aa r X i v : . [ qu a n t - ph ] J u l The step-harmonic potential
L. Rizzi
Dipartimento di Fisica e Matematica,Universit`a dell’Insubria, Via Valleggio 11, 22100 Como, Italy
O. F. Piattella, S. L. Cacciatori, and V. Gorini
Dipartimento di Fisica e Matematica, Universit`a dell’Insubria,Via Valleggio 11, 22100 Como, Italy andINFN, sezione di Milano, Via Celoria 16, 20133 Milano, Italy
Abstract
We analyze the behavior of a quantum system described by a one-dimensional asymmetric poten-tial consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integralrepresentation method, which allows us to classify the independent solutions as equivalence classesof homotopic paths in the complex plane. We then consider the propagation of a wave packetreflected by the harmonic barrier and obtain an expression for the interaction time as a functionof the peak energy. For high energies we recover the classical half-period limit. . INTRODUCTION The harmonic oscillator plays a central role in physics because it is exactly solvable andprovides a simple model for a host of physical phenomena. Besides the simplest case of theone-dimensional free oscillator, modifications of the harmonic oscillator are of interest. Atypical variant is to confine the oscillator in a box. This system has been investigated inone dimension and for arbitrary dimensions. The purpose of these investigations is tocalculate the corrections to the energy levels caused by the presence of infinite barriers at afinite distance. This analysis has also been done for spherically symmetric potentials, suchas the hydrogen atom. In addition, the behavior of a wave packet propagating in a genericpower-law one-dimensional potential well has been considered in terms of its “collapse andrevival,” namely of its scattering over the well and its subsequent reforming.
The truncatedharmonic oscillator has been used to study how the presence of discrete levels in the energyspectrum affects tunneling through such a well. In the present paper we study the bound states and the propagation of a wave packet in aone-dimensional potential consisting of a half-space harmonic oscillator plus a step. In Sec. IIwe solve the Hamiltonian eigenvalue equation using the integral representation method,which allows us to classify the independent solutions as equivalence classes of homotopicpaths in the complex plane. In Sec. III we calculate the energy of the bound states andcompare them with the standard harmonic oscillator and the half-space harmonic oscillatorwith an infinite barrier. In Sec. IV we study the properties of the propagation of a wavepacket which, coming from infinite distance, is reflected by the harmonic barrier. Ouranalysis is based mainly on the investigation of the delay time in the reflection, which canbe interpreted as the duration of the interaction with the harmonic barrier. We express theinteraction time as a function of the peak energy and study its asymptotic behavior. Weshow that in the high energy limit the delay approaches the classical value, namely the halfperiod of the harmonic oscillator. Finally, we comment on how this behavior changes in thepresence of a stronger or weaker confinement.The problem that we address has been studied by Mei and Lee to test the adequacy ofa perturbation scheme on an exactly solvable model. Our analysis, which has a differentpurpose, has the advantage of being based on the integral representation method which canbe applied to a wider class of problems. In addition, we do not confine ourselves to the study2f the bound states, but also investigate the motion of continuous spectrum wave packets. II. THE STEP-HARMONIC POTENTIAL
Consider a particle subject to the one-dimensional potential: U ( x ) = U ( x ≥ κx ( x < , (1)where κ and U are real positive constants (see Fig. 1). U U ( x ) x III
FIG. 1: The step-harmonic potential.
The proper and improper eigenfunctions of the Hamiltonian operator are ordinary solu-tions of the eigenvalue equation outside of the discontinuity of the potential. Such solutionsmust be continuous together with their first derivatives across the singularity.
We willsolve the eigenvalue equation for x > x < x > − ~ m d u ( x )d x + U u ( x ) = Eu ( x ) , (2)where m is the particle mass and E is the energy eigenvalue. The general solution of Eq. (2)has the form: u ( x ) = Ae ikx + Be − ikx ( E > U ) A ′ e kx + B ′ e − kx (0 < E < U ) , (3)where ~ k ≡ p m | E − U | .We must choose A ′ = 0. Otherwise, for 0 < E < U , u ( x ) would diverge exponen-tially for x → + ∞ and therefore it would neither belong to L ( R ) (i.e. the space ofthe square summable functions over R ), nor satisfy the eigenpacket condition for impropereigenfunctions. x < u ( y )d y + ( ǫ − y ) u ( y ) = 0 , (4)where y = αx, α ≡ r mκ ~ , ǫ ≡ E ~ r mκ , ω ≡ r κm . (5)We set u ( y ) = F ( y ) exp ( − y /
