An approach for calculating tunneling amplitudes from a nonlocalized initial state is presented. Generalizing the matching conditions and equations of motion to allow for complex momentum permits a description of tunneling in the presence of so-called classical motion. Possible applications of the method are presented

Bowcock, Peter

Gregory, Ruth

English

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Fermi Naiiorlal Accelerator Laboratory
FERMI LAB- Pub- 91/i 42 -A
April 1991
TUNNELLING FROM NON-LOCALISED INITIAL STATES
/JdqJv -/2_,o
Peter Bowcoek / 93 _ 9
Enrico Fermi Institute, University of Chicago, 5640 S.Ellis Ave, Chicago, IL 60637, U.S.A.
Ruth Gregory /9-- / O
NASA/FermiIab AJtrophynics Center, Fermi National Accelerator Laboratory
P.O.Boz 500, Batavia, IL 60510, U.S.A.
ABSTRACT
We present a new approach for calculating tunnelling amplitudes from a
non-localised initial state. Gener_sing the matcl_ng conditions and equations
of motion to allow for 'complex' momentum permits a description of tunnelling
in the presence of so-called classical motion. We comment on possible applica-
tions of the method.
(NASA-CR-18B44_) TUNNELLING FROM
NON-LQCALISFO INITIAL STAT6S (_ermi
Nation,_l Accelerator ldh.) i0 p CSCL 20H
N91-24869
Unc1_s
G]/12 0019360
V Operated by Universities Research Association Inc. under contract with the United States Department of Energy
https://ntrs.nasa.gov/search.jsp?R=19910015555 2020-03-19T18:05:59+00:00Z
That quantum mechanical probability density can permeate into regions in which a
classical particle is forbidden was (and perhaps still is) one of the more intriguing features of
the Schr6dinger equation. Rapidly however, it was turned into one of the major successes of
the early wave mechanics in providing a successful theoretical description of alpha particle
emission in radioactive decay. Since then, tunneLLing has found many applications, yet
our understanding of many aspects of the phenomenon remains limited. Motivated by
difilculties 1 in calculating tunnelling amplitudes in recent two-fleld inflationary models 2-4,
we re-examlned the quasi-classical approximation to the Schr6dinger equation to see if
one could incorporate more complicated evolution of the wave function without losing the
simplicity of that approach. We found that the problem reduces to two key issues: that
of incorporating complex momentum into a quasi-classical description, and the related
problem of matching conditions between different asymptotic regimes. Here we will show
how to resolve these questions, illustrating some results from a test potential. A more
detailed exposition may be found in a companion papeP.
First let us summarise the current status for calculating tunnelling amplitudes. Most
tunnelling calculations proceed via the quasi-classical approximation s, which extracts the
leading order "classical" behaviour of the system. If we assume the wave function takes
the form
: q,C ,t) = , (1)
then the Schrodinger equation gives
VG) 2 i_t V_ G E U(x) (2)2m
as the equation of motion for tr. The quasi-classical approximation drops the O(_t) term
in (2), which dearly requires that IVo-I' > nlV2o-I.
In tunnelling, we need to solve the Schr6dinger equation in regions where U > E,
therefore we set tr = ig and obtain
(Vg)' = 2m(_r- E) (3a)
=_ g = _/2rn(U- E)dI, (3b)
2
where 1is the path which minimises the integral and x! is the point of emergence of that
path into the classical r_gime. This geometric solution to the problem in more than one
dimension was first obtained by Banks, Bender and Wu 7 for the case of tunnelling from
a localised state, they called l the eJcape path. The calculation of 1 and g in practise
is facilitated by the Euclidean time prescription whereby we associate Vg with Pc, a
Euclidean momentum, and (3a) becomes a Hamiltonlan problem of particle motion in the
(inverted) potential -U. It is then a classical dynamics problem to calculate the trajectory
x(r), which interpolates between the initial position and the position of emergence from
the barrier.
This approach is powerful, however it does have one crucial restriction, namely, it
requires that space be divided into regions in which V<r is entirely real or entirely imaginary
so that we can identify p = V0" with some i("time"). Clearly we then need i to vanish on
the boundaries between these regions. In the context of the escape path, we need ±2 _ 0
at each end of the path. In general it requires two initial conditions to fix the start of
the escape path, which uses up the requisite number of boundary conditions, leaving no
additional freedom for the end point, which in general will not satisfy _2 = 0. Another
related problem is that we have no allowance for the transport of real momentum under a
barrier, which we know definitely does occur in the case of a continuous symmetry. There
