We consider a free motion of a quantum particle. Introducing an explicit measurement procedure for velocity, we demonstrate that the measured velocity is related to the group and phase velocities of the corresponding matter waves. We show that for long distances the measured velocity coincides with the matter wave group velocity.
Abstract We consider the motion of a quantum particle in a free space. Introducing an explicit measurement procedure for velocity, we demonstrate that the measured velocity is related to the group and phase velocities of the corresponding matter waves. We show that for long distances the measured velocity coincides with the matter wave group velocity. We discuss the possibilities to demonstrate these effects for the optical pulses in coherently driven media or for radiation propagating in waveguides.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
The velocity is a macroscopic concept. The question what is the meaning of velocity in Quantum Mechanics( QM) ,this problem itself is inno answersituation. In the semi-classical theory,for the reconstruction of the velocity definition,we have v→= ▽S / m[m is the particle's mass and ψ = Aexp( j S /h) ]. This method builds the relation of velocity and wave function,but it is not the necessary and useful concept of wave's speed. The velocity of light in vacuum is a common basic constant of physics,however QM also need this parameter obviously,i. e. the new wave mechanics is on need of this physical quantity.The process and result of people's practice show that,scientist has intended to bring forth in the measurement of c with very high accuracy does stand out,if he used the scalar equation c = fλ. Modern physics maintain that the wave velocities( phase velocityvp、group velocity vg) are scalars,not vectors.The wave has its special scientific features; the regular pattern is different from the Classical Mechanics( CM). It needs supports of the QM,but it is not identical with the serious QM. Now,it has not only a theory that can explain the wave problems perfectly,so it is support by the classical physics and quantum physics.The quantization process of EM fields explains the particle-wave duality mathematically. Post quantization,from the EM fields we obtained the photons. But scientist still is puzzled by a difficult question:what is photon. As the common knowledge,the matter waves of photon still a paradox in physics. In addition,the electronic probability waves corresponding the Schrdinger equation,but people says the photonic probability waves have not that wave equation. In other words,the physical meanings of matter wave's phase velocity are not clear,and why the phase velocity of de Broglie waves is faster-than-light,many explanations will be needed.Many instances similar to these examples make the particle-wave duality still the magic ghost in modern physics. Finally,the study on negative wave velocity( NWV) in recent years has a great deal of experience on wave sciences,and bring the new chance on theoretical thinking.
For the wave-particle duality of microscopic particles,the relationship of the velocity and the velocity of phase and the velocity of group for the microscopic particles is analyzed,it is pointed out that the velocity relating to the frequency and wavelength of two De Broglie expressions is not the velocity of group but the velocity of phase.On the contrary,the velocity in the defining expression of momentum is the real speed of particles,which doesn't equal to the velocity of phase but equals to the velocity of group.
The de Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which is the same treatment that historically led to this famous relation. In the same context of the Minkowski four-vector formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that both transform as ordinary velocities under a Lorentz transformation. We show that such a transformation rule is a necessary condition for the covariance of the de Broglie relation, and stress the pedagogical value of the Einstein-Minkowski-Lorentz relativistic context in the presentation of the de Broglie relation.
Velocity plays a central role in classical mechanics, yet quantum mechanics deals with the variables E,t,p and x. P nonrelativistically is mo v, but a given p value corresponds to various m1v1=m2v2=m3v3 and so does not determine v uniquely. (A p, E pair or a p,mo pair is needed for this.). Here we consider a quantum particle in a square well potential. If detection of a particle occurs through impulse= Integral F dt, then m1v1=p and m2v2=p yield the same impulse although the velocities may be completely different. If one seeks a “time-independent” scenario then one would expect the two to yield the same spatial probability distribution even though E is different for each. In such a case E1 and E2 distinguish between v1 and v2. For a particle in an infinite well v1 and v2 represent different frequencies and so this “frequency” information role is taken on by energy. What happens if one considers two different average momenta levels for the same single particle? In such a case mo is fixed and p1 and p2 correspond to different v1 and v2 i.e. to different frequencies. There is no time in a time-independent scenario so how does one represent the relative frequencies? One may do this spatially, we argue. If p1=n1 3.14/L and p2=n2 3.14/L where L is the well size then p1 has n1 crests and p2, n2 crests. Thus in one case relative velocity for the same p is distinguished by E1 and E2, but not by any spatial distribution. In the second for the same mo and different v1 and v2, there is both a different spatial distribution describing the relative frequencies, but also different E’s which also distinguish and are linked to frequency. This leads to the idea of probability crests with certain base widths proportional to 1/p. What happens in the case of a free particle? Such a particle should also be associated with crests linked to 1/p, but the particle is traveling through space. It is not restricted to L and so its probability density should mimic classical mechanics i.e. be a constant everywhere. To allow for this one considers a probability distribution of crests and troughs together with a second spatially shifted distribution indicating the direction and nature motion , but with the constraint that the modulus is 1. This suggests cos(px) and sin(px) for the two distributions. This is very different from the classical mechanical scenario of a bound state spatial density (1) which is proportional to dt with fixed dx i.e. Cdx/v(x) where v(x) is velocity. Such a picture only accounts for relative densities in space. Thus for two different constant velocities v1 and v2 which represent different frequencies of traversing a distance L, the spatial density is the same classically. Thus important information is lost and we argue that Cdx/v(x) is incomplete. In quantum mechanics the envelope of P(x)=W*(x)W(x)dx approaches the classical value for high n (energy level), but quantum mechanics contains much more information, namely the idea that different v for the same mo have different frequencies which must be manifested spatially.