De Broglie s wavelengths

For 1, de Broglie s wavelength is small enough compared to the classical collision radius b so that a wave packet can be constructed which, approximately, follows the classical Coulomb trajectory [3]. The opposite limit, where the Sommerfeld parameter Zie hv<, denotes the case of weak Coulomb interaction where the Born approximation may be expected to be valid. 

In order to reconcile this relativistic invariance with De Broglie s wavelength hypothesis, for waves at rest and in a moving frame,... 

De Broglie s work clearly shows that a moving electron can be considered as a wave. If it behaves in that way, a stable orbit in a hydrogen atom must contain a whole number of wavelengths, or otherwise there would be interference that would lead to cancellation (destructive interference). This condition can be expressed as... 

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. 

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. 

The observation that the wavelength of light is linked to the particle-like momentum of a photon prompted de Broglie to postulate the likelihood of an inverse situation whereby particulate objects may exhibit wave-like properties. Hence, an electron with linear momentum p could under appropriate conditions exhibit a wavelength A = h/p. The demonstration that an electron beam was diffracted by periodic crystals in exactly the same way as X-radiation confirmed de Broglie s postulate and provided an alternative description of the electronic stationary states on an atom. Instead of an accelerated particle the orbiting electron could be described as a standing wave. To avoid self-destruction by wave interference it is necessary to assume an... 

Neutrons can be described by de Broglie s wave-particle formalism and a wavelength k can be determined according to X = h/mv with h being Planck s constants and m and v being the mass and velocity respectively. The neutron s mean energy can be expressed as ... 

This equation, called de Broglie s equation, allows us to calculate the wavelength for a particle, as shown in Example 12.2. 

Notice from Example 12.2 that the wavelength associated with the ball is incredibly short. On the other hand, the wavelength of the electron, although quite small, happens to be of the same order as the spacing between the atoms in a typical crystal. This is important because, as we will see presently, it provides a means for testing de Broglie s equation. 

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 ... 

Two years after de Broglie s prediction, C. Davisson (1882-1958) and L. H. Germer (1896-1971) at the Bell Telephone Laboratories demonstrated diffraction of electrons by a crystal of nickel. This behavior is an important characteristic of waves. It shows conclusively that electrons do have wave properties. Davisson and Germer found that the wavelength associated with electrons of known energy is exactly that predicted by de Broglie. Similar diffraction experiments have been successfully performed with other particles, such as neutrons. 

---