## Abstract

Potential scattering of free-electron wave packets is considered in the framework of non-stationary quantum-mechanical theory. The general expression for the average angle of scattering is obtained. The traditional quantum-mechanical plane-wave approximation and classical results are shown to be incorporated in the results derived.

© Optical Society of America

## 1. Introduction

Potential scattering of particles is a pretty well-known and studied phenomenon. Its
new features can arise, when quantum-mechanical states of particles have the form of
wave packets. Such a formulation is often mentioned in many papers and books [1], but, as far as we know, almost never is used as a practical
instrument for calculation of experimentally measurable parameters. Meanwhile, the
modern techniques of strong-field photoionization of atoms allows to produce very
nice electron wave packets in a controllable way [2], and this is one of the motivations for re-considering
seriously the problem of wave packet scattering. Another motivation is based on the
observation that sometimes the classical and standard stationary quantum-mechanical
theory of potential scattering give strongly different results. Indeed, if one
calculates the average angle of scattering, θ¯, the classical
mechanics [3] predicts that, in the approximation of small deviations,
θ¯ is of the first order in the potential *U*(* r*), θ¯ ~

*U*. On the other hand, the first non-zero contribution to θ¯ determined by the standard stationary quantum-mechanical theory [4] is determined by the result of the first Born approximation, which is squared in

*U*, θ¯ ~

*U*

^{2}. This is a direct contradiction between the predictions of the classical and standard quantum-mechanical theories, which, as far as we know, has never been explicitly formulated and which requires explanations. The natural explanation to this controversy arises from the non-stationary quantum-mechanical theory of wave-packet scattering which is briefly described below. In this letter, the theory of potential scattering of wave-packets is briefly outlined. The lowest-order quantum-mechanical angle of scattering is found and its relationship and analogy with the predictions of the classical theory are established and discussed.

## 2. Non-stationary wave-packet quantum-mechanical theory of scattering

Inevitably, the theory of wave-packet scattering has to be constructed as a
non-stationary theory. In principle, such a theory has to take into account
irreversible spreading of wave packets. Moreover, the results obtained below can be
referred as the transient ones and they are essentially different from the
asymptotic results of the stationary theory. Non-stationary wave-packet theory of
scattering is formulated as the initial-value problem, and this formulation is
different from the usual *S*-matrix approach in the stationary theory
of potential scattering.

In the non-stationary approach, an electron is assumed to be described
quantum-mechanically by its wave function ψ(* r*,

*t*) obeying the time-dependent Schrödinger equation

with an initial condition Ψ(* r*,

*t*=0) = Ψ

^{(0)}(

*), where Ψ*

**r**^{(0)}(

*) is an unperturbed initial wave function of an electron (at*

**r***t*= 0). It is assumed that at

*t*=0 an electron is located far from a target atom. The exact time-dependent wave function of an electron Ψ(

*,*

**r***t*) can be expanded in a series (integral) of plane waves

where *E*_{p}
= **p**^{2}/2*m* is the free-electron energy corresponding
to the momentum * p* and the probability amplitudes

*C*

_{p}(

*t*) obey the equation

and the initial conditions

In Eq. (3) and below, *Ũ*(* q*) is the Fourier-transform of the potential

*U*(

*),*

**r**In the framework of perturbation theory with respect to the potential *U*(* r*)

where ${C}_{\mathit{p}}^{\left(0\right)}$ =
*C*_{p}
(0) and

Let us assume now that the initial electron wave function Ψ^{(0)}(* r*) is characterized by a Gaussian distribution over momentum

where Δ*r*
_{0} is the size of the unperturbed wave
packet at *t* = 0, **p**_{0} is the momentum of the “center of mass”,
and **r**_{0} is its initial position. In the problem of scattering, in
analogy with the classical picture, the component of the initial-position vector **r**_{0}, perpendicular to **p**_{0}, **r**_{0⊥}, determines the impact parameter ρ = |**r**_{0⊥}|.

The general consideration of this paper is valid for any
Δ*r*
_{0} and *t*, and it takes
into account the spreading effects completely. However, some of our specific
calculations below and the main qualitative conclusions are based on the assumption
that the characteristic wave-packet spreading time

is much longer than any other characteristic times of the problem under
consideration, *t*_{spr}
>>
*t*_{c}
, where *t*_{c}
includes both the interaction time and a time it takes for a scattered electron to
reach the detector. From this point of view, the results obtained in the case of
*t*_{spr}
>>
*t*_{c}
can be referred to as the transient ones and they
can be essentially different from asymptotic longtime results occurring at
*t*_{c}
>>
*t*_{spr}
.