is also the problem of picking an initial position from which to integrate, since we can only
localise a particle at the expense of information about its momentum. We clearly need a
more general picture of tunnelling.
In order to solve the Schr6dinger equation when the momentum is complex, we rewrite
(2) in terms of the real variables y and g, where _ = f + ig:
(VI) 2 _ (Vg) 2 + _V_g = 2raCE- tr)
2Vf.Vg - hV2f = O.
(4a)
(4b)
In the quasi-classical approximation we neglect the O(t_) terms in these equations, this will
be valid unless
(5)
3
i.e. unless IVf[, IVg] is small or ]V2fl, IV2gl large, this latter situation occurs at caustics
of the motion. The simplest way to visualise this is to consider a plane wave scattering
oiT some barrier. The integral curves of Vf trace out the path that a family of classical
particles would follow in that potential. Clearly, neighbouring, initially parallel, trajecto-
ries will at some stage cross, and at such a place, referred to as a caustic, V2f becomes
unbounded.
Having identified the boundary between different asymptotic regions, we now focus on
the turning surface to obtain the matching conditions. We first consider a local coordinate
system along the caustic .[z,9 i} where z is orthogonal distance from the caustic, and the
yi are cartesian coordinates in the surface of the caustic. Provided the caustic is not too
strongly curved, the potential is roughly a function of z, and we may deduce that the true
(continuous) solution of the Schr6dinger equation is
_b(z,7/')= eiP,"/nAi _-2/3(2raU, =),/3 z + _ (6)
where Ai(z) is the Airy function. That the curvature of the causticnot be too large trans-
lates to a bound on V_U': noting that (6) implies that the quasi-classicalapproximation
breaksdownwhenI=I< )-,/3,weobtainalimitof << Equa-
tion(6) glvcs us the inltalconditions for f and g for integratingout beyond the caustic.
Clearly the paralld momenta, Pl axe conserved across the boundary, i.e.Vif, Vig con-
served. The orthogonal components are determined on eithersideof the boundary by using
equation (4a) and demanding that real(V f) and imaginary (Vg) momenta be orthogonal
(equation(4b)). This fixesthe initialconditions for the momentum on the other side of the
boundary. We now need to propagate f and g out from the boundary, i.e.solve (4a,b).
In order to facilitatethe solution of (4a) we assume E ._ U, which implies that Vg is
dominant under the barrier,however, we do not wish to neglect Vf as an O(h) correction,
otherwise we would use existingtechnlqucs. Instead we want to consider a situationwhere
_ _ _ 1, and iterativelysolve
(Vg)" = 2mU - (2mE - (V f)_) (7_)
vy.vg= 0, (7b)
4
bearing in mind that (V f) 2, E are of the same order, and small compared with (Vg) 2, FT.
The first step is to find the leading behaviour, that is, to solve (Vg) 2 = 2mU. In
order to do this, we use existing techniques. We regard the solution under the barrier as
being composed of a family of escape paths {l(a)} originating from the turning surface at
(!/_..),0), and propagating under the barrier until a caustic is reached. Along each escape
path we have a solution for g determined by (3b) with E = 0. This will give us a leading
order solution to the problem. Now we use the information from (7b), that f is constant
along integral curves of g, to set f = f(_0_..)(x)) where _o_,,)(x) = V_.) are the equations
for the family of escape paths 1(..). Finally, we input this solution for f back into (7a) to
obtain the correct form of g to order E/U. Once we have the solution under the barrier
we may match across into the far 'classical' r_gime and follow a similar procedure to the
above to complete the solution of the Schr6dinger equation.
To recap: this method solves the SchrSdinger equation in the stationary quasi-classical
approximation for tunnelling in potentials with non-localised minima. The restrictions on
the type of potentials it can deal with are that E << U under the barrier and that the
transverse derivatives of U at the matching boundary be bounded by h-4/3..,rr 2/3=. This
method is therefore ideal for problems involving scattering off a 'wall'.
As a simple illustration of the method, consider a plane wave, e_k'x/h, scattering off
a wedge potential in two dimensions, U(x) = VO(z)®(a - z - eV), see fig. 1. Here, our
initial conditions at z = 0 are g = 0, f = /%V,, g,, = 2_V and f,. = 0. Integrating
out is straightforward since the potential is constant and gives g = V_-_--V_. Imposing
f =const. along integral curves of g, which in this case are V = V,, simply gives / = k,v,
then inputting this back into (7a) gives g = X/2mV- kt2z = _z. In this example our
iterative procedure under the barrier is now complete. At the far edge of the barrier the
appropriate boundary conditions are now non-standard, the momentum parallel to the far
edge being complex: Pll = (P_ -ie_;)T, where T is the tangent vector to the far edge
