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6.8.4 Collisional dynamics of propagating resonances |
6.9 Validity of the gradient expansion |
6.10 Hydrodynamical and thermodynamical limits |
7 Space-time delays and measurements |
7.1 Speed of the propagating wave packet |
7.2 Measurements and resulting time delays and advances |
7.3 Apparent superluminality in neutrino experiments as a time advance effect |
8.2 Quantum mechanics |
8.2.1 One dimensional quantum mechanical motion |
8.2.2 Three dimensional scattering problem |
8.3 Quantum field theory |
8.4 Quantum kinetics |
A Virial theorem for infinite classical motion in central potential |
B Relations for wave functions obeying Schrödinger equation |
C Asymptotic centroids of the wave packets |
D Relations for the sojourn time |
E H-theorem and minimum of the entropy production |
Many definitions of time, as a measure of a duration of a process, are possible in classical mechanics because for the measuring of the time duration any process is suitable, which occurs at a constant pace. Naively thinking, a response of a system to an external perturbation should be delayed in accordance with the ca... |
Time delays and possible time advancements in quantum mechanical phenomena have been extensively discussed in the literature, see Refs. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and references therein. In spite of that many questions still remain not quite understood. Worth mentioning is the Hartman effect [13] , that th... |
Additionally to the mentioned time delays other relevant time quantities were introduced, and the differences between the averaged scattering time delay δts and the Wigner scattering time delay δtW were discussed in Ref. [3, 19, 20] , see also Refs. [4, 5, 6] and references therein. Based on these analyses authors of R... |
The appropriate frame for the description of non-equilibrium many-body processes is the real-time formalism of quantum filed theory developed by Schwinger, Kadanoff, Baym and Keldysh [22, 23, 24, 25] . A generalized kinetic description of off-mass-shell (virtual) particles has been developed based on the quasiclassical... |
Recent work [40] suggested a non-local form of the quantum kinetic equation, which up to second gradients coincides with the KB equation and up to first gradients, with the BM equation. Thus, the non-local form keeps the Noether 4-current and Noether energy-momentum conserved at least up to first gradients. Second adva... |
In this paper we study problems related to determination of time delays and advancements in various phenomena. In Sect. 2 we discuss how time delays and lesser time advancements arise in the description of oscillations in classical mechanics and in classical field theory of radiation. In Sect. 3 we consider time delays... |
Starting from Sect. 5 we use units ℏ=c=1. Where necessary we recover c and ℏ. |
In this section we introduce a number of time characteristics of the dynamics of physical processes. We demonstrate how a time shifted response of a system to an external perturbation appears in classical mechanics and classical electrodynamics. We show that there may arise as delays as advancements in the system respo... |
Let us introduce some definitions of time, as a measure of duration of processes in classical mechanics, which will further appear in quantum mechanical description. |
For measuring of a time duration any process is suitable, which occurs at constant pace. For example to measure time of motion one can use a camel moving straightforwardly with constant velocity →v, then t=l/v, where l≃Nl0 is the distance passed by the camel, N is number of its steps, l0 is the step size. Such a simple... |
Another way to measure time is to exploit the particle conservation law. One of the oldest time-measuring devices constructed in such a manner is a clepsydra or a water clock. Its usage is based on the principle of the conservation of an amount of water. Water can be of course replaced by any substance, which local den... |
We will call this quantity a dwell time since similar definition of a time interval is used in quantum mechanics in stationary problems. |
In one dimensional case the time particles dwell in some segment of the z axis open at the ends z1 and z2, through which particles flow outside the segment, can be found as |
t(1,cl)d=∫z2z1ρdz|j(z1)+j(z2)|, (2.2) |