2) and obtain from Eq. (4) the following equation for F ( y ) F ′′ ( y ) − yF ′ ( y ) + ( ǫ − F ( y ) = 0 , (6)which is the Hermite equation. The solutions of Eq. (6) are entire functions and can be found by the method of integrationby series. We prefer to employ the integral representation method. We start by lookingfor solutions of Eq. (6) that have the form F ( y ) = Z γ d t f ( t ) e − t +2 ty , (7)where γ is a path in the complex plane C and f is a suitable function which is holomorphicin a region which contains the graph of γ . We substitute Eq. (7) into Eq. (6) and obtain Z γ d t [4 t + ( ǫ − f ( t ) e − t +2 ty − Z γ d t (cid:18) dd t e ty (cid:19) tf ( t ) e − t = 0 . (8)Equation (8), after integration by parts of the second integral, can be written as h − tf ( t ) e − t +2 ty i ∂γ + Z γ d t [( ǫ + 1) f ( t ) + 2 tf ′ ( t )] e − t +2 ty = 0 . (9)From Eq. (9) it follows that Eq. (7) is a solution of Eq. (6) if h tf ( t ) e − t +2 ty i ∂γ = 0 and f ( t ) = t − ǫ +12 . (10)Therefore, we can write a solution of Eq. (6) in the form F ( γ ) ( y ) = Z γ d t t − ǫ +12 e − t +2 ty , (11)where γ must be chosen according to the first condition in Eq. (10) and such that the integralin Eq. (11) is well defined. The classification of the appropriate γ ’s allows us to classify allsolutions of Eq. (6).The integrand in Eq. (11) is singular at t = 0. For ǫ = 2 n + 1 ( n = 0 , , . . . ), the point t = 0 is a pole of order n + 1, otherwise it is a branch point.In the following we distinguish two classes of paths, which correspond to two linearlyindependent solutions of Eq. (6). 4 Γ Γ Re(t)Re(t) Γ FIG. 2: Possible paths for ǫ = 2 n + 1 ( n = 0 , , . . . ). A. The case ǫ = 2 n + 1 In this case we can rewrite Eq. (7) as F ( γ ) n ( y ) = Z γ d t e − t +2 ty t n +1 , (12)where the integrand is holomorphic on C but the origin. Possible choices of γ for whichthe contour condition in Eq. (10) holds are shown in Fig. 2; Γ and Γ have a real partthat goes to infinity, Γ is a closed path circling the origin, and Γ is a closed path thatdoes not contain the origin. By virtue of Cauchy’s theorem, F (4) n = 0, and because thepaths can be deformed so that Γ + Γ = Γ , the other three solutions satisfy the relation F (1) n + F (3) n = F (2) n , where F ( j ) n is the solution corresponding to the path Γ j ( j = 1 , , , , namely F (2) n ( y ) = I d t e − t +2 ty t n +1 , (13)corresponds to the Hermite polynomial of order n . By completing the square in the integrandof Eq. (13), we find F (2) n ( y ) = e y I d t e − ( t − y ) t n +1 . (14)We take advantage of Cauchy’s formula and rewrite Eq. (14) as F (2) n ( y ) = 2 πin ! ( − n e y d n d y n ( e − y ) = 2 πin ! H n ( y ) , (15)where H n ( y ) is the Hermite polynomial of order n . m(t) Re(t)Γ Γ Γ FIG. 3: Possible paths for ǫ = 2 n + 1. Note that in this case Γ cannot be closed at infinity. B. The case ǫ = 2 n + 1 In the generic case ǫ ∈ R , ǫ = 2 n + 1, we rewrite Eq. (11) as F ( γ ) ǫ ( y ) = Z γ d t e − t +2 ty t β , (16)where β ≡ ( ǫ + 1) /
2. If β is not a positive integer, t = 0 is a branch point for the multivaluedfunction t β . In this case we must cut the complex plane, for example along the positive realaxis. In the latter case, the classes of possible paths are depicted in Fig. 3.In the following we show that the solutions corresponding to Γ and Γ ( F (1) ǫ and F (3) ǫ )diverge as e y for y = ±∞ and therefore the corresponding eigenfunction u ( y ) cannot beeither proper or improper. We are thus left with the solution corresponding to Γ [ F (2) ǫ ] whichagain diverges as e y for y → + ∞ . It also diverges for y → −∞ , but the corresponding u ( y )and its derivatives vanish more rapidly than any polynomial thanks to the presence of theexp( − y /
2) factor.According to standard results in the theory of integrals depending on a parameter, it iseasy to show that F ( γ ) ǫ ( y ) is an entire function. Furthermore, the derivatives of F ( γ ) ǫ areobtained by differentiating with respect to y under the integral sign of Eq. (16). Thus weobtain the relation d m F ( j ) ǫ d y m = 2 m F ( j ) ǫ − m . (17)We next address the asymptotic behavior of two independent solutions, for example, F (1) ǫ and F (2) ǫ . 6 he Γ solution . We rewrite Eq. (16) for the path Γ by introducing the variable z = t − y : F (1) ǫ ( y ) = e y Z Γ d z e − z ( y + z ) β . (18)The branch point is now z = − y and the cut is shifted as well. We extract | y | from theintegral in Eq. (18) : F (1) ǫ ( y ) = e y | y β | G ( y ) , (19)where G ( y ) ≡ Z Γ d z e − z [sgn( y ) + z | y | ] β . (20)An elementary calculation shows thatlim y →±∞ G ( y ) = lim y →±∞ Z Γ d z e − z [sgn( y ) + z | y | ] β = − √ π sgn( y ) β , (21)where we have taken the limit under the integral sign by virtue of the dominated convergencetheorem (see Appendix B). The asymptotic behavior of F (1) ǫ ( y ) for y → ±∞ is therefore F (1) ǫ ( y ) ∼ −√ π e y y β . (22)The corresponding eigenfunction u (1) ( y ) = F (1) ǫ ( y ) exp ( − y /
2) cannot be either proper orimproper.
