3
Transmission across
potential wells and
barriers
The physics of transmission and tunneling of waves and particles across differ-
ent media has wide applications. In geometrical optics, certain phenomenon
of light propagation cannot be explained without taking the wave nature of
light [51]. Thus interference, diffraction and ”tunnelling” of light through re-
gions which are forbidden in geometrical optics can be understood through
wave nature of light. Another example is the well-known phenomenon of to-
45
tal internal reflection. When light impinges upon the interface between two
media with refractive indices n1 and n2, it suffers total internal reflection when
θi ≥ sin−1(
n1n2
), where θi is the angle of incidence. According to the geometrical
optics description, there is no penetration of light into the rarer medium. How-
ever, it is known that, in reality, the light does propagate into the rarer medium,
even under total internal reflection conditions. This is possible by tunnelling
when the thickness of the rarer medium is comparable to the wavelength of
the light. The penetration of light into “forbidden” regions is used in a vari-
ety of optical devices such as prism couplers, directional couplers, switches,
etc. Similarly, in quantum mechanics the penetration of particle into forbidden
region can be explained by means of the wave nature associated with it.
The general problem of particle tunneling can be classified into two in-
teresting categories [51] both of which involve the propagation of a particle
with energy E through a classically forbidden region of potential energy V(x),
where E < V(x). In the first category, there is a region in space where the
particle can be represented by a “free” state and its energy is greater than the
background potential energy. As the particle strikes the barrier there will be a
finite probability of tunneling and a finite probability of being reflected back.
In the second category the particle is initially confined to a quantum well (ex-
ample: tunnelling through two or more barriers with wells in between). The
particle initially confined to a ”quantum well” region is a quasi-bound (QB) or
sharp resonant state. In the quasi-bound state, though the particle is primarily
confined to the quantum well, it has a finite probability of tunnelling out of the
well and escaping. The key differences between the two cases is that in the
first problem the wave function corresponding to the initial state is essentially
46
unbound, whereas in the second case it is primarily confined to the quantum
well region. However, the phenomenon of resonances is more general and can
occur during transmission across well, barrier, and potential pockets created by
two well separated barriers. These are generally outlined only briefly in some
of the books (see for example[52]) in literature. It may further be pointed out
that transmission and tunneling across potential barriers in one dimension is a
fundamental topic in the under graduate courses on quantum mechanics. But,
most of the text books on introductory quantum mechanics confine to trans-
mission and tunneling across the potential barriers and approximate methods
to calculate the transmission coefficients using WKB type approximation (see
for example [31]). Similarly resonance tunneling is briefly outlined by indicat-
ing the reflectionless transmission at specific energies . Therefore, a systematic
study [11, 12] of resonant or quasi-bound states, their widths, and their relation
to the variation of T is desirable.
S -matrix theory in three dimensional (3D) potential scattering provides a
unified way for the analysis of resonances and bound states generated by the
potential through its analytical structure [10]. These are done using the study of
pole structure of S -matrix in complex momentum or complex energy planes. In
chapter 2 we have outlined how a somewhat similar procedure can be adopted
in one dimensional transmission problem using the pole structure of reflec-
tion amplitude and transmission amplitude. In three-dimensional systems, a
potential with an attractive well followed by a barrier is useful for analyzing
bound states, sharp resonances generated by potential pockets, and broader or
above barrier resonant states generated by wide barriers [53]. In contrast, in
one dimension we usually study a set of two or more well separated potential
47
barriers [12, 54]. The gap between two adjacent potential barriers provides
a pocket for the formation of sharp resonant states, and these generate sharp
peaks in the transmission coefficient. In addition, ordinary potential wells and
barriers can also generate resonances but they will be broader. It is also impor-
tant to study the resonance generated by purely attractive well. The motivation
of the present chapter is to explore the nature of broader resonances and related
subtle features of the transmission across purely attractive well and the corre-
sponding repulsive barrier to obtain a deeper understanding of transmission
across a potential in one dimension. The study of sharper quasi-bound states
[12] generated by twin symmetric barriers will be described in next chapter.
