arXiv:nucl-th/0111076v36Jan2002
1
TWO-POTENTIAL APPROACH TO MULTI-DIMENSIONAL
TUNNELING
S.A. GURVITZ
Department of Particle Physics, Weizmann Institute of Science,
Rehovot 76100, Israel
We consider tunneling to the continuum in a multi-dimensional potential. It is
demonstrate that this problem can be treated as two separate problems: a) a
bound state and b) a non-resonance scattering problem, by a proper splitting
of the potential into two components. Finally we obtain the resonance energy
and the partial tunneling widths in terms of the bound and the scattering state
wave functions. This result can be used in a variety of tunneling problems. As
an application we consider the ionization of atomic states by an external field.
We obtain very simple analytical expressions for the tunneling width and for the
angular distribution of tunneling electrons. It is shown that the angular spread of
electrons in the final state is determined by the semi-classical traversal tunneling
time.
Contents
1 Introduction 2
2 General formalism 2
3 Ionization of bound state in semi-classical approximation 7
4 Partial width and the tunneling time 9
5 Total width and angular distribution of ionized particles 10
6 Summary 12
Acknowledgments 12
References 12
2 S.A. Gurvitz
1 Introduction
This paper is devoted to the memory of Michael Marinov, my friend and
collaborator. Within the scope of his broad scientific activity Michael paid spe-
cial attention to different aspects of quantum mechanical tunneling, as for in-
stance to the coherent and quantum oscillation effects in tunneling processes,1,2
and to the traversal tunneling time.3,4
In particular Michael was interested in
a semi-classical description of multi-dimensional tunneling. Our frequent dis-
cussions of this subject were very illuminating and helpful to me, especially
in relation with the two-potential approach to multi-dimensional tunneling,
which I shall present below.
It is well known that propagation through classically forbidden domains
has been extensively studied since the early days of quantum mechanics. Yet,
its exact treatment still remains very complicated and often not practical, in
particular for the multi-dimensional case. The treatment of the tunneling prob-
lem can be essentially simplified by reducing it to two separate problems: a
bound state plus a non-resonance scattering state. This can be done consis-
tently in the two-potential approach (TPA).5-7
This approach provides better
physical insight than other existing approximation methods, and it is simple
and very accurate. It was originally derived by us for the one-dimensional
case. Here we present an extension of TPA to many degrees of freedom and
its application to ionization of atomic states.
2 General formalism
Consider a quantum system in a multi-dimensional space, described by
the Schršodinger equation (we have adapted units where h = 1)
i

t
|(t) = [K + V (x)]|(t) , (1)
where x = {x1, x2, . . . , xn}, K = 2
/2m is the kinetic energy term and V (x)
is a multi-dimensional potential. The latter contains a barrier that divides the
entire space into two domains (the "inner" and the "outer" one), such that the
classical motion of the system in the inner domain is confined. Yet this does not
hold for the quantum mechanical motion of this system. It is well known that
any quantum state with a positive energy, localized initially inside the inner
domain, tunnels to continuum (the outer domain) through the barrier. If the
time-dependence of this process is dominated by its exponential component,
we have a quasi-stationary (resonance) state. Below we describe a simple and
consistent procedure for treatment of such multi-dimensional quasi-stationary
Two-potential approach to multi-dimensional tunneling 3
states, which represents a generalization of the TPA, developed in Refs. 5­7
for one-dimensional case.
Let us first introduce a hyper-surface  inside the potential barrier, which
separates the whole space into the inner and the outer regions, as shown
schematically in Fig. 1. Respectively, the potential V (x) can be represented as
1

