J. Modern Physics, 2010, 1, 124-136
doi:10.4236/jmp.2010.12018 Published Online June 2010 (http://www.SciRP.org/journal/jmp)
Copyright © 2010 SciRes. JMP
Exact Analytical and Numerical Solutions to the
Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting
Non-Exponential Decay at All Times
Athanasios N. Petridis1, Lawrence P. Staunton1, Jon Vermedahl1, Marshall Luban2
1Department of Physics and Astronomy, Drake University, Des Moines, USA;2Ames Laboratory and Department of Physics and
Astronomy, Iowa State University, Ames, USA.
Email: athan.petridis@drake.edu
Received March 2nd, 2010; revised April 7th, 2010; accepted April 30th, 2010.
Abstract
The departure at large times from exponential decay in the case of resonance wavefunctions is mathematically
demonstrated. Then, exact, analytical solutions to the time-dependent Schrödinger equation in one dimension are
developed for a time-independent potential consisting of an infinite wall and a repulsive delta function. The exact
solutions are obtained b y means of a superposition of time-indep endent solutions spanning the given Hilbert space with
appropriately chosen sp ectral functions for which the resultin g integrals can be evaluated exactly. Square-integrability
and the boundary cond itio ns are sa tisfied . The simplest of the obta ined so lutio ns is presen ted and the pro bab ility fo r the
particle to be found inside the potential well as a function of time is calculated. The system exhibits non-exponential
decay for all times; the probability decreases at large times as 3
t
. Other exact solutions found exhibit power law
behavior at large times. The results are generalized to all normalizable solutions to this problem. Additionally,
numerical solutions are obtained using the stagge red leap-fro g algorithm for select p otentia ls exhibiting the prevalence
of non-exponential decay at short times.
Keywords: Non-Exponential , Decay, Exact , Solutions
1. Introduction
The law of exponential decay is typically discussed in
association with atomic transitions or resonances in
scattering amplitudes. Even though the approximations
made in order to arrive at exponential decay of excited
states or resonances are well understood the mistaken
impression that this law is universal and exact often
prevails. This perception is reinforced by experiments
often done in student laboratories geared towards
studying the half-lives of radioactive nuclei or unstable
particles and, very importantly, by numerous research
publications and data tables in which exponential decay
is tacitly assumed. The fact that these experiments
measure counting rates during only finite time intervals
and are focused on decays of quasi-stationary states is
usually not discussed, let alone studied in detail.
The history of this particular problem is quite interes-
ting. Early on Khalfin [1] used dispersion relations to
show that even quasi-stationary states with spectral
functions that have a lower bound in their energy
spectrum must decay non-exponentially at large times.
Winter [2] examined the infinite wall plus repulsive delta
function potential and obtained a single implicit solution
in the form of an integral for the special case in which
the initial wavefunction is an eigenfunction of the
infinite square well of the same width and as a result it is
a near-resonance (quasi-stationary) state of the actual
potential. His analytic approximation to the integral in
the limit of low barrier transmittance (large strength of
the delta function) proved that the survival probability
exhibits exponential decay in the (intermediate) time
interval-when the dominant quasi-stationary resonance
prevails inside the well-while at very large times it
decays following the power law 3
t. By means of
numerical studies the same author found oscillations in
the probability current at times before the power law sets
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
125
in if the initial state has a relatively wide energy spec-
trum.
The purpose of this article is to demonstrate explicitly
the existence of systems that exhibit non-exponential
decay at all times by developing exact, analytical, closed
form solutions to the time-dependent Schrödinger equa-
tion for a one-dimensional potential and non-quasis-
tationary initial states as well as to illustrate non-
exponential decay using numerical solutions to specific
problems for which analytical solutions are not obtaina-
ble. The clear advantage of the analytical approach
without any approximations is that it yields an equation
for the survival probability of the initial state that can be
studied for any time interval and that is unequivocally
non-exponential. The conclusions are easily generalized
and the long-time behavior of the solutions is predicted
and shown to follow an asymptotic power law. It is, thus,
established that for a large class of systems, non-expo-
nenential decay is the rule rather than the exception.
This paper also elucidates and generalizes previous
research work. Recently there has been increasing
interest in the time dependent Schrödinger equation and,
in particular, in the decay of physical systems. The
equivalence of exponential decay of a perturbed energy
eigenstate with Fermi's golden rule when the final
density of states is energy-independent and with the
Breit-Wigner resonance curve has been long known and
presented in several papers [3] and textbooks [4].
Dullemond [5] has verified this behavior for a simple but
exactly solvable model and found, however, that if
final-state energy-dependence is introduced into this
model a non-exponential decay pattern will dominate at
large times.
Oleinik and Arepjev [6] have shown that tunneling of
electrons out of a finite potential well when a long-range
electric field is suddenly switched on follows a 3
t
probability decay law at large times. Specific systems
that may exhibit non-exponential decay include systems
with non-local interactions [7], certain closed many-body
systems [8], quasi-particles in quantum dots [9], polarons
[10], and non-extensive systems [11]. Petridis et al., [12]
have studied numerically a variety of systems in which
the initial wave function is mostly or entirely set in a
finite potential well and have observed rich behavior,
including non-exponential decay into the continuum.
Non-exponential decay was experimentally observed
for the first time by Wilkinson et al., [13] in the
tunneling of ultra-cold sodium atoms initially trapped in
an accelerating periodic optical potential created by a
standing wave of light. Kelkar, Nowakowski, and Khem-
chandani [14] have reported evidence for the non-
exponential alpha decay of Be
8. Rothe, Hintschich, and
Monkman [15] have clearly measured non-exponential
time-dependence in the luminescence decay of dissolved
organic materials after pulsed laser excitation.
Time-dependent quantum mechanical problems are
usually addressed using time-dependent perturbation
theory, adiabatic or sudden approximations as well as
several numerical techniques. Exact analytical solutions
to certain problems are highly desirable, especially in
cases when the approximate methods may be inadequate
to describe all aspects of the solutions or when numerical
treatments do not explicitly reveal their mathematical
properties.
Burrows and Cohen [16] have developed exact
solutions for a double-well quasi-harmonic potential
model with a time-dependent dipole field. Cavalcanti,
Giacconi, and Soldati [17] have solved the problem of
decay from a point-like potential well in the presence of
a uniform field and have indicated that, due to an
infinitely large number of resonances, there may be
deviations from the naively expected exponential time-
dependence of the survival probability.
In this article a well established method for solving
time-dependent quantum mechanics problems is used to
develop exact, analytical, closed-form solutions to the
infinite wall plus repulsive delta function potential. The
large-time non-exponential decay for three solutions to
this system is established and the asymptotic power law
behavior is explicitly demonstrated to be 3
t
for the
first two and
4
t for the third. It is also proven that this
result, (or a higher negative power of t), is valid for all
square-integrable solutions to this system. Furthermore
numerical solutions are developed for finite-range po-
tentials and shown to exhibit a rich, non-exponential
decay behavior, including oscillations.
2. The Exponential Decay Approximation
The time-dependent wavefunction,
)
,
(
t
x
, can be
expressed as a superposition of fixed energy states,
)( x
E
, each evolving in time as iEt
e,
,)()(=),( dEexEtx iEt
E