This chapter (see ref. [11]) is organized as follows: Section 3.1 is about the
formulation of bound states, resonant states and transmission coefficient T in
1D. In section 3.2 we investigate the resonance states, its correlation with os-
cillatory structure of T and the corresponding box potential for 1D rectangular
potential. In section 3.3 we study the transmission across a potential having
a combination of well and barrier and in section 3.4 we analyze T across an
attractive potential for which T does not show any oscillations. In section 3.5
we summarize the main conclusions of our study.
3.1 Bound states, resonance states and
transmission coefficient
The basic equation that we use to study the bound states, resonant states, trans-
mission and reflection is the one dimensional time independent Schrodinger
48
equation for a particle of mass m with potential U(x) and total energy E:
d2ϕ(x)dx2 + (k2 − V(x))ϕ(x) = 0, (3.1)
where k2 = 2mE/~2, V(x) = 2mU(x)/~2 and α2 = k2 − V(x). For convenience
we use the units 2m = 1 and ~2=1 such that k2 denotes energy E. In our
numerical calculations we have used Å as the unit of length and Å−2 as the
unit of energy.
(a) Bound state eigenvalues from bound state wave function
Stationary states of the system which correspond to discrete energy levels and
describe the bound state motion are called bound states. For bound states,
the energy is real and its wave function is square integrable. If a potential
can generate bound states, then the solution of Eq.(3.1) have exponentially
decaying tails at the boundaries. For potentials vanishing as |x| → ∞, the bound
state energies are negative. In particular, for short-range potentials satisfying
plane wave or free particle asymptotic boundary conditions, the bound state
wave function has the asymptotic behavior:
ϕ(x) −→|x|→∞
e−kb |x|, (3.2)
and the corresponding bound state energy Eb is given by Eb = (ikb)2 = −k2b .
We see that the bound states can be thought of as negative energy states with
zero width.
(b) Resonance state eigenvalues from resonance state wave function
For evaluation of the resonant states, we seek a solution of Eq.(3.1) with
complex energy k2R = (kr − iki)2 = (k2
r − k2i ) − i2krki = ER = Er − iΓr/2 with
49
Er > 0 and Γr > 0 such that the wave function behaves asymptotically as:
ϕ(x) −→x→∞
ei(kr−iki)x
ϕ(x) −→x→−∞
e−i(kr−iki)x. (3.3)
This behavior implies a positive energy state Er with width Γr and lifetime ~/Γr
and |ϕ(x)| diverges exponentially as eki |x| but decays exponentially in time be-
cause the time-dependent part of the solution of the Schrodinger wave equation
is e−iEt/~. Smaller width Γr corresponds to a long lived resonance state. If the
widths are very narrow, resonant states behave similarly to bound states in the
vicinity of the potential and may be considered as QB state. However, unlike
the bound states the QB or resonance states are not normalizable because of
the exponential divergence stated above indicating the significant presence of
resonance state wave function in the entire domain.
(c) Resonance states and transmission coefficient
In one dimensional quantum transmission across a potential, the transmission
coefficient can be obtained by solving the Schrodinger wave equation for E > 0
and then by applying the appropriate boundary conditions for the continuity of
ϕ(x) and ϕ′(x) at the boundaries. A smooth potential can be approximated by
a large number of very small step potentials, and the transmission and reflec-
tion coefficient R can be obtained by using transfer matrix technique. This
method can be used for potentials that are not analytically solvable. Alterna-
tively, one can adopt a numerical integration of the Schrodinger equation using
well known procedures like Runge kutta method.
50
For the study of R and T we seek a solution of ϕ(x) satisfying the conditions:
ϕ(x) −→x→−∞
Aeikx + Be−ikx,
ϕ(x) −→x→∞
Feikx, (3.4)
such that T = |F/A|2 and R = |B/A|2, where A, B and F are amplitudes of
incident, reflected and transmitted wave respectively. In one dimension scat-
tering potential, the S -matrix is given by the ratio of F/A. The pole of F/A
in the complex k-plane or in the complex energy E-plane generated by zero of
A corresponds to the resonant state. The resonance state energy generated by
the potential can be estimated from T by studying it as a function of energy E.
In the vicinity of resonance, T shows peaks due to enhanced transmission at
resonance energy. The peak position of T corresponds to resonance energy Er
and its width corresponds to the width of the resonance state Γr.