To find the average angle of scattering, let us use the following two-step procedure.
First, let us average the well-defined quantum-mechanical operator
*p̂*_{x}
over the wave function
Ψ (* r*,

*t*). The

*x*-axis is assumed to be lying in the (

**p**_{0},

**r**_{0}) plane perpendicular to

**p**_{0}(directed along the

*z*-axis). For spherically symmetric potentials

*U*(

*r*), (

**p**_{0},

**r**_{0}) is the scattering plane, i.e., the average transverse momentum

*p¯*

_{y}= 0, where 0

*y*is the third Cartesian axis perpendicular to both 0

*x*and 0

*z*. In terms of the probability amplitudes

*C*

_{p}(

*t*),

*p¯*

_{x}is given by

The ρ-dependent average angle of scattering θ¯ is determined as

θ¯(ρ) is the analogue of the classical angle of scattering [5] for an individual electron-atom pair of interacting objects.

By substituting into Eq. (10) the probability amplitude
*C*_{p}
(*t*) (7), we get the first-order
average angle of scattering of a wave packet by a single atom

$$=\frac{1}{{\pi}^{3/2}{p}_{0}}\mid \underset{0}{\overset{t}{\int}}\mathit{dt\prime}\frac{1}{{\left[\Delta r\left(\mathit{t\prime}\right)\right]}^{3}}\int d\mathit{r}\frac{\partial U}{\partial x}\mathrm{exp}\left\{-\frac{{\left(\mathit{r}-{\mathit{r}}_{0}-{\mathit{v}}_{0}t\prime \right)}^{2}}{{\left[\Delta r\mathit{\left(}\mathit{t\prime}\right)\right]}^{2}}\right\}\mid .$$

This expression can be reduced to the “classical-like” form [3] with the effective potential
*U*_{eff}

The effective potential *U*
_{eff}(* r*,

*t*) is determined as [6]

where |${\mathrm{\Psi}}_{c\mathit{.}m\mathit{.}}^{\left(0\right)}$
(* r*′,

*t*)|

^{2}is the squared absolute value of the zero-order electron wave function of the wave packet in its own center-of-mass frame

Eq. (13) looks very similar to the classical angle of scattering,
found in the approximation of small deviations [3]. Except for the replacement of
*U*(*r*) by *U*_{eff}
(* r*,

*t*), the only other difference between Eqs. (13) and the classical one concerns the limits of integration over

*t*: 0 and

*t*instead of -∞ and +∞. But, as we assume that

*t*′ = 0 and

*t*′ =

*t*correspond to, respectively, some instants of time long before and long after scattering, these limits can be substituted by -∞ and +∞ to reduce formally Eq. (13) to the form of the classical expression. But of course, in fact, there is no complete identity between Eqs. (13) and the classical expression, and the difference between them is concentrated in the difference between

*U*(

*r*) by

*U*

_{eff}(

*,*

**r***t*). It should be noted that the effective potential identical to that of Eq. (14) has been introduced earlier in theory of atomic emission in a very strong laser field in the so-called Barrier-Suppression Wave-Packet-Spreading regime of ionization [6]. An origin of the effective potential has rather simple reasons and qualitative explanation. In the wave-packet state, the electron charge-density is spread around its “center of mass” in accordance with the distribution law |${\mathrm{\Psi}}_{c\mathit{.}m\mathit{.}}^{\left(0\right)}$(

*,*

**r***t*)|

^{2}(15). If such a “charged cloud” interacts with other dot-like charges located at some point

*′, the total interaction energy is given by a sum (integral) of contributions from all the parts of the “electron cloud” [*

**r***U*(

*-*

**r***′)] with the weight function |${\mathrm{\Psi}}_{c\mathit{.}m\mathit{.}}^{\left(0\right)}$(*

**r***,*

**r***t*)|

^{2}, in accordance with the definition (14). For the specific case of a pure Coulomb potential

*U*

_{c}(

*r*)=- α/

*r*, the effective potential is known [6] to have the form

where Erf denotes the error function [7]. As at large arguments the error function approaches one, at
large distances, *r* >>
Δ*r*(*t*), the effective potential of Eq.(16) coincides with the Coulomb potential,
*U*_{eff}
(* r*,