s Outer region
1
Inner region
x
x
s2
2
Figure 1: An example of the separation between the inner and the outer regions for a two-
dimensional potential. S1 and S2 denote the equipotential lines for the energy E0, Eq. (2).
The shaded space correspond to the classically forbidden region. The separation line  is
taken as a circle.
a sum of two components, V (x) = U(x)+W(x). The first component U(x) is
defined in such a way that U(x) = V (x) in the inner region, and U(x) = U0 in
the outer region, where U0 is the minimal value of V (x) on the hyper-surface,
U0 =min V (x  ). Respectively, the potential W(x) = 0 in the inner region,
and W(x) = V (x) - U0 in the outer region.
We start with the bound state 0(x) of the "inner" potential,
(K + U)|0 = E0|0 , (2)
where E0 < U0. The potential W(x) is switched on at t = 0. Then the state
|0 becomes the wave packet
(x, t) = b0(t)0(x)e-iE0t
+ bk(t)k(x)e-iEkt dk
(2)n
(3)
where b0(t), bk(t) are the probability amplitudes of finding the system in the
corresponding eigenstates |0,k of the Hamiltonian H0 = K + U. For sim-
plicity we assume that the potential U(x) contains only one bound state |0
4 S.A. Gurvitz
corresponding to the energy E0, Eq. (2). The amplitudes b0(t), bk(t) can be
found from the Schršodinger equation it| = (H0 + W)| , supplemented
with the initial conditions: b0(t) = 1, bk(0) = 0.
If the state |(t) is a metastable one, the probability of finding the system
inside the inner region drops down exponentially: |b0(t)|2
 e-t
, where  is
the width of this state. This implies that  is determined by the exponential
component of the amplitude b0(t). The latter corresponds to a pole in the
complex E-plane of the Laplace transformed amplitude ~b0(E):
~b0(E) =

0
eiEt
b0(t) dt (4)
The amplitude ~b0(E) can be found from the Schršodinger equation by using the
Green's function technique. We obtain:6
~b0(E) =
i
E - E0 - 0|W|0 - 0|W ~G(E)W|0
(5)
where the Green's function ~G is given by
~G(E) = G0(E)(1 + ~W ~G(E)), (6)
and
G0(E) =
1 - 
E + U0 - K - U
,  = |0 0| . (7)
Here ~W(x) = W(x) + U0 is the outer component of the potential V (x). Note
that Eqs. (5)- (7) represent the standard perturbation theory, except for the
"renormalized" distortion potential, i.e. W(x) is replaced by ~W(x) in the
Green's function ~G, Eq. (6).6
It follows from Eq. (5) that a pole of ~b0(E) (at E = Er - i/2), that
determines the energy Er and the width  of a quasi-stationary state, is given
by the equation
E = E0 + 0|W|0 + 0|W ~G(E)W|0 . (8)
One can easily show6
that Eq. (8) determines in fact, the poles in the total
Green's function G(E) = (E - K - V )-1
at complex E. Since their position
depends only on the potential V (x), the resonance energy Er and the width ,
given by Eq. (8) are independent of the choice of the separation hyper-surface,
.
One can solve Eq. (8) perturbatively by using the standard Born series for
the Green's function
~G = G0 + G0
~WG0 + G0
~WG0
~WG0 +    . (9)
Two-potential approach to multi-dimensional tunneling 5
Substituting Eq. (9) into Eq. (8) we obtain the corresponding series for Er -
i/2. Yet, such a series converges very slowly.
For this reason we proposed in Ref. 6 a different expansion for ~G, which
converges much faster than the Born series. This is achieved by expanding ~G
in powers of the Greens function G ~W ,
G ~W (E) =
1
E - K - ~W
, (10)
corresponding to the outer potential ~W, instead of G0, Eqs. (7), (9), which
corresponds to the inner potential U. One then obtains from Eqs. (6), (10)
~G = G ~W + G ~W (U - U0) ~G - ~G ~W (1 + ~W ~G) (11)
Iterating Eq. (11) in powers of G ~W and then substituting the result into
Eq. (8) we obtain the desirable perturbative expansion for the resonant energy
and the width. Using simple physical reasonings and direct evaluation of higher
order terms for one dimensional case, we demonstrated in Refs. 6, 7 that this
expansion converges very fast, so that ~G is well approximated by the first term
G ~W . In this case Eq. (8) becomes:
E = E0 + 0|W|0 + 0|WG ~W (E)W|0 , (12)
Solving Eq. (12) for complex E = Er - i/2 we find the resonance energy Er
and the width  with a high accuracy (of order /U0).
Let us assume that that ||,   E0, where  = Er - E0 is the energy
shift. In this case we can solve Eq. (12) iteratively by replacing the argument
E in the Green's function by E0:
E = E0 + 0|W|0 + 0|WG ~W (E0)W|0 . (13)
Equation (13) can be significantly simplified. Indeed, the Green's function G ~W
satisfies the Schršodinger equation
E0 - K - ~W(x) G ~W (E; x, x
) = (x - x
) , (14)
which can be rewritten as
W(x)G ~W (E0; x, x
) = E0 - U0 -
2
2m
G ~W (E0; x, x
) - (x - x
) . (15)
Substituting Eq. (15) into Eq. (13) we find that the corresponding contribution
from the kinetic energy term (2
/2m) can be transformed as
0(x)2
G ~W (E0; x, x
) = (0G ~W ) - (G ~W 0) + G ~W 2
0 (16)
6 S.A. Gurvitz
In addition, the potential W(x) is zero in the inner region, so that the integra-
tion over x, x
in the matrix elements (13) takes place only in the outer region,
Rout. Then using Eq. (16) and the Gauss theorem we find
Rout
0(x)2
G ~W (E0; x, x
)dx =