3.2 Resonance states and transmission generated
by rectangular potential
As a simple analytically solvable example, we consider the attractive square
well potential of width 2a given by:
V(x) =
−V0, V0 > 0, |x| < a,
0, |x| > a.(3.5)
51
The corresponding box potential of attractive square well potential is:
V(x) =
−V0, V0 > 0, |x| < a,
∞, |x| > a,(3.6)
which generates the eigenvalues:
En = −V0 + n2π2/(2a)2, n = 1, 2, ..... (3.7)
The potential in Eq.(3.6) can be considered to be infinitely repulsive for |x| > a,
we expect that the states for E < 0 have higher energies than the corresponding
square well given by Eq.(3.5). These potentials can generate bound and reso-
nance states. Since the potential V(x) is symmetric in x, the states generated
will have definite parity.
• The resonance state positions and widths are evaluated by solving the
Schrodinger equation Eq.(3.1), satisfying the boundary conditions given
by Eq.(3.3). Because we have chosen the potential to be symmetric with
respect to the origin, we expect both even and odd wave functions. The
former behave as cos(αx) near the origin and the latter as sin(αx). Thus
for even resonant states, the matching of the wave function and its deriva-
tive at x = a leads to the condition αtan(αa) = −ik. As a result the even
resonance state energy and widths are obtained from the complex roots
kn of the equation:
kcos(αa) − iαsin(αa) = 0. (3.8)
52
The roots kn are just below the real k-axis in the complex k-plane such
that Re(kn) > |Im(kn)|. This condition signifies a positive energy resonant
state with finite width. The corresponding equation for the energies and
widths of odd states is:
ksin(αa) + iαcos(αa) = 0. (3.9)
By incorporating the asymptotic condition Eq.(3.2), a similar procedure
can be used to obtain the bound state energy eigenvalues En.
• The transmission and reflection coefficient generated by the attractive
rectangular potential Eq.(3.5) is obtained by solving the Eq.(3.1) with
boundary condition given by Eq.(3.4). The following expressions are
obtained [23]:
FA=
4αke−2ika
(α + k)2e−2iαa − (α − k)2e2iαa , (3.10)
BA= −2i
(k2 − α2)4kα
sin(2αa)e2ika
(FA
), (3.11)
BA=
2i(α2 − k2)sin(2αa)[e−2iαa(α + k)2 − e2iαa(α − k)2]
, (3.12)
BF=
2i(α2 − k2)sin(2αa)4kα
, (3.13)
T =
∣∣∣∣∣∣FA∣∣∣∣∣∣2 = (4αk)2
(4kα)2 + 4(α2 − k2)2sin2(2αa), (3.14)
53
R =
∣∣∣∣∣∣BA∣∣∣∣∣∣2 = 4(α2 − k2)2sin2(2αa)
(4kα)2 + 4(α2 − k2)2sin2(2αa)= 1 − T, (3.15)
where α2 = V0 + k2. For a repulsive rectangular potential α2 = k2 − V0.
3.2.1 Comparison of the states generated by an attractive
square potential and the corresponding box potential
We evaluate the eigenstates generated by the potential given by Eq.(3.5) by
solving Eq.(3.1) satisfying the boundary condition Eq.(3.3) for resonance states
and Eq.(3.2) for bound states and compare these eigenstates with the corre-
sponding box potential Eq.(3.6). The eigenstates generated by the potential
Eq.(3.5) is also examined from the oscillatory structure of T as a function of
energy. The oscillatory behavior of T is due to the sine term in Eq.(3.14). In
Figure. 3.1 we show the variation of T with energy for a set of potential pa-
rameters. As E → ∞, T → 1 as it should. In our numerical calculations we
use Å as the unit of length and Å−2 as the unit of energy (i.e we set 2m = 1,
~ = 1), and choose parameters such that the numerical results and figures show
the physical features we seek to highlight. In Table 3.1 we summarize the nu-
merical results obtained using V0 = −3 and a = 5. In this case the attractive
square well has six bound states, and the corresponding box potential has only
five negative energy states. We also give the positions of the peaks of T . From
Table 3.1 it is clear that the peaks positions of T are related to the energies
of the corresponding resonant states, even though there is a small difference
in the numerical values. This difference is primarily due to the large widths
associated with the resonances generated by attractive well.
54
Figure 3.1: Variation of transmission coefficient T as a function of energy Efor an attractive square well potential Eq.(3.5 ) with V0 = −3 and a = 5.