*t*) ≈ - α/

*r*. At small arguments,

*x*<< 1, Erf(

*x*) ≈ 2

*x*/√π and, hence, at small distances,

*r*<< Δ

*r*(

*t*),

*U*

_{eff}(

*,*

**r***t*) (15) approaches

In contrast to the Coulomb potential, the effective potential *U*_{eff}
(* r*,

*t*) has no singularity at

*r*=0 [6]. It is worth noticing that for an infinitely extended wave packet (Δ

*r*→∞), the effective potential

*U*

_{eff}(

*,*

**r***t*) (16) turns zero identically. In the case of a pure Coulomb potential

*U*

_{C}(

*r*)=-α/

*r*, the effective potential

*U*

_{eff}(

*r*) is given by Eq. (16). With this effective potential and in the approximation of a non-spreading wave packet, Δ

*r*(

*t*) ≈ Δ

*r*

_{0}, the final result appears to be given by a very simple and nice formula:

where ${\mathrm{\theta}}_{\mathit{\text{cl}}}^{\left(1\right)}$ = 2α/[${\mathit{\text{mv}}}_{0}^{2}$ρ) is the angle of scattering calculated from the classical theory and for the Coulomb potential.

In the limit of a narrow wave packet, Δ*r*
_{0}
<< ρ, θ¯^{(1)} = ${\mathrm{\theta}}_{\mathit{\text{cl}}}^{\left(1\right)}$. Appositely, in the limit of a very wide wave packet,
Δ*r*
_{0} >>
ρ, the first-order quantum-mechanical angle of scattering
quantum-mechanical angle of scattering θ¯^{(1)} (18)
falls as 1/Δ${r}_{0}^{2}$:

In the case of a short-range potential *U*(*r*), the
first-order angle of scattering can be shown to be independent of
Δ*r*
_{0} at small
Δ*r*
_{0}
(Δ*r*
_{0} <<
*d*, ρ) and coinciding with the corresponding classical
expressions. In the limit of a large width of the wave packet,
Δ*r*
_{0} >> ρ
and Δ*r*
_{0} >>
*d* (where *d* is the atomic size), the exponential
factor on the right-hand side of Eq. (12) can be approximated by one to give

Eq. (20) shows that in the wide-packet limit, the first-order
average angle of scattering as a function of
Δ*r*
_{0} falls as
1/Δ${r}_{0}^{4}$ , i.e., much faster
than in the case of a pure Coulomb potential

It should be noted that, whereas the general equations (12) and (13) for θ¯^{(1)} are valid at any
Δ*r*
_{0}, the specific results of Eqs. (18)–(21) are derived under the assumption of a
non-spreading wave packet, Δ*r*(*t*)
≈ Δ*r*
_{0}. This assumption can be
reformulated as the condition that the spreading time
*t*_{spr}
(9) is much longer than the time of scattering
(interaction), *t*_{int}
. The latter, for a short-range
potential can be estimated as *d*/*v*
_{0} to
give

Both assumptions, Δ*r*
_{0} <<
*d* and *t*_{spr}
>>
*t*_{int}
, can be rewritten in the form of the
inequalities

where *ƛ*_{dB}
=
*ħ*/*mv*
_{0} is De-Broglie
wavelength of an electron. These are the conditions under which the
quantum-mechanical expression for the first-order angle of scattering
θ¯^{(1)} [(13) and (18)] coincide exactly with the
corresponding classical expressions. For the long-range Coulomb potential, the
quantum-mechanical expression for the ρ-dependent first-order angle of
scattering (18) coincides with the classical one as long as
Δ*r*
_{0} <<
ρ. In this case the effective interaction time can be determined as
ρ/*v*
_{0}. The substitution of
*d* by ρ in inequalities (23) determines the conditions
under which spreading of the wave packet can be ignored and the wave packet itself
can be considered as a narrow one for scattering by the long-range Coulomb
potential:

## 3. Discussion

It should be noted, that for the electron-atom scattering, probably, classical
effects of the first-order in *U* are hardly observable, because
it’s rather difficult to construct a well localized and long-living
electron wave packet with a size smaller than an atomic size. Such a wave packet
would spread during a very short time on the order of the atomic time (~
10^{-16} s). Vice-versa, in the case of scattering from a focused light
field, such a situation is quite realistic, because the focal size is usually much
larger than the atomic radius. For instance, for electrons with
*v*
_{0} ~ 10^{8} cm/s and for the
focal waist *d* ~ 10^{-2} cm, inequalities (24)
give