[ G ~W n0 - 0nG ~W ] d
+
Rout
G ~W 2
0dx (17)
Since the wave function of the (inner) bound state in the outer region satisfies
the Schršodinger equation with the potential U0, the last term in Eq. (17) can
be rewritten as
Rout
G ~W 2
0dx = 2m(U0 - E0)
Rout
G ~W 0dx , (18)
Using Eqs. (15), (17), (18) we then obtain
0(x)W(x)G ~W (E0; x, x
)dx =
1
2m

[ G ~W n0 - 0nG ~W ] d - 0(x
)
(19)
Substituting Eq. (19) into Eq. (13) we find that the latter can be rewritten as
E = E0 -
1
2m x
0(x)

n G ~W (E0; x, x
)W(x
)0(x
)dd
, (20)
where

 means the gradient on the right minus the gradient on the left.
The x
integration can be carried out in the same way by using Eqs. (15)-
(18). Again, the 2
0(x
) term cancels as well as the -function contribu-
tion. Finally we obtain that the resonance energy and the width are expressed
through a double integral over the hyper-surface 
E = E0 -
1
(2m)2
x,x
0(x)

n G ~W (E0; x, x
)

n 0(x
)dd
. (21)
The resonance width  can be easily obtained from Eq. (21) by taking the
spectral representation for the Green's function,
G ~W =
|k k|
E0 - Ek - i
dk
(2)n
, (22)
Two-potential approach to multi-dimensional tunneling 7
where Ek = k2
/2m and k are the eigenstates of the "outer" Hamiltonian
corresponding to non-resonance scattering states.
[K + ~W(x)]k(x) = Ekk(x) , (23)
Using Eqs. (21), (22) one obtains the total width as an integral over the partial
width k
 = k
dk1    dkn-1
(2)n-1
|k|=k0
(24)
where k0 =

2mE0 and
k =
1
4mkn x
0(x)

n k(x)d
2
|k|=k0
. (25)
Equations. (24) and (25) represent our final result for the total and partial
tunneling widths of the resonance state with energy Er = E0, Eq. (2) . a
The
latter is the energy of a pure bound state 0 in the inner potential U(x). Both
k and E0 are independent of the hyper-surface  (up to the terms of the order
of /U0), providing that  is taken far away from the equipotential surfaces
V (x) = E0 .7
Equations. (2), (24), (25) considerably simplify the multi-dimensional tun-
neling problem, by reducing it to the bound state problem. The latter can
be treated with standard techniques. Some examples of applications of our
method are given below.
3 Ionization of bound state in semi-classical approximation
Consider a three-dimensional potential u(r), which contains a bound state
0(r) with the energy 0 < 0. Let us superimpose an additional potential v(x),
depending on the coordinate x. We assume that v(x) ionizes the bound state
0, i.e. u(r) + v(x) < 0 for x  . We also assume that u(r) and v(x) do
not overlap, such that u(r)  0 for x > x1 and v(x)  0 for x < x2, where
x1 < x2. Now taking the separation hyper-surface  as a plane, x = x0 where
x1 < x0 < x2, we can identify u(r) and v(x) with the inner and the outer
potentials, U and ~W, Eqs. (2), (23). Then the bound state wave function
0(r) coincides with the inner wave function 0; Eq. (2).
Providing that v(x) is a function of x only, the outer wave function,
Eq. (23) can be written as a product k(r) = exp(iqb)k(x), with r = (x, b),
a Equation (25) resembles the Bardeen formula8 for energy splitting of two degenerate levels
in a double-well potential.
8 S.A. Gurvitz
k = (k, q) and k2
= 2m(u0 + 0), where u0 = -v(x  ). As a result,
Eq. (25) for the partial width q reads
q =
1
4mk
eiqb
0(x, b)