Table 3.1: Transmission across an attractive square well of depth V0 = −3 andrange a = 5 [see Eq.(3.5)]. In all tables and figures the energy unit is Å−2 andlength unit is Å. The positive energy box states En coincide with the maximaof T .Sequence Energy Peak energies Bound state and Half-width Γn/2number eigenvalues En of T resonance state energies of eigenstates
1 -2.90 -2.92 02 -2.61 -2.68 03 -2.11 -2.29 04 -1.42 -1.75 05 -0.53 -1.08 06 0.55 -0.39 07 1.84 0.55 0.46 0.308 3.32 1.84 1.72 0.629 4.99 3.32 3.18 0.9110 6.87 4.99 4.84 1.2011 8.94 6.87 6.70 1.5012 11.21 8.94 8.75 1.81
55
Figure 3.2: Comparison of bound and resonant states of an attractive squarewell in Eq.(3.5) for potential parameters V0 = −3 and a = 5 with the corre-sponding states for the box potential (3.6). The abscissa is the sequence num-ber of the states. It may be pointed out that in the case of the bound states ofattractive square well n−1 signifies the ‘principal’ quantum number associatedwith energy eigenvalues.
In Figure.3.2 we demonstrate a correlation between the bound and reso-
nance states of the attractive square well and the corresponding box energy
eigenvalues given by Eq.(3.7). Along the x-axis we give the sequence number
of the resonant states and bound states generated by the box, as they occur,
starting with the lowest energy state. These numbers can be interpreted as
quantum numbers for the bound states. From Table 3.1 we see that the sets of
positive energy states generated by the potential in Eq.(3.5) and the peak posi-
tions of T for the corresponding well are the same, but the results for negative
energy bound states for the potential given by Eq.(3.5) and the corresponding
results for the box potential differ.
56
Correlation between peaks of T and the resonance states
It is well known that in 3D potential scattering the bound states and resonant
states can be associated with the poles of the S -matrix in complex k or, equiv-
alently, complex-E plane [10]. For transmission across a potential in one di-
mension, it is natural to examine whether the resonant states we have described
correspond to the complex poles of the transmission amplitude F/A. From Eq.
(3.11) we see that the pole structure of F/A and the reflection amplitude B/A
are the same. The only condition to be satisfied is that for real positive energies
R + T = 1, which means that whenever T has a peak R has a minimum. If we
use Eq.(3.11) for F/A and a convenient numerical procedure such as an itera-
tive method, we can calculate the zeros of the denominator of Eq.(3.10), which
will generate the poles of F/A and B/A. From this calculation we can verify
that all the resonance states listed in Table 3.1 correspond to the complex poles
of F/A.
The reason for relating the complex poles of F/A to resonances are as follows:
The amplitude A/F can be understood as the coefficient of an incoming inci-
dent wave eikx as x→ −∞ when the outgoing transmitted wave eikx for x→ ∞
has amplitude unity. Similarly, B/F can be understood as the coefficient of the
reflected wave as x → −∞ for the same condition. If A/F = 0 for a complex
k = kr − iki in the lower half of k-plane with kr > ki, the resulting wave func-
tion satisfies the asymptotic condition given in Eq.(3.3) signifying a resonant
state. However, A/F = 0 implies a corresponding pole in F/A, signifying that
F/A has complex poles corresponding to a resonance. This correspondence
is analogous to three dimensional potential scattering for which the complex
zeros k = kr − iki of the coefficient of incoming spherical wave component of
57
the regular solution of the scattering problem represent the resonant states and
consequently are identified as the pole position of the corresponding partial
wave S -matrix S l [10]. This result demonstrates that resonances have similar
interpretations in one and three dimensions. If let A/F = 0, we obtain the
condition satisfied by the zeros of A/F:
α + kα − k
= ±e2iαa , (3.16)
which corresponds to the complex poles of the transmission amplitude F/A.
It is straightforward to verify that Eq.(3.16) implies Eqs.(3.8) and (3.9), which
generate the resonance energies and widths. This verification and the numerical
example we have described demonstrate that the complex poles of the trans-
mission amplitude F/A for unit incident wave signify resonances and generate
the peaks of the oscillations of T for an attractive square well potential.