of the barrier. Orthogonallty of the real and imaginary momenta and (7a) requires that
P± = (Px + ie_;_x )N+O(ta) • Setting
_=z-a+ev , T/=V-¢(z-a) (8)
5
(see fig. 2) we may integrate out from the boundary to obtain
fo., = Pl ( + P, 77
Pl
(9)
Note some interesting features about this solution. The incoming and outgoing real mo-
menta are not parallel, although they subtend the same angle with each side of the barrier.
Note too that g is spatially dependent, decreasing in regions where the barrier is thinner.
However, the really curious features of scattering off this potential are most clearly ex-
hibited if we consider a Gausslan wave packet scattering off the barrier. For the sake of
definiteness we take the momentum to be peaked at (Pl,P,) with spread A_2, hitting the
origin at t = 0. Such an incident wave packet might reasonably be supposed to represent
a particle scattering off the barrier. For the trajectory of the outgoing peak we obtain
= p,(1 + 1 + (1 -- 3 (10)
with a transmission amplitude of e -''ou'/t'. In other words, a particle 'incident' on the
barrier at _ = z = !1 = 0 emerges at T/_ = 2--_(1 + 1 -2aMi/,¢)_). One can also calculate
the time of emergence, f_ - "_P2 fl _ 2tt_tt/t¢)3 The particle emerges considerably
-- 2Ahpx_ k_
'downstream' from where one might expect it, neither at x = (a, 0), nor at x : (u, ca), but
at x = (a--eT/e, _/e). Furthermore, the time at which it emerges is potentially large compared
with _ and even, if P3 < 0, potentially negative! In fact, both these peculiarities arise
from a simple physical reason, the fact that the most energetically favourable time for the
Gaussian to tunnel is not necessarily when the peak hits the barrier. Tunnelling amplitudes
depend exponentially on the size of the barrier, therefore it is more favourable to tunnel
where the barrier is thinner, hence the dependence of the transition amplitude on zo,t. On
the other hand, the probability density along z - 0 is damped by an exponential factor
depending upon how far away that point is from the peak of the Gaussian. Clearly there
will be a pay off between these two factors which may mean that it is more energetically
favourable for the fringe of the Gausslan to tunnel, rather than its peak.
6
Apart from giving an initial step in the problem of calculating two-field tunnelling
amplitudes, our method (and illustrative example) may also have applications in other
areas of tunnelling theory. For instance, one of the outstanding problems in tunnelling
is defining a unanimously acceptable tunnelling time. From the uncertainty principle, we
know such a time, _T, should be of order _, but when using stationary states in a time
independent potential, finding tT is not so straightforward, indeed quite a controversy
exists s. Our example suggested that one of these definitions, the extrapolated phase time 9
is not a good definition of tunnelling time as it stands. Unfortunately, we can shed no light
on what is!
While we have only applied this method to a small range of examples s, the results
we have so far obtained are encouraging: the modifications seem to give quite dii_erent
qualitative pictures for scattering. The esssential step forward from the Banks, Bender
and Wu approach was to identify a general boundary between different WKB r4gimes. It
can only be hoped that a similar generalisation of Coleman's field theoretic procedure 1° is
possible.
Acknowledgements.
We would like to thank Je_ Harvey and Rocky Kolb for useful discussions. We would
also like to thank the Aspen Center for Physics and the Institute for Theoretical Physics
at Santa Barbara for hospitality while this work was in progress. This work was supported
in part by the DoE, grant DEFG02-90-ER-40560, and the NSF, PHY900036 (PB), and by
the DoE and NASA, grant NAGW-1340, at Fermilab (RG).
References.
[1] E.Wcinberg, Phys. Rev. D40 3950 (1989).
R.Holman, E.Kolb, S.Vadas, Y.Wang and E.Welnberg, Ph_/_. Left. 237B 37 (1990).
[2] D.La and P.St_nhardt. Phys. Rev. Left. 62 376 (1989).
[3] P.Steinhardt and F.Aeeetta, Ph_/s. Rev. Left. 64 2740 (1990).
[4] K.F.Adams and K.Freese, Ph_/s. Rev. D43 353 (1991).
[5] P.Bowcock and R.Gregory, Multi-dimen,_ionaI tunnelling and complez momentum.
(EFI 91-19, FERMILAB-PUB-91/103-A).
7
[6] L.Land_tuand E.Lifshitz, Quantum Mechanics (Pergamon Press).
[7] T.Banks, C.Bender and T.Wu, Phys. Rev. D8 3346 (1973).
T.Banks, C.Bender and T.Wu, Phys. Rev. D8 3366 (1973).
[8] E.Hauge and J.Stovneng, Reg. Mod. Phys. 61 917 (1989).
[9] E.H.Hauge, J'P'Falck and T.A.Fjeldy, Phys. Rev. BS6 4203 (1987).
[10] S.Coleman, Phys. Rev. D15 2929 (1977).
C.Callan and S.Coleman, Phys. Rev. D16 1762 (1977).
x-a-Ey
x=O
Figure 1. The barrier U(x) -- VO(x)(_(a - x - ey).
9
out
Figure 2. Trajectory of Gaussian peak scattering off barrier.
10


Tunnelling from non-localised initial states

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