x
k(x) - k(x)

x
0(x, b) d 2
b
2
x=x0
, (26)
where k = 2m(u0 + 0) - q2. Respectively, the total width, Eq. (24), is
given by
 = q
d 2
q
(2)2
(27)
Since k(x) is a non-resonant wave function, it falls down exponentially
inside the barrier and can be evaluated by using the standard semi-classical
approximation.9
One can write
k(x) =
k
|~pq(x)|
e
-
xf
x
|~pq(x
)|dx
. (28)
Here ~pq(x) = 2m[0 - v(x)] - q2, and xf is the outer classical turning point,
~pq(xf ) = 0.
The inner wave-function describes the bound state and therefore it de-
creases exponentially under the barrier, as well
0(x, b)  e
-
(x2+b2)1/2
xi
|~p(x
,b)|dx
. (29)
Here |~p(x, b)| = 2m[0 - u(x, b)], and xi is the inner classical turning point.
It follows from Eq. (29) that 0(x, b) decreases very quickly with b so that only
small b contribute in the integral (26). Thus, we can approximate the inner
wave function and its derivative as
0(x, b) 
xi
x
0(xi, 0) e
-
x
xi
|~p(x
,0)|dx
-|~p(x,0)|b2
/2x
, (30)
0(x, b)
x
 -|~p(x, 0)| 0(x, b) . (31)
Substituting Eqs. (28), (30), (31) into Eq. (26) we can perform the b-integration
by using
eiqb
0(x0, b)d 2
b =
2xi
|~p(x0, 0)|
0(x0, 0) e
-
x0
xi
|~p(x
,0)|dx
-x0q2
/2|~p(x0,0)|
.
(32)
Two-potential approach to multi-dimensional tunneling 9
Since x0  xi the exponent in Eq. (32) can be approximated as
x0
xi
|~p(x
, 0)|dx
+
x0q2
2|~p(x0, 0)|

x0
xi
[~p2
(x
, 0) + q2
]1/2
dx
(33)
Now using Eqs. (32), (33), and k/x  |~pq(x)|k, we obtain the follow-
ing formula for the partial width,
q =
42
x2
i 2
0(xi, 0)
m(2m|0|)1/2
e
-2
xf
xi
pq(x)dx
, (34)
where
pq(x) = 2m[u(x, 0) + v(x) - 0] + q2 , (35)
is the absolute value of the imaginary momentum in the direction of the ex-
ternal field.
4 Partial width and the tunneling time
In contrast with the one-dimensional tunneling, the final state in a multi-
dimensional tunneling is highly degenerate. Hence, a tunneling particle can
be found in different final states with the same total energy. In our case these
are all the states with different lateral momenta q, leading to the angular
distribution of a tunneling particle. Using our result for the partial width,
Eq. (34), one easily finds that this distribution is of a Gaussian shape. Indeed,
by expanding the exponent of Eq. (34) in powers of q2
we obtain
xf
xi
pq(x)dx  S(E0, xi, xf ) +
q2
2m
xf
xi
mdx
p(x)
, (36)
where p(x) = pq=0(x) and
S(0, xi, xf ) =
xf
xi
p(x)dx (37)
is the action in the classically forbidden region. Then Eq. (34) reads
q =
42
x2
i 2
0(xi, 0)
m(2m|0|)1/2
e-2S(0,xi,xf )-q2
/m
, (38)
where
 =
xf
xi
mdx
|p(x)|
, (39)
10 S.A. Gurvitz
is the Bšuttiker-Landauer tunneling time.10
The latter is assumed to be an
amount of time, which a particle spends traversing a tunnel barrier. It ap-
pears, however, that such a quantity is not uniquely defined in one-dimensional
tunneling.11,3
Yet, the situation is different for the multi-dimensional case. Our
result, Eq. (38), shows that the Bšuttiker-Landauer traversal tunneling time 
is related to a well defined quantity, namely to the lateral spread of a tunnel-
ing particle (r0).4
Indeed, the latter corresponds to the inverse average lateral
momentum, r0 = 1/ < q >, where < q >= m/ as obtained from Eq. (38).
The Gaussian shape of angular distribution of tunneling particles is a gen-
eral phenomenon, which can be understood in the following way.4
It is well
known that the most probable tunneling path corresponds to the minimal ac-
tion. In our case this path is along the direction of the external field. Therefore
any momentum q, normal to this direction raises the action by a term propor-
tional to the corresponding energy q2
/2m. This leads to the Gaussian fall-off
of the related probability distribution.
5 Total width and angular distribution of ionized particles
Substituting Eq. (38) into Eq. (27) and performing the q-integration, we
obtain the following expression for the total width
 =
 x2
i 2
0(xi, 0)
 (2m|0|)1/2
e-2S(E0,xi,xf )
. (40)
Using this result, Eq. (38) can be rewritten in a more compact form as
q =
4
m
  e-q2
/m
(41)
Let us consider two exemplifying applications of Eqs. (40), (41).
(a) Hydrogen atom
Consider the Hydrogen atom in the ground state under the electric field E.
We use the atomic units in which m = 1, the ground state energy 0 = -1/2
and the ground state wave function 0(r) = exp(-r)/