3.2.2 Numerical analysis of resonance states generated by
repulsive rectangular potential and its corresponding
box potential
For our study of resonance states generated by a repulsive potential, we have
taken a repulsive rectangular potential of width 2a given by:
V(x) =
V0, V0 > 0, |x| < a,
0, |x| > a .(3.17)
58
The corresponding box potential is:
V(x) =
V0, V0 > 0, |x| < a,
∞, |x| > a ,(3.18)
which can generate the eigenvalues:
En = V0 + n2π2/(2a)2, n = 1, 2, ..... (3.19)
Transmission across such a potential is the most commonly studied example in
introductory quantum mechanics. Our primary interest is in the interpretation
of the oscillations of T for energies E > V0 with the resonance states generated
by the potential Eq.(3.17). In Figure. 3.3 we show the variation of T with E
for the potential Eq.(3.17). Based on our interpretation of the oscillations of
T generated by an attractive square well, we can understand these oscillations
in a similar manner. Resonances generated by a reasonably flat barrier are
well studied for nucleus-nucleus collisions [53, 55]. Barrier top resonances are
broader states compared to the very narrow resonant states that are generated
by the potential pockets sandwiched between wide barriers. However if the
barrier is flatter and wider, a number of narrower resonance states are generated
for energies above the barrier. This condition is satisfied in our example of the
rectangular barrier with the choice a = 5. Unlike the attractive well, there are
no bound states associated with the rectangular barrier in Eq.(3.17).
We used the conditions given in Eq.(3.3) to search for resonance states gen-
erated by the barrier at energies E > V0. These states along with the maxima
of T are listed in Table 3.2. The expression T for E < V0 is given by Eq.(3.14)
59
Figure 3.3: Variation of T with E for the repulsive rectangular barrier [Eq.(3.17)] with V0 = 3 and a = 5.
with α2 = k2 − V0. It is clear that the peaks of the oscillations of T at above
barrier energies correspond to the barrier top resonances. As before, there is a
small difference between the peak positions of T and the corresponding reso-
nance energies due to the fact that we are dealing with broad states. Note that
the numerically computed T has contributions from background terms in addi-
tion to pole terms. When the imaginary part of the pole position of transmission
amplitude is small, the resonance contribution dominates T in the vicinity of
resonance energy, and the peak position of T and the corresponding resonance
position are closer. However, when a resonance is broad, the background term
becomes more significant and there is a shift of the peak position of T with
respect to the corresponding resonance energy. The numerical results shown in
Tables 3.1 and 3.2 demonstrate this effect.
60
To make the comparison between the attractive well and the corresponding
barrier more complete, we examine the variation of the barrier top resonance
position En with n and compare it with the corresponding box potential posi-
tive energy eigenvalues Eq.(3.19). We show this in Figure 3.4. The correlation
between the box states and barrier top states is close as in the case for an attrac-
tive square well. Figure 3.2 looks different because of the single sequencing
of bound and resonant states generated by the well. The positive energy states
generated by Eq.(3.17) and the corresponding peak position of T match very
well. The reason for this common feature can be understood as follows.
The expression for T in Eq.(3.14) holds for an attractive square well with
α2 = k2 + V0. The same expression is valid for the barrier for k2 > V0 with
α2 = k2 − V0. We have taken the width 2a of the well and the barrier to be
large (a = 5) to generate a greater number of states. For both the well and
the barrier, the maxima of T are governed by the zeros of sin(2αa). Thus for
an attractive well, the peaks of T are at positive energies, signifying that the
resonances occur at En = [(n2π2/(2a)2) − V0] > 0. Such a close correlation is
not present for the negative energy states, as is clear from Table 3.1. For the
barrier, the peaks of T are at En = [(n2π2/(2a)2) + V0] for n=1,2,.... Both sets
of En are eigenvalues of the corresponding box potentials. In contrast, resonant
state energies and widths are generated from the complex poles of T , and their
positions are slightly shifted from the maxima of T because the resonances are
broader. Tables 3.1 and 3.2 summarize these results.
61
Table 3.2: Comparison of transmission peaks and barrier resonance energiesfor the square barrier potential Eq. (3.17) with V0 = 3 and a = 5.