(1)
where )( x
E
are fixed-energy (stationary) solu- tions
to the Schrödinger equation for the given Hamil- tonian
and )(E
is an energy distribution or “spectral
function”. It is important that this integral converge and
the resulting wavefunction is square-integrable for the
given boundary conditions (i.e., it belongs to the related
Hilbert space).
If the energy is non-negative and its distribution in the
above integral has a dual-pole (resonance) structure in
the complex plane, that is
,
)(
1
=
))((
1
=)(22
0
*
00 EEEE
E

(2)
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
126
where  iE00 =
, and 0
<<<0 E, then )(E
is
strongly peaked at 0
E and essentially only )(
0
x
E
contributes, i.e., to a good approximation
.
)(
)(=),( 22
0
0
0dE
EE
e
xtx iEt
E
(3)
With the substitution  )/(= 0
EEu , the integral
becomes
.
1
)(=),( 2
/
0
0
0du
u
ee
xtx tui
E
tiE
E


(4)
Defining 0>= t
for forward evolution and
0>/= 0E
the above expression can be re-written as,

,),(),()(=),(
0
0

iSC
e
xtx
tiE
E
(5)
where
.
1
)(sin
=),(,
1
)(cos
=),( 22 du
u
u
Sdu
u
u
C 

 
(6)
With uu =
' the first integral is
.
1)(
)(cos
=
1
)(cos
1
)(cos
=),(
'
2'
'
22
du
u
u
e
du
u
u
du
u
u
C




(7)
Similarly the second integral is
'
'
22 '2
sin()sin( )sin()
(,)== ,
11()1
uu u
Sdududu
uu u


 
 

 
 (8)
since the integrand in the first term is odd in u and
vanishes as ||u. The wavefunction, therefore,
becomes (dropping the primes on u)
.
1
)(=),( 2
0
0
du
u
e
e
e
xtx iu
tiE
E
(9)
At this stage the exponential-integral function
dy
y
e
zE y
z
=)(
1 (10)
is useful. Clearly,
.=
)(
1
z
e
d
z
zdEz
(11)
Upon defining the function

,)()(
2
=),( 11


iuEeiuEe
i
uf  (12)
its derivative is calculated to be

du
iudE
e
du
iudE
e
i
du
df )()(
2
=11



iuiu
eiu 11
2
=
.
1
=2u
eiu
(13)
Therefore,

.)()(
2
=
|),(=
1
11
2


iEeiEe
i
ufdu
u
eiu

(14)
Using the well-known expansion,
,...
21
1=)(2
1

zzz
e
zEz (15)
and keeping only the two first terms for large ||
i
,
the wavefunction becomes

