De Broglie wave packet

Under these conditions the de Broglie wave packet is sequentially given as... 

We are going now to further explore the influence of the non-constant amplitude dependency A k) and to see its consequences in the de Broglie wave packet evolution. However, worth using the momentum representation from de Broglie quantification in order to better emphasize on the quantum (through Planck constant dependency) influence. That is we consider the general wave function ... 

A directly application and of an extreme importance of the de Broglie wave packet is to consider its normalization by noting that the wave function in the real space and the amplitude in the reciprocal space (or of the impetus by the de Broglie quantification) are conjugate size in the sense of the Fourier mutual transforms. 

Nevertheless, one fundamental consequence of dealing with (r) as a generalized function is that it can be considering as patterning a special (generalized in the sense of resumed) wave function abstracted from the de Broglie wave-packet, namely... 

Finally, employing (postulating the equivalence of) the actual founded phase for classical stationary wave-condition with the quantum stationary action related phase of the de Broglie wave-packet... 

It is worth observing that the practical rule (4.548) is indeed consistent since recovering in Eq. (4.549) the kernel of the Gaussian de Broglie wave-packet—for the wave behavior of a quantum objeet—as expeeted. As a consequence, the result (4.550) may be therefore eonsidered as a viable analytical expression for characterizing the complementary particle nature of the quantum manifestation of an object. 

The mysterious phase velocity of the de Broglie wave and the group velocity of the amplitude wave, c2/ > c, refer to the, by now familiar superluminal motion in the interior of the electron. As many authors noted and Molski(1998) recently reviewed [86] an attractive mechanism for construction of dispersion-free wave packets is provided in terms of a free bradyon4 and a free tachyon that trap each other in a relativistically invariant way. It is demonstrated in particular how an electromagnetic spherical cavity may be... 

From above de Broglie wave-function packet one can identify at f = 0 the functions... 

On the femtosecond time scale, an entirely new domain emerges. First, a wave packet can be prepared, as the temporal resolution is sufficiently short to "freeze" the nuclei at a given intemuclear separation. Put in another way, the time resolution is much shorter than the vibrational (and rotational) motions such that the wave packet is prepared, highly localized with a de Broglie wave length of 0.1 A, with the structure frozen. Second, this synthesis is not in violation of the... 

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. 

The group velocity of de Broglie matter waves are seen to be identical with particle velocity. In this instance it is the wave model that seems not to need the particle concept. However, this result has been considered of academic interest only because of the dispersion of wave packets. Still, it cannot be accidental that wave packets have so many properties in common with quantum-mechanical particles and maybe the concept was abandoned prematurely. What it lacks is a mechanism to account for the appearance of mass, charge and spin, but this may not be an insurmountable problem. It is tempting to associate the rapidly oscillating component with the Compton wavelength and relativistic motion within the electronic wave packet. 

Some of the greatest physicists of all time, Einstein, de Broglie and Schrodinger amongst them, never did like this situation. God does not play dice was Einstein s famous remark (see, e.g.. Pais (1991), p. 425). In order to save the idea of determinism, de Broglie searched for hidden parameters and Schrodinger studied the time evolution of wave packets. 

Gor kov and Eliashberg investigated the problem of a size-induced metal-insulator transition in terms of the location of the gas of conduction electrons in a metal through the (finite) size-induced confinement of the electron wave packet. The de Broglie wavelength X of electrons is given by ... 

In 1903, Marie Curie, her husband and Henri Becquerel received the Nobel Prize in physics Marie won another Nobel prize (chemistry) in 1911. In 1900, Max Planck had postulated that light energy must be emitted and absorbed in discrete particles, called quanta. In Paris in 1924, Victor de Broglie concluded that if light could act as if it were a stream of particles, particles could have the properties of waves. Both quanta and waves are central to quantum physics. Quantum theory states that energy comes in discrete packets, called quanta, which travel in waves. The principle of wave-particle duality states that all subatomic particles can be considered as either waves or particles. Light is a stream of photon particles that travel in waves. 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

Recently, there has been a renewed interest in the de Broglie-Bohm formulation of quantum mechanics as a numerical tool to perform multidimensional wave packet calculations [13-15]. It has also been used to visualize the motion of quantum mechanical wave packets by trajectories and to study the transition from quantum mechanics to classical mechanics [16-18]. Carlsen and Goscinski [16], for instance, have studied fractional and full revivals of circular Rydberg wave packets in the hydrogen atoms using this formulation. 

However, a moving particle of atomic or electronic mass does not obey Newtonian mechanics. Instead, It behaves as a wave packet with a wavelength given by the de Broglie expression... 

This can be realized, for instance, since one uses Gaussian momentum function (or spectral function if it viewed in terms of associated wave-vector k, throughout the de Broglie relationship) as the driving amplitude for the wave-packet ... 

Wentzel, Kramers and Brillouin) description of the quantum wave-packet and of the associated de Broglie wavelength ... 

By recognizing the generating distribution functions for the (de Broglie) free wave-packets, see Section 2.2.2,... 

The present algorithm may be naturally supplemented with the analysis of the wave-particle duality. This is accomplished by means of considering further averages over the quantum fluctuations for the mathematical obj ects exp(-/lx) and exp(-A x ) that are most suited to represent the waves and particles, due to their obvious shapes, respectively. Such computations of averages are best performed employing the Fourier -transformation as resulted from the de Broglie packet, Eq. (4.537) with Eq. (4.537), equivalently rewritten successively as (Putz, 2009 Putz, 2010c) ... 

Louis de Broglie proposed that electrons, known as particles or corpuscles with mass and negative charge, can be considered as waves. He derived the wavelength X for a particle by applying Einstein s theory of relativity on a wave packet. His conclusion was that X should be related to the momentum p as... 

The formation of wave packets depends on the interaction of a complementary bradyon-tachyon pair. The bradyonic group velocity corresponds to the de Broglie wavelength of the packet, Adu = hlmv. The tachyonic component defines the internal structure of the wave packet with Compton wavelength Ac = h/mc. The two components are said [20] to be trapped in a relativistically invariant way. We note that VbVt = = 1/eoMo, where the group velocity of the tachyon Vt... 

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