Sequence Energy eigenvalues Peak energy Resonance Resonancenumber n En of T energies Er half-width Γr/2
1 3.10 3.10 3.10 0.022 3.40 3.40 3.38 0.093 3.89 3.89 3.86 0.194 4.58 4.58 4.53 0.335 5.47 5.47 5.40 0.506 6.55 6.55 6.46 0.707 7.84 7.84 7.72 0.928 9.32 9.32 9.19 1.16
Figure 3.4: Comparison of the barrier top resonant state energies of the repul-sive rectangular potential [Eq. 3.17] with V0 = 3 and a = 5 with the corre-sponding states for the box potential [Eq. 3.18]. The abscissa is the sequencenumber of the states.
62
3.3 Effect of adjacent well on barrier
transmission
Now we wish to discuss an interesting aspect of barrier tunneling. In most
applications for studying the transmission probability across a potential bar-
rier one uses the well known barrier penetration formula deduced using WKB
approximation. In this approximation, one explores only the barrier region of
the potential between classical turning points and the rest of the potential on
either side of the barrier play no role. This cannot be strictly true because, in
principle, transmission and tunneling should depend on the the full potential.
In particular if there is a deep well on either or both sides of the barrier, the
resonances generated by these states can be expected to affect the behaviour of
T . This feature was explored earlier in Ref.[14]. In this section we demonstrate
the role of adjacent well in the transmission across a rectangular barrier having
an attractive well on one side. The WKB approximation based expression for
T is given by (see also Eq.(2.60)):
T = exp[− 2
x2∫x1
√(V(x) − k2)dx
]. (3.20)
Here x1 and x2 > x1 are the turning points, which define the classically forbid-
den region x1 < x < x2 of the barrier at E = k2.
This semiclassical approach to tunneling incorporates the potential between
the turning points at a given energy but ignores the potential elsewhere result-
ing in the expression for T given by Eq.(3.20), which is independent of V(x) in
the domains x < x1 and x > x2. In the light of our discussion on the transmis-
63
sion across a well, it is interesting to see the difference that an adjacent well
makes in tunneling across a barrier. For this purpose we consider the potential:
V(x) =
0, x < −a,
−V0, V0 > 0, − a < x < a,
V1, V1 > 0, a < x < b,
0, x > b.
(3.21)
This potential is a combination of an attractive well followed by a barrier. In
Figure 3.5 we show the variation of T for energies below and above the barrier
for V0 = 3, a = 5, V1 = 6, and (b − a) = 0.5. We have kept (b − a) small to
reduce the excessive damping of T by the barrier at energies below V0. Figure
3.5 also gives the variation of T generated only by an attractive well [Eq.(3.5)]
and the repulsive barrier Eq.(3.17) with width (b − a). The variation of T for
the potential given by Eq.(3.21) generates oscillatory structures related to the
resonant states of the well. If we used a WKB approach or equivalently re-
stricted only to the potential between classical turning points, we would have
obtained a smoothly increasing curve up to E = V0 and missed the oscillatory
features. However, T for a barrier only gives the overall variation of T , im-
plying that the WKB approach provides a reasonable approximation of T but
does not incorporate finer details. The variation of T by only a potential well
without barrier deviates farther from the results obtained by using Eq.(3.21).
One can conclude that if the potential has interesting structures like potential
well beyond the turning points, it is preferable to use more accurate method to
calculate transmission coefficient instead of WKB based approximations.
64
Figure 3.5: Plot of T (E) for an attractive square well Eq.[3.5] with V0 = −3,a = 5 as indicated by W; the repulsive rectangular barrier [Eq.(3.17)] withV0 = 6 and a = 0.25 is indicated by B, and the combination of attractive squarewell and repulsive barrier [Eq.(3.21)] with V0 = −3, V1 = 6, a = 5, and b = 5.5is indicated by W-B.
3.4 Transmission across a well with no oscillations
From our study, we find that an attractive square well can generate bound states
and resonance states. As a result T across an attractive square potential shows
oscillations with E > 0. This tempted us to examine whether T always shows
oscillations with E for an attractive well. Reference [53] studied the nature
of the resonances generated by the potential pocket sandwiched between well
separated barriers and also the barrier top resonances in three-dimensional scat-
tering for rectangular type potentials and for more smoothly varying potentials,
and demonstrated that the nature of the results in both cases is similar. To have
a sharper resonance above the barrier, it is necessary to have a broader barrier.