. In these units the
inner and the outer potentials are: u(r) = -1/r and v(x) = -Ex. Then the
action S, Eq. (37) reads
S =
xf
xi
1 -
2
x
- 2Exdx 
xf
xi

1 - 2Exdx -
xf
xi
dx
x

1 - 2Ex
. (42)
Two-potential approach to multi-dimensional tunneling 11
Although the main contribution to the action is coming from large x, we have
keep in (42) also the next term of expansion (see Ref. 12). Performing the
integration in Eq. (42) we obtain for the action
S 
1
3E
+ ln
Exi
2
-
xi
2
, (43)
where we assumed that Exi  1. Respectively for the "tunneling" one obtains
 = 1/E. Using these results we easily find from Eq. (40) the well-known result
for the ionization width of the Hydrogen atom12
 =
4
E
e-2/3E
. (44)
Respectively, the angular distribution of ionized electrons, Eq. (41), is
q =
4
E
e-q2
/E
. (45)
(b) Potential well with short range forces
The bound s-state wave function of a short range potential at distances
larger then the radius of the well a has the following form
0(r) =
A


r
e-r
, (46)
where  = (2|0|)1/2
and A is a dimensionless constant depending on the
specific form of the well12
. For instance, if a  1/, then A = (2)-1/2
.
Similar to the previous case we use the same units but now with  = 1. The
action and the tunneling time are evaluated straightforwardly from Eqs. (37),
(39) taking into account that the initial turning points are xi = a  0. One
finds
S =
1/2E
0

1 - 2Exdx =
1
3E
(47)
 =
1/2E
0
dx

1 - 2Ex
=
1
E
(48)
Using x2
i 2
0(xi, 0) = A2
one obtains from Eq. (40)
 = A2
E e-2/3E
. (49)
12 S.A. Gurvitz
This coincides with the result obtained by Demkov and Drukarev,12,13
using a
more complicated technique.
The angular distribution of ionized particles, Eq. (41), is
q = 42
A2
e-2/3E
e-q2
/E
. (50)
The same result was obtained earlier in Ref. 4.
6 Summary
In this paper we extended the two-potential approach to quantum tunnel-
ing to a multi-dimensional case. We found simple expressions for the energy
and the partial widths of a quasi-stationary state, initially localized inside a
multi-dimensional potential. The partial width, which determines the distribu-
tion of tunneling particles in the final state, is given by an integral of the bound
and non-resonant scattering wave functions over the hyper-surface taken inside
the potential barrier.
We applied our results for the ionization of atomic states by an external
field depending only on one of the coordinates. In this case we obtained very
simple semi-classical expressions for the total ionization width and the angular
distribution of tunneling electrons, valid for any initial state. We demonstrated
that in the case of the Hydrogen atom and also for a short range potential our
general expression for the total width produces the results, existing in the
literature, but obtained with a more complicated calculations.
We also demonstrated that angular distribution of tunneling electrons in
the final state falls off as a Gaussian function of the lateral momentum. The
exponential is determined by the traversal semi-classical tunneling time.
Acknowledgments
I am grateful to M. Kleber, W. Nazarewicz, P. Semmes and S. Shlomo for
helpful discussions. I would like to acknowledge the kind hospitality of Oak
Ridge National Laboratory, while parts of this work were being performed.
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