1
)(),( 2
0
0


i
tiE
Eei
e
e
xtx
.)(=22
0
00
0


E
e
t
i
e
e
x
tiE
t
tiE
E
(16)
Thus, the probability density for times large relative to
1/222
0)(
E is
])(sin
12
11
[|)(||),(|
0
22
0
2
22
0
2
2
2
2
2
0
2
tE
E
e
t
Et
extx
t
t
E





(17)
and it has a term decaying exponentially with constant
2Γ plus a 2
t term dominating at very large times as
well as an “intermediate” decaying oscillatory term. In
this example not only is the rise of exponential decay
shown to emerge for a spectral function exhibiting a
resonance (dual-pole) structure but the departure from
this behavior at large times is clearly elucidated, having a
power-law dependence. It is noteworthy that the non-
exponential behavior is related to the cut-off in the
energy interval. If the energy were to vary over the entire
real axis then the residue theorem would yield exponen-
tial decay. The short time behavior is very complicated
as Equation (9) indicates and it is also not exactly
exponential.
3. Infinite Wall and Delta-function Potential
The method to be employed to address the problem of an
infinite wall plus a delta-function potential is standard
and consists of the following steps: a) The time-
independent solutions to Schrödinger equation are found
subject to the boundary conditions of the problem. These
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
127
are stationary solutions (energy eigenfunctions) that span
the Hilbert space of the given Hamiltonian. b) Since any
finite or infinite, discrete or continuous linear combina-
tion of the stationary solutions (basis functions), as long
as it is square-integrable, is also a solution belonging to
the given Hilbert space, exact analytical solutions can be
developed by a superposition of the eigenfunctions with
energy-dependent spectral functions multiplied by the
standard oscillatory time-dependence of the stationary
states. It is, obviously, necessary that the superposition
integral over the energy converge. Spectral functions for
which the resulting integrals are tractable are chosen here.
The convergence as well as the square-integrability
(normalizability) of the resulting wave functions are
verified. c) The survival probability, i.e., the probability
for finding the particle inside the potential well is
calculated and its properties are studied analytically.
The problem is defined by the one-dimensional
repulsive potential,


,<0)(
0<
=)(
0xLxV
x
xV
(18)
with 0>L and 0>
0
V. The steps outlined above are
followed.
a) The solutions to the time-independent Schrödinger
equation,
),(=)()(
)(
2
1
2
2xExxV
dx
xd EE
E
(19)
(with particle mass 1=m, 1=, and 0E for this
potential) are,
(0) ()=0,0("0"),
Exx region (20)
),""(0),(sin=)( 1
)( IregionLxpxCx
I
E (21)
),""(
),(cos)(sin=)( 32
)(
IIregionxL
pxCpxCx
II
E

 (22)
where Ep 2= and 1,2,3
C are constants in x. These
functions obey the boundary conditions
),(=)( )()(LLII
E
I
E (23)
),(2=)()( )(
0
)()(LVL
dx
d
L
dx
dI
E
II
E
I
E
(24)
while the boundary conditions at 0=x are automa-
tically satisfied. The energy eigenfunctions, E
, are not
required to vanish at infinity since time-dependent func-
tions, ),(tx
, produced by Equation (1) for large x are
acceptable solutions. Selecting C1 as the overall norma-
lization constant, the boundary conditions at Lx =
yield
,)(cos)(sin
2
1= 0
12
pLpL
p
V
CC (25)
),(
sin
2
=2
0
13 pL
p
V
CC
(26)
rendering 2
C and 3
C functions of the energy. The
choice of 2
C or 3
C as the normalization constant
would introduce an energy-dependence in 1
C and
would effectively amount to different choices of spectral
functions.
The linearly independent energy eigenfunctions
obtained are orthogonal under the inner product
,)()(
lim
)()(=),(
2
*
1
)(
0
2
*
1
0
21
dxxxe
dxxx
Lx
L
L



(27)
with all wavefunctions in the defined Hilbert space
identically vanishing for 0x. The orthogonality
relation is
),()(=),( '
'ppEw
E
E
(28)
where Ep2= , '' 2= Ep and
2
3
2
2|)(||)(|
2
=)(ECECEw
.)(2sin2)(2cos22
2
|=| 0
2
0
2
0
2
2
2
1pLpVpLVVp
p
C
(29)
The Dirac
-function representation used is
.
)(
1
lim
=)(22'
0
'

pp
pp (30)
b) The solution to the time-dependent Schrödinger
equation,
),,()(
),(
2
1
=
),(
2
2txxV
x
tx
t
tx
i

(31)
can be written as the energy-convolution integral,
,)()(=),(0dEexEtx iEt
E
(32)
with )(E
a spectral function such that this integral is
convergent for all
x
and all t and the resulting
wavefunction is square-integrable. Note that square-
integrability of ),( tx
also requires E to be real. The
overall normalization constant is, then, calculated from
*
0(,) (,)=1.xtxtdx