where ρ(z) is the particle density and j(z)=v(z)ρ(z) is a 1D flux density. Obviously, for a particle flux from a hole at z=z2 (at j(z1)=0) with constant density ρ and constant velocity v we then have t(cl)d=l/v with l=z2−z1. If ρ depends on t, the definitions (2.1), (2.2) become inconvenient, since t(cl)d is then a non... |
Another relevant time-quantity reflecting a temporal extent of a physical process can be defined as follows. Consider the motion of a classical particle in an arbitrary time-dependent one-dimensional potential U(z,t). The particle trajectory is described by the function z(t)∈C, where C is the space region allowed for c... |
t(cl)soj(z1,z2,τ)=τ∫0dtθ(z(t)−z1)θ(z2−z(t))=τ∫0dtz2∫z1dsδ(s−z(t)). (2.3) |
Such a temporal quantity can be called a classical sojourn time. What is notable is that exactly this time has a well defined counterpart in quantum mechanics. |
Now consider particle motion in a stationary field U(z). Using the equation of motion dz/dt=v(z;E), where v(z;E)=√2m(E−U(z)) is the particle velocity and E, the energy, for an infinite motion we can recast the sojourn time (2.3) as |
t(cl)soj(z1,z2,τ)=z(τ)∫z(0)dzv(z;E)z2∫z1dsδ(s−z)=min{z2,z(τ)}∫max{z1,z(0)}dzv(z;E) (2.4) |
provided the interval [z1,z2] overlaps with the interval [z(0),z(τ)] . If the particle motion is infinite one can put τ→∞ . For finite motion the integral would diverge in this limit and τ must be kept finite. It is convenient to restrict τ by the half of period τ≤P/2, which depends on the energy of the system and is g... |
P(E)=2z2(E)∫z1(E)dzv(z;E), (2.5) |
where now z1,2(E) are the turning points, given by equation U(z1,2)=E . For τ>P/2 the sojourn time contains a trivial part, which is a multiple of the half-period, t(cl)soj(z1,z2,τ)=nP/2+t(cl)soj(z1,z2,τ−nP/2), where n is an integer part of the ratio 2τ/P. |
Following (2.4), the classical sojourn time t(cl)soj(z1,z,τ(z1,z)) can be rewritten through the derivative of the shortened action |
t(cl)soj(z1,z,τ(z1,z))=∂Ssh(z1,z,E;U)∂E, (2.6) |
Ssh(z1,z,E;U)=z∫z1pdz=z∫z1√2m(E−U(z))dz. |
Taking z=z2 we get |
t(cl)soj(z1,z2,P/2)=P/2, (2.7) |
provided z1,2 are the turning points. |
For an infinite motion with E>maxU(z), following (2.4) we can define a classical sojourn time delay/advance for the particle traversing the region of the potential compared to a free motion as |
δtclsoj=t(cl)soj(−∞,∞,∞;U)−t(cl)soj(−∞,∞,∞;U=0)=√m2+∞∫−∞(1√E−U(z)−1√E)dz. (2.8) |
Calculating t(cl)soj(−∞,∞,∞) we extended the lower limit in the time integration in (2.3) to −∞ . The classical sojourn time delay/advance (2.8) for infinite motion can be then rewritten as |
δt(cl)soj=∂(Ssh(E;U)−Ssh(E;0))∂E, (2.9) |
where Ssh(E;U)=∫+∞−∞pdz. |
The definition (2.9) of the time delay is similar to the definition of the group time delay δtgr appearing in consideration of waves in classical and quantum mechanics. In the later case the Ψ-function of quasi-classical stationary motion is expressed as Ψ∝eiSsh(z1,z,E;U)/ℏ. With the help of a classical analog of the p... |
ℏδ(cl)(z1,z,E;U)≡Ssh(z1,z,E;U), (2.10) |
we now introduce the group time |
t(cl,1D)gr(z1,z,E;U)≡ℏ∂δ(cl)(z1,z,E;U)∂E. (2.11) |
t(cl,1D)gr(z1,z2,E;U)=ℏ∂δ(cl)(z1,z2,E;U)∂E=P/2, (2.12) |
provided z1,2 are turning points. |
For one-dimensional infinite motion, introducing δ(cl)=Ssh(−∞,∞,E;U)/ℏ≡Ssh(E;U)/ℏ and δ(cl)free=Ssh(E;0)/ℏ, we can write the group time delay respectively the free motion as |
δt(cl,1D)gr=ℏ∂(δ(cl)−δ(cl)free)∂E=δt(cl,1D)soj. (2.13) |
Moreover, one may introduce another temporal scale — a phase time delay |
δt(cl)ph=ℏδcl/E. (2.14) |
Also, from Eq. (2.8) we immediately conclude that in 1D the time shift is negative (advance), δtclsoj<0, for an attractive potential U<0 and it is positive (delay) for a repulsive potential U>0. |
Extensions of the definitions of the full classical sojourn time and classical sojourn time delay/advance concepts to the three-dimensional (3D) motion are straightforward. In analogy to Eq. (2.3) the time a particle spends within a 3D volume Ω during the time τ can be defined as |
Consider now a radial motion of a particle in a central stationary field decreasing sufficiently rapidly with the distance from the center. Using the symmetry of the motion towards the center and away from it, we can choose the moment t=0, as corresponding to the position of the closest approach to the center. Then for... |
δt(cl)W=2limt→∞(t(r,U)−r(t,U=0)/v∞), (2.16) |
