periodicity, he hypothesized that not only did light have a wave nature, but so must electrons. De Broglie pictured electrons as particles embedded in standing waves around a nucleus. If the two modes of existence were to mesh, he had somehow to relate wavelength to mass or momentum. His famous formula M = h/p (where M is the wavelength, p is the particle momentum, and h is Planck s constant) had the right dimensions, but needed to be confirmed by experimental tests. 

De Broglie s formula was useful for knowing a particle s wavelength, but a more comprehensive understanding of the electron s motion and location while bound to a nucleus was needed. In order to be able to... 

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