In principle, creation and control of such wave packets are realizable with the help
of modern experimental techniques. For example, electron wave packets can be
produced in a controllable way be means of photoionization or multiphoton ionization
of atoms by a laser field. In the case of a not too strong laser field and short
pulses, the energy width of wave packets is expected to be of the order of the
inverse pulse duration τ, Δ*E* ~
1/τ, whereas the mean energy is on the order of the light frequency
ω. This corresponds to the uncertainty of the electron momentum Δ_{p} ~ τ^{-1} ω^{-1/2} and the
width of a packet in space Δ*r* ~ τ
ω^{1/2}. E.g., for ω ~ 0.1 and
τ ~ 10^{5} (~ 30 ps) we get
Δ*r* ~ 10^{5} in atomic units or
Δ*r* ~ 10^{-3} cm. This size falls
into the range of parameters determined by inequalities (24). The spreading time of
such a packet is of the order of 10^{-6} s. During this time an electron
with a speed *v* ~ ω^{1/2} ~
10^{8} cm/s crosses a very large distance about 1 m. A scheme of an
experiment for observing scattering of wave packets by a focused light field can be
similar to the well-known experiment [2] with two laser foci. In this experiment, electrons produced
in one of the foci were scattered by the second one and the energy distribution of
scattered electrons was investigated. Under similar conditions, by measuring the
average angle of scattering for different relations between
Δ*r*
_{0} and the focal waist
*d*, one has to find a big difference between the cases
Δ*r*
_{0} < *d* and
Δ*r*
_{0} > *d*. A
much larger classical angle of scattering has to be observed in the first of these
two cases. Transition from the case Δ*r*
_{0}
< *d* to Δ*r*
_{0}
> *d* can be provided either by changing the conditions of
focusing in the second focus or by changing the pulse duration of the first laser,
which is supposed to ionize atoms and to produce wave packets.

In addition to explanations given above, it may be reasonable to mention an
alternative interpretation of the results derived. According to this interpretation,
the non-zero first-order term θ¯^{(1)} ≠ 0
(12) in the perturbation-theory expansion of the average angle of scattering arises
owing to interference between the incoming and scattered parts of the wave function.
Such interference exists only in the transient case when the initial size of the
wave packet Δ*r*
_{0} is much smaller than the size
of a target *d* and the wave-packet spreading time is much longer
than the time it takes for the scattered electron to reach the detector. These
results can be essentially different from the asymptotic infinitely-long-time
results. In terms of the interference interpretation, it is clear why
θ¯^{(1)} vanishes in the plane-wave limit. In this
case simply there is nothing to interfere with for partial plane waves scattered at
any non-zero angle with respect to the incident plane wave. To formulate explicitly
the interference interpretation of the results described one has to investigate the
structure of the wave function after scattering under these or other assumptions
about its initial size, time of scattering, distance from the atom after scattering,
etc.

The results obtained indicate a significant difference between two cases, which sometimes are referred to as equivalent ones: scattering of a stream of well-localized particles, which are incoherent to each other, and scattering of a particle, which has the wave function close to a single plane wave. According to our results, the average angle of scattering in these two cases is given either by the classical expression, or by the well-known result derived in the first Born approximation of the traditional quantum-mechanical theory of scattering, respectively.

It should be noted, finally, that the problem discussed can be related to even much more general and fundamental features of particles in beams. Indeed, practically under any conditions electrons have some degree of localization determined by the size of the their wave functions (wave packets). This is a quantum mechanical size, hardly recognizable in the routine experiments. In particular, in the case of electron beams, produced by electron guns, the only size, which is usually measured and mentioned, is the transverse size of the beam. The above-discussed quantum-mechanical size of electrons can be either equal or much smaller than the size of the beam. The quantum-mechanical size of wave packets remains a hidden parameter of electrons hardly ever seen and discussed. In our opinion, scattering of electrons by ponderomotive potentials of focused lasers and measuring, e.g., the verage angle of scattering can be used to measure the above-discussed quantum-mechanical size of electrons and to make conclusions on the size of the wave packets describing electrons in beams.

## References and links

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**3. **L.D. Landau and E.M. Lifshitz, “Mechanics” (Permagon Press, Oxford-New York1976).

**4. **L.D. Landau and E.M. Lifshitz, “Quantum Mechanics”, (Permagon Press, Oxford-New York1977).

**5. **M.V. Fedorov, S.P. Goreslavsky, and V.S. Letokhov, “Ponderomotive forces and stimulated Compton scattering on free electrons in a laser field”, Phys. Rev. E , **55**, 1015 (1997) [CrossRef]

**6. **M.V. Fedorov and J. Peatross, “Strong-field photoionization and emission of light in the wave-packet-spreading regime”, Phys. Rev. A **52**, 504 (1995). [CrossRef]

**7. **I.S. Gradstein and I.M. Ryzhik, “Table of Integrals, Series and Products” (Academic, New York1980).