In the light of this result we might expect that the general pattern of resonances
65
we have found for the attractive rectangular well and rectangular barrier is ap-
plicable even when they are replaced by smoother potentials. However there
can be exceptions. Such an exception is given by the behavior of T for the
modified Poschl-Teller type potential [56]:
V(x) = −β2λ(λ − 1)cosh2βx
. (3.22)
This potential has bound states:
En = −β2(λ − 1 − n)2; n ≤ λ − 1. (3.23)
The expression for T in this case is given by:
T =p2
1 + p2 , (3.24)
where
p =sinh(πk/β)
sin(πλ). (3.25)
An interesting feature of this potential Eq.(3.22) is that T = 1 for integer
λ ≥ 1. Our interest here is to examine T for a typical set of λ and β. In Figure
3.6 we show the variation of T for three sets of λ and β. No oscillatory structure
is present, which implies that no resonant states are generated by this attractive
potential. This property is also evident from Eq.(3.24). There is also another
important case of the attractive Coulomb potential U(r) = −γ/r, with γ > 0,
which generates an infinite number of bound states but no resonant states. In
this case the bound states associated with the poles of the S -matrix for the l th
66
Figure 3.6: Plot of T (E) for the Poschl-Teller potential Eq.(3.22)
partial wave are given by [10]:
S l =Γ(l + 1 − iη)Γ(l + 1 + iη)
. (3.26)
Here η = γ/2k is the Rutherford parameter associated with the Coulomb po-
tential. The repulsive Coulomb potential with γ < 0 is another example that
does not generate resonant states.
Our results show that there can be attractive wells which can generate res-
onances and there are examples of other potentials which do not and knowl-
edge of this feature therefore is important for the satisfactory application of
WKB approximation for tunneling. Hence we give some additional informa-
tion available on this problem. The problem of mathematically proving the
occurrence or non occurrence of resonances for a given potential is non trivial
and remains unresolved. However, some details are known. For example, non
analyticity features in a potential causes multiple zeros in reflection coefficient
67
[57]. Similarly the top/bottom curvature and higher localization of a potential
profile can be crucial in generating resonances [58]. More interestingly the
rectangular well/barrier seems to be the best illustration which displays these
oscillations. On the other hand potentials like Gaussian, Lorentzian, exponen-
tial, parabolic or Eckart do not entail multiple oscillations in T as they are less
localized. The WKB formula for R and T works well for them. Higher order
potential like V(x) = −V0x2n,±V0e−x2n, ±,V0
1+x2n , n=2,3,4... can become reflection-
less at certain discrete energies [58, 59].
3.5 Summary
Now we summarize below the main conclusions about our study of quantum
transmission and eigenstates generated by potentials like wells and barriers:
• By taking an attractive square well and its repulsive counterpart, we have
shown that the oscillations in T as a function of energy corresponds to
the broad resonant state energies obtained by using boundary conditions
satisfied by the resonant state wave function.
• We found that the resonant energies and their widths can be obtained in
terms of the complex poles of the transmission amplitude or reflection
amplitude in the lower half of the complex E-plane in the vicinity of
the real axis. This result is similar to the corresponding behavior of S -
matrix poles in three-dimensional scattering and hence provides a unified
understanding of transmission in one dimension and potential scattering
in three dimensions in terms of the pole structure of the corresponding
amplitudes.
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• For an attractive square well and the rectangular barrier, the positions of
the maxima of T are practically same as the energies of the eigenstates of
the corresponding one-dimensional box potentials. The resonance state
energies are also close to the peak positions of T , this confirms the close
correlation between the oscillatory structure of T at resonance energies.
• From the study of transmission across a potential that is a combination
of an attractive well and a repulsive barrier, we found that T exhibits os-
cillations even at energies below the barrier, in contrast to the WKB type
barrier expression for T , which does not manifest these features. Hence
simple WKB type expressions for T may not be a reasonable approxima-
tion if the total potential has physical features such as an attractive well
outside the barrier region.
• The oscillations of T across an attractive potential are likely to be found
for most potentials. But these features can have exceptions for example
the attractive modified Poschl-Teller type potential which can generate
bound states but no resonance states. Hence the study of T across this
potential as a function of energy does not show any oscillations.
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