When a Gaussian pulse propagates through a barrier its velocity is taken to be that with which the peak moves. When a uniform amplitude pulse propagates its velocity is taken to be that of the front of the main part of the pulse. By using a method due to Brillouin (1960) it is shown that the uniform amplitude pulse has quite different velocity characteristics from that of a Gaussian pulse, and indeed it propagates with a constant velocity-which is more in keeping with physical intuition.
When a Gaussian pulse propagates through a barrier its velocity is taken to be that with which the peak moves. When a uniform amplitude pulse propagates its velocity is taken to be that of the front of the main part of the pulse. By using a method due to Brillouin (1960) it is shown that the uniform amplitude pulse has quite different velocity characteristics from that of a Gaussian pulse, and indeed it propagates with a constant velocity-which is more in keeping with physical intuition.
Some recent theoretical studies tended to employ analytically-continuous gaussian, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, gaussian functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the superluminal interpretation of transition times.
Some recent theoretical studies have tended to employ analytically-continuous {\\em gaussian}, or infinite-bandwidth step pulses to examine tunneling process. The stationary phase method is often employed to this aim. However, {\\em gaussian} functions do not have a well-defined front, such that their speed of propagation becomes ambiguous. Also, infinite bandwidth signals cannot propagate through any real physical medium (whose transfer function is therefore finite) without pulse distortion, which also leads to ambiguities in determining propagation velocity during the tunneling process. In this manuscript, we appoint some incompatibilities with the application of the method of stationary phase in deriving tunneling times which, eventually, can ruin the {\\em superluminal} interpretation of transition times.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
Tunneling without distortion is not a propagation phenomenon but a quasistatic process in which the slowly varying envelope of the input pulse modulates the amplitude of a standing wave in the barrier. Because the envelope has a bandwidth that is small compared to the cutoff frequency of the barrier it does not propagate; it merely stands and waves. The envelope of the reflected and transmitted fields can adiabatically follow the envelope of the input pulse with only a small delay proportional to the energy stored in the barrier. The Hartman effect, in which the delay becomes independent of barrier length, is explained on the basis of saturation of stored energy with increasing length. Because the delay is not a propagation delay, it should not be linked to a velocity of propagation. True evanescent waves have zero speed and not infinite velocity as often asserted.
It is well known that uniform classical motion may be described by a constant velocity ie. one which does not change in time. A different constant number, however, is needed for each velocity so uniform motion in general is not being represented by a single number in this scenario. In (1) we noted how a single number such as energy E could represent systems linked to different physical theories. For example, a classical particle may move according to .5mvv + V(x) = E where v is velocity. On the one hand a single number E describes each point (x,t), but at each point there is different detailed motion such as potential energy, velocity etc. Similarly in a Maxwell-Boltzmann gas exp(- ( .5mvv + V(x))/T) is proportional to probability. For a given E1 = .5mvv + V(x) there are many v,x points which have the same E1 and thus probability. Finally for a quantum bound state there is a single En (energy) even though there is a potential energy V(x) and average kinetic energy -1/2m d/dx dW/dx / W (W=wavefunction) at each x. In this note we ask whether a single number may apply to all uniform motion with this constant being equal then to different details in each system. The reason we ask this question is that we think a given constant velocity, say 25 m/s, does not characterize uniform motion. The problem we argue is one of resolution of x and t. For example, 25 m/s may characterize very erratic motion from x=0 to x=25 m in a period of 1 sec. This simply means the average speed is 25 m/s, not that the motion is uniform. To bypass this argument one may suggest perfect x, t resolution for all speeds. In such a case erratic motion may only exist in an extremely tiny dx or dt which approaches 0. This suggests there exists a spacetime (x,t) which may analyze all uniform motion with infinitesimal precision i.e is all-seeing. It is not clear that there exists any physical mechanism for this. We argue instead that velocity is a property of a moving object, not of spacetime. An object undergoes force over distance and receives a velocity which is one of its properties. In such a case infinitesimal resolution of t and x may not apply to all velocities. In fact if v velocity is a property of the object, x,t resolution may also be associated properties. Thus details of a particular uniform motion are not simply v (which yields p and E=pp/2m nonrelativistically), but also x and t resolution. Is it possible that these details may combine to create a single constant representing all uniform motion? We argue that this is possible and that this constant is in fact the free particle action -Et+px which holds both relativistically and nonrelativistically. For t proportional to 1/E and p proportional to 1/p (same proportionality constant) one has a single constant, Action=0, describing all uniform motion. We however, arrive at the form -Et+px without using notions of action or Lagrangian.
Propagation of a Gaussian wave packet in an absorbing medium is examined in order to shed light on the physical basis of group velocity which exhibits a singular behavior. It is found that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range. Fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fourier-component waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long traveling distance. The results provided a unified picture of wave-packet propagation in an absorbing medium.