(33)
The first choice of spectral function to be considered is
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
128
,=)( 2
1
EK
eE
(34)
with
K
a positive constant. This offers the advantage
that the integrals above can be evaluated in closed form
and the resulting wave function is square-integrable even
without the presence of the convergence factor that
appears in Equation (27). The time-dependent solution is,
then,
0,=),(
(0) tx
(35)
,)(
2
=),( 3/22
)
2
2(
2
1
)(
itKexCtx itK
x
I
(36)
3/22
2
2
2
2
2
1
)( )(
2
=),(

itKeCtx itK
xLxL
II
,)()( 00
2
)
2
2(
2
)2(
0
2
)
2
2(
2


xitVVKeVitKe itK
xL
itK
x
(37)
where
1/2
2
0
3/2
2
2
2
0
3/2
3
0
3/2
2
2
3
3/2
122
28
=

 K
Ve
K
V
K
VLe
K
CK
L
K
L

(38)
is the overall normalization factor obtained by means of
Equation (33).
c) The probability density ),(),(= *
t
x
t
x
can
be calculated for the interior (region I) and the exterior
(region II) of the potential well. It is presented in Figure
1 at six times starting from 0=t, in increasing order.
The initial wavefunction is not entirely localized inside
the well. As time progresses the wavefunction spreads
and tunnels through the potential barrier in both
directions. The interference of the wave that propagates
outwards through the barrier and the wave that is outside
creates the observed ripples. Inside the well there are no
ripples because the wavefunction is forced to be odd in
x
, having a node at 0=x. The centroid of the pro-
bability density in region II at 0=t is always located
at L2, regardless of the value of
K
.
The survival probability is, then, defined to be
.),(),(=)( *
0dxtxtxtPL
in

(39)
This yields the closed-form result
22
3/2 42
2
1342342
2
()= erf
88
KL
Kt
in KL KL
Pt Ce
KKt KKt










(40)
A plot of the survival probability versus time is given in
Figure 2. (0)
in
P is controlled by
K
. It decreases as
K
increases, i.e., as the momentum spectrum becomes
sharper. For example, if 3=L, (0)
in
P is 0.9615 for
0.1=K, 0.5 for 0.5=K, and 0.1468 for 1.2=K. A
physical interpretation of this effect is that at 0=t
some decays have already happened. On the other hand
the decay becomes slower as
K
increases. The
expansion of in
P in inverse powers of time includes
only odd terms with alternating signs. At large times the
leading term, that has a positive sign, is proportional to
3
t, a clearly non-exponential behavior.
4. Corrections to the Exponential Decay Law
The law governing the decay of physical systems is
typically assumed to be a simple exponential time-
dependence of the number )(tN of the systems that
have not decayed until time t, i.e., ()= (0)NtN
e( )
x
pt
, where
is the decay constant. As
mentioned earlier this simple law is consistent with the
Breit-Wigner curve and Fermi's golden rule if the final
density of states is energy independent. It refers to the
survival probability of a given initial energy resonance
(quasi-stationary state). For the choice of spectral
function given by Equation (34) the initial state is not a
resonance state. If a very large number of systems is
assumed to be initially described by ,0)(
x
and a
system is said to have decayed if the particle has exited
the potential well, then the number of surviving systems
is proportional to the probability in
P, i.e.,
.
(0)
)(
=
(0)
)(
in
in
P
tP
N
tN (41)
The differential decay law is
,)()(= dttNtdN
(42)
where,
is, in general, dependent on time. Substi-
tution from Equation (41) gives
))].((l[=
1
=)( tPn
dt
d
dt
dP
P
tin
in
in

(43)
In the case studied, Equation (40) yields
,
)](e2)[(
4
=)( 2
24
3
2
zrfzetK
tze
tz
z

(44)
where 24
/= tKKLz . This function is plotted versus
time in Figure 3.
The decay parameter
peaks in time. Its maximal
value, max
, is smaller as
K
or L increases but does
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
129
Figure 1. The probability density for a potential consisting of an infinite wall and a repulsive delta function and using the
spectral function given by Equation (34) at six times (from the upper panel in the left colume to the lower panel in the right
column, = 0.0,0.3,0.6,0.9,1.2,1.5t). In this plot =3L, 0=1V and =1/2K
not depend on 0
V
. The peak and the small time interval
around it correspond to an almost exponential decay.
This, however, cannot be directly associated with the
dominant (lowest energy) resonance that this potential
accommodates. Resonances in the energy can be iden-
tified as the maxima of the function [18]
,
)8(sin8)]8(cos[122
2
=
||||
||
=)(
0
2
0
2
3
2
2
2
1
ELEVELVE
E
CC
C
Eg

