Transmission across potential wells and barriers

Transcription

1 3 Transmission across potential wells and barriers The physics of transmission and tunneling of waves and particles across different 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 regions which are forbidden in geometrical optics can be understood through wave nature of light. Another example is the well-known phenomenon of to- 45

2 tal internal reflection. When light impinges upon the interface between two media with refractive indices n 1 and n 2, it suffers total internal reflection when θ i sin 1 ( n 1 n 2 ), where θi is the angle of incidence. According to the geometrical optics description, there is no penetration of light into the rarer medium. However, 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 variety 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 interesting 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 (example: 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

3 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 transmission 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 indicating 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 reflection 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

4 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 important 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 corresponding 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 oscillatory 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, transmission and reflection is the one dimensional time independent Schrödinger 48

5 equation for a particle of mass m with potential U(x) and total energy E: d 2 ϕ(x) dx 2 + (k 2 V(x))ϕ(x) = 0, (3.1) where k 2 = 2mE/ 2, V(x) = 2mU(x)/ 2 and α 2 = k 2 V(x). For convenience we use the units 2m = 1 and 2 =1 such that k 2 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 k b x, (3.2) and the corresponding bound state energy E b is given by E b = (ik b ) 2 = k 2 b. 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 kr 2 = (k r ik i ) 2 = (kr 2 ki 2) i2k rk i = E R = E r iγ r /2 with 49

6 E r > 0 and Γ r > 0 such that the wave function behaves asymptotically as: ϕ(x) ϕ(x) e i(k r ik i )x x e i(k r ik i )x. (3.3) x This behavior implies a positive energy state E r with width Γ r and lifetime /Γ r and ϕ(x) diverges exponentially as e k i x but decays exponentially in time because the time-dependent part of the solution of the Schrödinger wave equation is e iet/. Smaller width Γ r 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 Schrödinger 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 reflection coefficient R can be obtained by using transfer matrix technique. This method can be used for potentials that are not analytically solvable. Alternatively, one can adopt a numerical integration of the Schrödinger equation using well known procedures like Runge kutta method. 50

7 For the study of R and T we seek a solution of ϕ(x) satisfying the conditions: ϕ(x) ϕ(x) Ae ikx + Be ikx, x Fe ikx, (3.4) x 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 scattering 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 E r and its width corresponds to the width of the resonance state Γ r. 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 0, V 0 > 0, x < a, V(x) = 0, x > a. (3.5) 51

8 The corresponding box potential of attractive square well potential is: V 0, V 0 > 0, x < a, V(x) =, x > a, (3.6) which generates the eigenvalues: E n = V 0 + n 2 π 2 /(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 resonance 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 Schrödinger 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 derivative 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 k n of the equation: kcos(αa) iαsin(αa) = 0. (3.8) 52

9 The roots k n are just below the real k-axis in the complex k-plane such that Re(k n ) > Im(k n ). 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 E n. 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]: F A = 4αke 2ika, (3.10) (α + k) 2 e 2iαa (α k) 2 e2iαa ( ) B A = α 2 ) F 2i(k2 4kα sin(2αa)e2ika, (3.11) A B A = 2i(α 2 k 2 )sin(2αa) [e 2iαa (α + k) 2 e 2iαa (α k) 2 ], (3.12) B F = 2i(α2 k 2 )sin(2αa), (3.13) 4kα T = F A 2 = (4αk) 2 (4kα) 2 + 4(α 2 k 2 ) 2 sin 2 (2αa), (3.14) 53

10 R = B A 2 = 4(α 2 k 2 ) 2 sin 2 (2αa) (4kα) 2 + 4(α 2 k 2 ) 2 sin 2 (2αa) = 1 T, (3.15) where α 2 = V 0 + k 2. For a repulsive rectangular potential α 2 = k 2 V 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 corresponding 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 parameters. 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 numerical results obtained using V 0 = 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

11 Figure 3.1: Variation of transmission coefficient T as a function of energy E for an attractive square well potential Eq.(3.5 ) with V 0 = 3 and a = 5. Table 3.1: Transmission across an attractive square well of depth V 0 = 3 and range a = 5 [see Eq.(3.5)]. In all tables and figures the energy unit is Å 2 and length unit is Å. The positive energy box states E n coincide with the maxima of T. Sequence Energy Peak energies Bound state and Half-width Γ n /2 number eigenvalues E n of T resonance state energies of eigenstates

12 Figure 3.2: Comparison of bound and resonant states of an attractive square well in Eq.(3.5) for potential parameters V 0 = 3 and a = 5 with the corresponding states for the box potential (3.6). The abscissa is the sequence number of the states. It may be pointed out that in the case of the bound states of attractive square well n 1 signifies the principal quantum number associated with energy eigenvalues. In Figure.3.2 we demonstrate a correlation between the bound and resonance 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 positions 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