where r(t,U=0) is the particle's radial coordinate for free motion. Factor 2 counts forward and backward motions in radial direction. We will call this time delay, the Wigner time delay. One can see that this time is equivalent to a classical sojourn time delay, δt(cl)W=δt(cl)soj, defined similarly to Eq. (2.8). Using ... |
δt(cl)soj=δt(cl)W=1E∞∫0(2U(r(t))+r(t)U′(r(t)))dt, (2.17) |
where the integration goes along the particle trajectory r(t). The result holds for potentials decreasing faster than 1/r. We see that in 3D-case there is no direct correspondence between the signs of the potential and the time shift δt(cl)soj. For a power-law potential U=a/rα, α>0, we have a delay, δt(cl)W>0, for a(2−... |
Now, using that in a central field [43] |
t(r)=∫rr0drvr,vr=√v2∞−2U(r)m−M2m2r2, (2.18) |
where r0=r(vr=0) is the turning point, 111If there is no turning point, one puts r0=0. and M is the angular momentum, we can rewrite the limit in Eq. (2.16) as |
limr→∞(t(r)−r/v∞)=limr→∞(r∫r0drvr−rv∞). (2.19) |
For a central potential the shortened action is Ssh(r0,r,E,U)=∫rr0prdr, Ssh(E,U)=∫∞r0prdr, and the classical analog of the phase shift is given by |
ℏδcl(v∞,M)−ℏδcl(v∞,M,U=0)=limr→∞[∫rr0prdr−∫rr0pr(U=0)dr],pr=mvr. (2.20) |
Then, similarly to Eq. (2.9) we can define the group time delay, as the energy derivative of the phase acquired during the whole period of motion (forward and backward), and from comparison with Eq. (2.19) we have |
δt(cl,3D)gr≡2ℏ∂(δ(cl)−δ(cl)free)∂E=δt(cl)W. (2.21) |
As we see, compared to the one-dimensional case (2.13) (where integration limits in expression for Ssh are from −∞ to ∞), in the three-dimensional case (2.21) for the delay in the radial motion there appears extra factor 2. In sect. 3 we shall see that such a delay undergo divergent waves, whereas scattered waves are c... |
Moreover, for systems under the action of external time dependent forces there appear extra time-scales characterizing dynamics. Above we considered undamped mechanical motion. Below we study damped motion. We consider several examples of such a kind, when mechanical trajectories can be explicitly found. We introduce t... |
Consider a particle with a mass m performing a one-dimensional motion along z axis in a slightly anharmonic potential under the action of an external time-dependent force F(t) and some non-conservative force (friction) leading to a dissipation. The equation of motion of the particle is |
¨z(t)+E2Rz(t)+Γ˙z(t)+Λz2(t)=1mF(t), (2.22) |
where ER is the oscillator frequency and Γ>0 is the energy dissipation parameter. The anharmonicity of the oscillator is controlled by the parameter Λ. Within the Hamilton or Lagrange formalism, Eq. (2.22) can be derived, e.g., with the help of introduction of an artificial doubling of the number of degrees of freedom,... |
−^Stϕ(t)=J(t),−^St=d2dt2+E2R+Γddt,J(t)=F(t)−1mΛϕ2(t), (2.23) |
with the differential operator ^St and the source term J, which depends non-linearly on ϕ and on the external force F(t). |
In absence of anharmonicity, Λ=0, solution of Eq. (2.22) can be written as |
z(t;Λ=0)=z0(t)−+∞∫−∞dt′G0(t−t′)w(t′),w(t′)=1mF(t′), (2.24) |
where the Green's function G0(t−t′) satisfies the equation |
^StG0(t−t′)=δ(t−t′). (2.25) |
The quantity z0(t) in Eq. (2.24) stands for the solution of the homogeneous equation ^Stz(t)=0 with initial conditions of the oscillator, namely, its position z0(0) and velocity ˙z0(0) (both are encoded in the oscillation amplitude a0 and the phase α0): |
z0(t)=a0exp(−12Γt)cos(ωRt+α0), (2.26) |
where ωR=√E2R−14Γ2. Two time-scales characterize this solution: the time of the amplitude quenching - the decay time |
t(cl)dec=2/Γ (2.27) |
and the period of oscillations P=2πωR, see Eq. (2.5). The value t(cl)dec describes decay of the field (ϕ=mz variable). The ϕ2 quantity is damping on two times shorter scale. Note that in quantum mechanics we ordinary consider damping of the density variable, |Ψ|2. The definition of the sojourn time (2.4) provides a rel... |
In the Fourier representation Eq. (2.24) acquires simple form |
z(ω;Λ=0)=z0(ω)−G0(ω)w(ω), (2.28) |
where w(ω) is the Fourier transform of the external acceleration w(t), |
w(ω)=+∞∫−∞dte+iωtF(t)m. (2.29) |
The Fourier transform of Eq. (2.25) yields the Green's function |
G0(ω)=+∞∫−∞eiωtG0(t)dt=1ω2−E2R+iΓω. (2.30) |
This Green's function has the retarded property having poles in the lower complex semi-plane at ω=±ωR−i2Γ. As a function of time, it equals to |
G0(t)=e−12ΓtωRsin(ωRt+π)θ(t),θ(t)={0,t<01,t≥0. (2.31) |
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