(45)
plotted in Figure 4 for 3=L and 1=
0
V. It can be
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
130
seen that the resonances are not exactly of the Breit-
Wigner shape, therefore they do not decay exactly
exponentially. The dominant (lowest peak energy)
resonance has a width at half maximum of 0.1
corresponding to a “life-time” of 10. In a resonant
decay the width in energy is expected to be equal to the
value of the decay constant. Clearly, the width here is
very different from 1.3
max
(Figure 3). The reso-
nance peak energy and width depend only on the strength
and the geometry of the potential, while max
also
depends on the spectral function. The choice )(
1E
used
Figure 2. The survival probability for a potential consisting
of an infinite wall and a repulsive delta function and using
the spectral function given by Equation (34) versus time
(solid line). In this plot =3L, 0=1V and =1/2K. The
dashed line represents the exponentially decaying function,
()=
f
t ex
p
()abt, fitted to data points, calculated from
the actual solution, in the range =2t to 4. The 2
per
degree of freedom is of order 6
10-
Figure 3. The decay parameter
for a potential
consisting of an infinite wall and a repulsive delta function
and using a spectral function that is exponential in the
energy versus time. In this plot =3L and =1/2K.
There is no dependence on 0
V
Figure 4. Energy resonances for the infinite wall plus
repulsive delta function potential for =3L and 0=1V
here does not give this resonance a large weight (as
opposed to Winter’s choice which involves an initial
state very close to the resonance for large 0
V
). The
lower energy components of the wavefunction indeed
dominate and tunnel through the barrier at a slow rate
smearing the resonance effect. Therefore, the limited
quasi-expo- nential behavior observed in this study is not
of a resonance nature.
The expansion of
in inverse powers of time
includes only odd terms with alternating signs. At large
times the leading term, that has a positive sign, is
proportional to 1
t, affirming the non-exponential
behavior. At very large times the change of
with
time is rather slow. A fit to in
P
at large times with an
exponential curve in a finite time interval (as it is done in
experiments) gives a very small value of 2
per degree
of freedom (of order 6
10) so that the distinction
between in
P at large times and a simple exponential
decay function is numerically minute (Figure 2).
5. Generalization
Exact, closed-form, analytical solutions to the time-
dependent Schrödinger equation for the potential con-
sisting of an infinite wall and a repulsive delta function
have been obtained by the authors of this article for other
spectral function choices. For example, the choice
2
1cos 2
2
()=2
L
iE
EEL



(46)
yields a square-integrable wavefunction. In the absence
of the delta function at Lx = this would produce an
effectively square density pulse at 0=t located
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
131
between 0=x and /2=Lx . Due to the actual boundary
conditions at Lx= this spectral function also produces
a cusp centered at Lx 2= . The survival probability is
readily expressible in terms of Fresnel sine and cosine
integrals [19]. Its asymptotic large time behavior is
3
t.
A question that naturally arises at this point is whether
the asymptotic time behavior can be generalized to other
possible solutions to this problem. This question was first
addressed by Khalfin [1] specifically for the case of
quasi-stationary initial states. Here a detailed answer is
provided for non-resonance cases employing the general
requirements of convergence and square-integrability.
There is a one-to-one correspondence between spectral
functions and square-integrable wavefunctions. This can
be seen upon projecting the wavefunction at =0t on
an energy eigenfunction and employing the orthogonality
condition of Equation (28):
.,0)()(
)(
1
=)(*
0dxxx
Ew
EE

(47)
Given an initial wavefunction the corresponding
spectral function can, in principle, be constructed.
Schrödinger’s time-dependent equation then produces
the wavefunction at any later (or earlier) time.
Convergence of the energy superposition integral in
region (II) requires that the spectral function be finite at
0E. In addition, in order for ),( tx
to be
square-integrable, )( E
must vanish at large energies.
This requirement can be made precise by inserting
Equation (32) into Equation (33) and applying Equation
(28) to obtain
1.=|)(|)( 2
0dEEEw
(48)
Inspection of the function )(Ew , given in Equation
(29), leads to the conclusion that |)(| E
must vanish
for E faster than E1/ due to a constant term
in )(Ew .
Assuming that)(E
satisfies the convergence condi-
tions and has no resonance structure, its contribution to
the energy superposition integral giving ),(
)( tx
I
, in
region (I), comes mostly from low energies. Again, this
situation must be contrasted to the case studied by
Winter [2]. Then at any
x
in region (I) the wave-
function can be approximated as
()
()
10
(,)(0)2.
Et
max
IiEt
x
tC ExedE