13 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, equivalently, complex-e plane [10]. For transmission across a potential in one dimension, 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 iterative 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 incident wave e ikx as x when the outgoing transmitted wave e ikx 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 = k r ik i in the lower half of k-plane with k r > k i, the resulting wave function 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 = k r ik i of the coefficient of incoming spherical wave component of 57

14 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 + k α 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 transmission amplitude F/A for unit incident wave signify resonances and generate the peaks of the oscillations of T for an attractive square well potential 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 0, V 0 > 0, x < a, V(x) = 0, x > a. (3.17) 58

15 The corresponding box potential is: V 0, V 0 > 0, x < a, V(x) =, x > a, (3.18) which can generate the eigenvalues: E n = V 0 + n 2 π 2 /(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 > V 0 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 generated by the barrier at energies E > V 0. These states along with the maxima of T are listed in Table 3.2. The expression T for E < V 0 is given by Eq.(3.14) 59

16 Figure 3.3: Variation of T with E for the repulsive rectangular barrier [Eq. (3.17)] with V 0 = 3 and a = 5. with α 2 = k 2 V 0. 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 resonance energies due to the fact that we are dealing with broad states. Note that the numerically computed T has contributions from background terms in addition 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

17 To make the comparison between the attractive well and the corresponding barrier more complete, we examine the variation of the barrier top resonance position E n with n and compare it with the corresponding box potential positive 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 attractive 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 α 2 = k 2 + V 0. The same expression is valid for the barrier for k 2 > V 0 with α 2 = k 2 V 0. 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 E n = [(n 2 π 2 /(2a) 2 ) V 0 ] > 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 E n = [(n 2 π 2 /(2a) 2 ) + V 0 ] for n=1,2,... Both sets of E n 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

18 Table 3.2: Comparison of transmission peaks and barrier resonance energies for the square barrier potential Eq. (3.17) with V 0 = 3 and a = 5. Sequence Energy eigenvalues Peak energy Resonance Resonance number n E n of T energies E r half-width Γ r / Figure 3.4: Comparison of the barrier top resonant state energies of the repulsive rectangular potential [Eq. 3.17] with V 0 = 3 and a = 5 with the corresponding states for the box potential [Eq. 3.18]. The abscissa is the sequence number of the states. 62

19 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 barrier 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)): [ x 2 T = exp 2 x 1 ] (V(x) k2 )dx. (3.20) Here x 1 and x 2 > x 1 are the turning points, which define the classically forbidden region x 1 < x < x 2 of the barrier at E = k 2. This semiclassical approach to tunneling incorporates the potential between the turning points at a given energy but ignores the potential elsewhere resulting in the expression for T given by Eq.(3.20), which is independent of V(x) in the domains x < x 1 and x > x 2. In the light of our discussion on the transmis- 63

20 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, V 0, V 0 > 0, a < x < a, V 1, V 1 > 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 V 0 = 3, a = 5, V 1 = 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 V 0. 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 restricted only to the potential between classical turning points, we would have obtained a smoothly increasing curve up to E = V 0 and missed the oscillatory features. However, T for a barrier only gives the overall variation of T, implying 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

21 Figure 3.5: Plot of T(E) for an attractive square well Eq.[3.5] with V 0 = 3, a = 5 as indicated by W; the repulsive rectangular barrier [Eq.(3.17)] with V 0 = 6 and a = 0.25 is indicated by B, and the combination of attractive square well and repulsive barrier [Eq.(3.21)] with V 0 = 3, V 1 = 6, a = 5, and b = 5.5 is 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 scattering 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

22 we have found for the attractive rectangular well and rectangular barrier is applicable 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]: This potential has bound states: 2 λ(λ 1) V(x) = β cosh 2 βx. (3.22) E n = β 2 (λ 1 n) 2 ; n λ 1. (3.23) The expression for T in this case is given by: T = p2 1 + p 2, (3.24) where p = sinh(πk/β). (3.25) sin(πλ) 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

23 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 iη). (3.26) Here η = γ/2k is the Rutherford parameter associated with the Coulomb potential. 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 resonances and there are examples of other potentials which do not and knowledge of this feature therefore is important for the satisfactory application of WKB approximation for tunneling. Hence we give some additional information 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

24 [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, exponential, 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) = V 0 x 2n,±V 0 e x2n, ±,V 0 1+x 2n, n=2,3,4... can become reflectionless 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. 68

25 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 oscillations 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 approximation 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. 69