De Broglie’s relation

Matter is classically particulate in nature, but it also manifests wave character. The wave property of matter is related to its particle nature by de Broglie s relation A = hip, where A is known as the de Broglie wave length. 

However, right at the beginning of the twentieth century, Planck and Einstein again introduced the corpuscular view with the notion of the photon. The energy E of such a particle is given by de Broglie s relation... 

We could get the same answer in a different way, using de Broglie s relation A, = h/p (Problem 5-15). The wave representing the electron would have to vanish at the two walls, similar to the waves on a violin string. The longest possible wave we could fit into the box would have wavelength k = 2L. Such a wave would go through half a cycle between the two walls, and would be zero at each wall. 

De Broglie (1924) now made the step of also adding a wave phenomenon to the particulate nature of a moving electron, led by considerations of symmetry and by the need for a generalization of the fundamental concepts of nature. Similar to equation (4) De Broglie s relation is ... 

For what follows one can accept this equation as a fundamental natural law of the mechanics of electrons, the correctness of which has been tested many times by experiment, just as Newton s law of attraction is a fundamental law of macroscopic mechanics. De Broglie s relation is then a special result of this equation. 

Equation (2) is thus the characteristic time-independent wave equation which we shall now apply to an electron by means of De Broglie s relation ... 

Apart from being obtained directly from de Broglie s relation, as above, W can also be calculated according to the general method of p. 123). 

Introduction.— The theory necessary to carry out a fairly accurate electron-diffraction investigation was developed many years ago. The bases of the theory are to be found in papers by Rutherford, Debye, and Ehrenfest. Thus most of the theory was available when Davisson and Germer and Thomson made their famous experiments on the diffraction of electrons by crystals and so verified de Broglie s relation, and when Mark and Wierl carried out the first experiments on scattering of electrons by gases. An important contribution to the theory was also made by Fax n and Holtsmark in 1927, but their results were not applied to structure determinations by gas electron diffraction until about two decades ago. ... 

De Broglie s relation can be reached as follows. Einstein s relation for photons is... 

Earlier we saw that we needed a wave equation in order to solve for the standing waves pertaining to a particular classical system and its set of boundary conditions. The same need exists for a wave equation to solve for matter waves. Schrodinger obtained such an equation by taking the classical time-independent wave equation and substituting de Broglie s relation for A. Thus, if... 

A better model is afforded by considering the wave nature of moving particles. According to de Broglie s relation, a particle of mass m that moves with velocity v has a wavelength... 

We use an approximate proof that illustrates the connection between the particulate and wave nature of matter. We start with a classical wave and a classical particle and we use de Broglie s relations to connect the two. We then discuss what this connection means for quantum mechanical systems. 

---