(49)
The upper limit of the integration is chosen as follows:
the factor )(exp iEt oscillates more rapidly as a
function of the energy as t increases. At very large
times these oscillations eventually lead to a vanishing
contribution to the integral. Therefore, the integral can be
cut off at a point )(tEmax whose first order term in the
expansion in powers of t1/ is tymax/, where max
y is
constant in t. At low energies )( E
is replaced by its
(finite and non-zero) value at 0=E and the function
)2(sin Ex is replaced by its argument at a given
x
.
Then, the variable change Ety= yields
.2(0) 0
3/2
1
)(dyeytxC iy
max
y
I
(50)
For small max
y the integral is approximately
3/2
2 [(2/3)max
y5/2
(2 / 5)]
max
iy. The wavefunction in region
(I) is to the first non-vanishing order
,(0)),( 3/2
1
)(
tMxCtx
I
(51)
where M is a constant and the survival probability
(Equation (39)) decreases with time as 3
t. Therefore,
in order for the wavefunction to be square-integrable, the
spectral function must be finite at 0E and decrease
at large
E
faster than E1/ . Then, if 0(0)
,
necessarily, the survival probability asymptotically
decreases as 3
t.
This argument can be extended to any finite value of
x
including region (II) since the coefficients 2
C and
3
C
are at most of (1)O for small E. Therefore, the
integral of the probability density over any finite range of
x
is finite (even without the convergence factor present
in Equation (27)) and it decreases asymptotically as 3
t.
The constant
M
in Equation (51) can be exactly
evaluated if )(E
decreases at large
E
faster than
E1/ . Then if )(E
is analytic in the fourth quadrant of
the complex
E
-plane the contour integral of ()sinE
(2)
x
Ee( )
x
piEt
along a closed path, consisting of the
positive real axis from R to 0, the negative imaginary
axis from 0 to iR
and a quarter-circle, , of radius R,
is zero (Figure 5). The integral along is bounded by
a constant times k
R1/ with |=|ER and 1>k and,
consequently, vanishes in the limit R. Then the
integration over the real axis gives the same result as that
over the imaginary axis. The variable change iyE
=
with y real, then, yields

()
10
(,)=( )sin2.
Iyt
x
tiCiy xiyedy

 
(52)
For large times only small values of
y
contribute to
the integral. The spectral function is substantially
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
132
different from zero only close to the origin and can be
replaced by (0)
and be pulled out of the integral
while the sine function can be approximated by its
argument in a finite range of
x
. The remaining integral
is evaluated as a gamma function and gives
3/2/43
1
)( /2(0)),( 
texCtxiI

(53)
confirming the earlier result.
The survival probability, in
P, discussed thus far refers
to the presence of the particle inside the potential well.
As has been shown in the previous section the spectral
function of Equation (34) produces non-zero probability
density outside the well at 0=t for 0>K. If the
“interior” of the well is defined to extend to
x
much
larger than
L
2 (without moving the delta function from
Lx =) then at 0=t the probability to find the particle
“inside” can be arbitrarily close to unity. Specifically the
“extended” survival probability )(4
L
P
in can be defined
by extending the integral of Equation (39) to Lx 4= .
This integral has been evaluated analytically and is
plotted in Figure 6 as a function of time. As predicted
and verified by an expansion of )(4L
P
in
in inverse
powers of time, its asymptotic time dependence is 3
t
.
An interesting feature of this plot is the presence of a
step-wise behavior which can be attributed to inter-
ference between waves moving in opposite directions.
The spectral function
00
3
()i0
()=0o,
E
VE fEV
Etherwise

(54)
has also been investigated. This yields an exact, closed
form result which is square integrable [19]. In this case
0=(0)
3
so that the survival probability does not vary
as 3
t. Rather, it varies as 4
t. A variation of the above
analysis shows this to be the expected behavior. It should
be clear that the lowest order non-vanishing term in an
expansion of the spectral function about zero will control
the behavior.
6. Numerical Examples
The discussion in the previous sections indicates that if
the initial wavefunction is not near a resonance state of
the given potential, exponential decay of the survival
probability should not be expected. However, analytical,
closed form solutions can only be obtained for a small
number of potentials and initial states. A numerical
approach is, then, needed to study arbitrary potentials
and initial functions. To this end the time-dependent
Schrödinger equation can be solved using the staggered
leap-frog method on a grid of spatial points of lattice
constant
x
and with an appropriate time-step
t
.
The method consists of computing the wavefunction at
time tt
2 starting with the function at time t and
updating it with the Hamiltonian at tt  , as follows:
)].,()(
ˆ
[2),(=)2,( ttxxHtitxttx 

(55)
This method being time-symmetric can be made very
stable for a time step that is much smaller than the
spacial lattice constant and, on a fine grid, it is also very
accurate. The spatial derivative in the Hamiltonian,
)(/1/2=
ˆ22 xVdxdH  , is computed using a spatially
symmetric formula. The spatial grid is chosen to be
much larger than the dimensions of the problem and on
its edges reflecting boundary conditions are applied (i.e.,
Figure 5. The complex plane contour used to calculate the
integral over E.
Figure 6. The “extended” survival probability for a
potential consisting of an infinite wall and a repulsive delta
function and using the spectral function given by Equation
(34) versus time. In this plot =3L, 0=1V and =1
/
2K.
The step-wise behavior is due to interference of waves
moving in opposite directions
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
133
the wavefunction is forced to be 0 there). This ensures
that no probability density leaks out of the grid but
requires that the reflected waves not interfere with the
wavefunction in the region of interest. Therefore, when
such interference starts (inevitably) occurring at appre-
ciable levels the computation is stopped. The Schrö-
dinger equation is self-dispersive and does not obey
relativistic causality. As a result, very fast moving or
even superluminal components of the wavefunction can
occur and reflect on the grid boundaries. The stability of
the numerical solution is checked by evaluating the norm
of the wavefunction at regular intervals to ensure it is
equal to 1. This is achieved with
9
10 precision. Seve-
ral cases, such as free gaussian wavepackets (spreading
with time) or a harmonic oscillator potential with an
initial wavefunction that is a linear combination of
eigenstates, have been solved to verify that the method
accurately reproduces known analytical results.
The numerical technique is used to study the short-
time behavior of a wavefunction that is initially set in a
potential well of finite size and strength and then tunnels
through its walls. Two simple potential functions are
used to this end. The first one is a cut harmonic oscillator
potential,

,o0
2
||i)(
2
1
=)( 0
2
0
therwise
B
xxfxx
xV
(56)
and the second is a cut linear potential,

.o0
2
||i||
=)( 00
therwise
B
xxfxxk
xV (57)
The initial wavefunction is chosen to be a gaussian
with no initial central momentum. Results for the
survival probability, in
P
, defined as the integral of the
density inside the potential well, for the case of the cut
harmonic oscillator potential are shown in Figure 7.
Here c
T
indicates the classical period corresponding to
the infinite harmonic oscillator potential with 1=m.
There is a distinctive step-wise decay due to oscillations
of the wavefunction. Each time the probability drops
sharply a wavepacket is emitted on either side of the well.
The derivative of in
P
with respect to time is also shown
to illustrate that it approaches 0 periodically. The
qualitative features of the decay are not sensitive to the
ratio of the standard deviation of the gaussian to the
value of
B
. In the same manner results for the cut
Figure 7. Results for a cut harmonic oscillator potential given by Equation (56) (=0.0001
and B = 200) with an initially
gaussian wavepacket and 0 central momentum. Upper: the survival probability versus time exhibiting periodic flat regions;
Lower: the derivative of the survival probability. The negative peaks occur when wavepackets emitted from the potential. c
T
is the period for the infinite harmonic oscillator potential with spring constant, α. This behavior is similar to that seen with a
cut linear potential
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
134
Figure 8. Results for a cut linear potential given by Equation (57) with an initially gaussian wavepacket having 0 central
momentum. Upper: the survival probability versus time exhibiting periodic flat regions; Lower: the derivative of the survival
probability. The negative peaks occur when wavepackets are emitted from the potential. This behavior is similar to that seen
with a cut harmonic oscillator potential
Figure 9. Results for a cut harmonic oscillator potential given by Equation (56) (=0.0001
and B = 200) with an initially
gaussian wavepacket that is the ground state of the infinite potential, having non-zero central momentum =1.0p. Upper:
the survival probability versus time exhibiting periodic flat regions; Lower: the derivative of the survival probability. The
negative peaks occur when wavepackets emitted from the potential
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
135
Figure 10. The probability density for a cut harmonic
oscillator potential given by Equation (56) (=0.0001
and
B = 200) with an initially gaussian wavepacket that is the
ground state, 0
u, of the infinite potential, having non-zero
central momentum, =1.0
p
, captured at about 2.5
classical periods of the infinite potential. The classical
amplitude of oscillations is =100
c
A. All quantities are
expressed in natural units. Wavepackets are periodically
emitted from the non-zero potential region, propagate
outwards and spread out. The first emitted packet is
traveling to the right and at this time frame is centered at
3500x. The second emitted packet is traveling to the left
and at this moment is centered at 900x. The interior
wavefunction is hitting the left wall of the potential well at
this moment
linear potential are shown in Figure 8 with similar initial
conditions. Again the decay is non-exponential with a
step-wise behavior. To illustrate this result further an
initial gaussian with non-zero central velocity, 0
v, is set
in a cut harmonic oscillator potential. This is accom-
plished by multiplying the initial gaussian by )(expipx,
where 0
=
p
mv is the central momentum. The results
are shown in Figure 9. In this case in
P
decays in larger
steps. The emission of wavepackets is shown in Figure
10,where the probability density is plotted versus x at a
particular time.
7. Conclusions
Exponential time-dependence has been shown to be only
an approximation to any real decay process even in the
case of commonly encountered resonance states. For
resonances, at large times a 2
tdependence emerges
preceded by some oscillations. The time-dependent
Schrödinger equation for non-resonance initial states has
been solved utilizing the eigenfunctions for a given
Hamiltonian. It has been applied to the case of a potential
consisting of an infinite wall and a repulsive delta
function. Exact, analytical, normalized solutions have
been obtained in closed form. In the case specifically
exhibited, i.e. , the choice spectral function )(
1
E
(Equation (34)), the survival probability, which is exactly
detailed in Equation (40), exhibits a non-exponential
behavior at all times. At large times it decays as
3
t. To
ensure square- integrability the spectral function must be
finite at 0
E
and decrease to 0 at large energies
faster than E1/ . It was shown that this behavior
pertains to all square-integrable wavefunctions that are
solutions to this problem for which (0) 0
. Other
spectral functions result in decays varying as tn with n
greater than 3. With the appropriate choice of spectral
functions which, due to linear independence need not be
the same for waves propagating in different directions,
the method could be applied to a variety of potentials.
Numerical studies of finite potential wells show that
non-exponential decay prevails at short times and can
exhibit an interesting step-wise behavior. In conclusion
quantum mechanics predicts non-exponential decay for
all systems studied.
8. Acknowledgements
Ames Laboratory is operated for the U.S. Department of
Energy by Iowa State University under Contract No.
W-7405-Eng-82.
REFERENCES
[1] S. A. Khalfin, “Contribution to the Decay Theory of a
Quasi-Stationary State,” Soviet Journal of Experimental
and Theoretical Physics, Vol. 6, 1958, pp. 1053-1063.
[2] R. Winter, “Evolution of a Quasi-Stationary State,” Phy-
sical Review, Vol. 123, No. 4, 1961, pp. 1503-1507.
[3] V. Weisskopf and E. Wigner, “Berechnung der natür-
lichen Linienbreite auf Grund der Diracschen Lichtt-
heorie,” Zeitschrift für Physik, Vol. 63, No. 1-2, 1930, pp.
54-73.
[4] J. J. Sakurai, “Modern Quantum Mechanics,” The Benja-
min-Cummings Publishing Company, 1985.
[5] C. Dullemond, “Fermi’s ‘Golden Rule’ and Non-Expo-
nential Decay,” arXiv:quant-ph/0202105, 2003.
[6] V. P. Oleinik and J. D. Arepjev, “On the Tunneling of
Electrons out of the Potential Well in an Electric Field,”
Journal of Physics A, Vol. 17, No. 9, 1984, pp. 1817-1827.
Exact Analytical and Numerical Solutions to the Time-Dependent Schrödinger Equation for a
One-Dimensional Potential Exhibiting Non-Exponential Decay at all Times
Copyright © 2010 SciRes. JMP
136
[7] M. I. Shirokov, “Exponential Character of Decay Laws,”
Soviet Journal of Nuclear Physics, Vol. 21, 1975, pp.
347-353.
[8] V. V. Flambaum and F. M. Izrailev, Unconventional
decay law for excited states in closed many-body systems.
Physical Review E, Vol. 64, No. 2, 2001, pp. 026124-
026130.
[9] P. G. Silvestrov, “Stretched Exponential Decay of a
Quasiparticle in a Quantum Dot,” Physical Review B, Vol.
64, No. 11, 2001, pp. 113309-113313.
[10] L. Accardi, S. V. Kozyrev and I. V. Volovich, “Non-
Exponential Decay for Polaron Model,” Physics Letters A,
Vol. 260, No. 1-2, 1999, pp. 31-38.
[11] G. Wilk and Z. Wlodarczyk, “Nonexponential Decays
and Nonextensivity,” Physics Letters A, Vol. 290, No. 1-2,
2001, pp. 55-58.
[12] A. N. Petridis, L. P. Staunton, M. Luban and J.
Vermedahl, Talk Given at the Fall Meeting of the
Division of Nuclear Physics of the American Physical
Society, Tucson, Arizona, Unpublished, 2003.
[13] S. R.Wilkinson, et al., “Experimental Evidence for Non-
Exponential Decay in Quantum Tunnelling,” Nature, Vol.
387, 1997, pp. 575-577.
[14] N. G. Kelkar, M. Nowakowski and K. P. Khemchandani,
“Hidden Evidence of Nonexponential Nuclear Decay,”
Physical Review C, Vol. 70, No. 2, 2004, pp. 24601-
24605.
[15] C. Rothe, S. I Hintschich and A. P. Monkman, “Violation
of the Exponential-Decay Law at Long Times,” Physical
Review Letters, Vol. 96, No. 16, 2006, pp. 163601-
163604.
[16] B. L. Burrows and M. Cohen, “Exact Time-Dependent
Solutions for a Double-Well Model,” Journal of Physics
A, Vol. 36, No. 46, 2003, pp. 11643-11653.
[17] R. M. Cavalcanti, P. Giacconi and R. Soldati, “Decay in a
Uniform Field: An Exactly Solvable Model,” Journal of
Physics A, Vol. 36, No. 48, 2003, pp. 12065-12080.
[18] A. Messiah, “Quantum Mechanics,” Dover Publishers,
Mineola, 1999.
[19] The expressions for this wavefunction and the probability
density are very long and complicated. They are available
from the authors upon request.