The Γ solution . The solution corresponding to Γ [ F (2) ǫ ( y )] has the form F (2) ǫ ( y ) = Z Γ d t e − t +2 ty t β , (23)where, as shown in Fig. 3, Γ circles around the branch point in an anti-clockwise sense.This solution has different behavior for y → + ∞ and y → −∞ . Asymptotic behavior for y → + ∞ . By virtue of Cauchy’s theorem we can deform Γ tosplit the integral in Eq. (23) into a sum of two integrals over the paths Γ and Γ : F (2) ǫ ( y ) = Z Γ ∪ Γ d t e − t +2 ty t β . (24)We again introduce z = t − y and extract y β from the integral. We obtain F (2) ǫ ( y ) = e y y β G ( y ) , (25)7 δ Re(t) δ r Im(t) FIG. 4: A standard trick in contour integration. where G ( y ) ≡ Z Γ d z e − z (1 + zy ) β + Z Γ d z e − z (1 + zy ) β . (26)Thanks to the dominated convergence theorem we findlim y → + ∞ G ( y ) = − ie − iπβ √ π sin( πβ ) . (27)The asymptotic behavior of F (2) ǫ ( y ) for y → + ∞ is thus F (2) ǫ ( y ) ∼ − ie − iπβ √ π sin( πβ ) e y y β . (28)Note that, if ǫ = 2 n + 1, Eq. (28) is incorrect because G ( y ) → y → + ∞ . We alreadyknow that, in this case, F (2) ( y ) is as a polynomial of degree n . Asymptotic behavior for y → −∞ . By taking advantage of Cauchy’s theorem, we deformand split Γ in the 3 sub-paths shown in Fig. 4. We choose for simplicity r = 1; from Eq. (23)we obtain F (2) ǫ ( y ) = I β ( y ) − ie − iπβ sin( πβ ) Z ∞ d t e − t +2 ty t β , (29)where I β ( y ) ≡ i Z π d θ e i (1 − β ) θ e − cos(2 θ ) − i sin(2 θ )+2 y cos θ +2 iy sin θ . (30)For y < t > t − β e − t +2 ty < t − β e − t , (31)and, moreover, t − β exp( − t ) is integrable in [1 , + ∞ ). Therefore, the integral on the right-hand side of Eq. (29) vanishes for y → −∞ by virtue of the dominated convergence theorem.In contrast, for I β ( y ) we have | I β ( y ) | ≤ Z π d θ (cid:12)(cid:12)(cid:12) e i (1 − β ) θ e − cos(2 θ ) − i sin(2 θ )+2 y cos θ +2 iy sin θ (cid:12)(cid:12)(cid:12) = Z π d θ e − cos(2 θ ) e y cos θ , (32)8o that | I β ( y ) | ≤ πe | y |− . Therefore, for y → −∞ , the absolute value of F (2) ǫ ( y ) is domi-nated by 2 πe | y |− , and u II ( y ) = F (2) ǫ ( y ) e − y / (33)is rapidly decreasing. Hence it is square summable on the positive real axis. Thus, itfollows that for ǫ = 2 n + 1 the full-space harmonic oscillator does not admit proper orimproper eigenfunctions. More general theorems allow one to obtain our results indirectly,for example, by studying the asymptotic behavior of the power series expansion of thesolutions of Eq. (4). Here we have adopted a more direct approach. III. EIGENFUNCTIONS AND ENERGY LEVELS
From the results of Sec. II we can write the energy eigenfunctions as
E < U u ( x ) = AF ǫ ( αx ) e − α x ( x < Be − kx ( x >
0) (34)
E > U u ( x ) = CF ǫ ( αx ) e − α x ( x < De ikx + Ee − ikx ( x > , (35)where we have dropped the superscript from F (2) ǫ . The integration constants A, . . . , E mustbe chosen such that u ( x ) and its first derivative are continuous at x = 0 (the junctionconditions). A. The case
E < U If the energy is smaller than the step height U , the junction conditions imply that B − F ǫ (0) A = 0 (36a) kB + αF ′ ǫ (0) A = 0 . (36b)The condition for the existence of a nontrivial solution is the vanishing of the system de-terminant. We define J ( β ) ≡ F ǫ (0) [see Eq. (A1)]. In Appendix A we obtain the followingexpression for J ( β ) [see Eq. (A3)]: J ( β ) = sin( πβ ) ie iπβ Γ (cid:18) − β (cid:19) . (37)9he recurrence relation for the derivatives of F ǫ , Eq. (17), can be rewritten in terms of J ( β )as F ′ ǫ (0) = 2 J ( β − . (38)Equation (38) implies that the junction conditions can be rewritten as − αJ ( β −
1) = kJ ( β ) , (39)or, equivalently, as Γ (cid:18) − β (cid:19) Γ (cid:18) − β (cid:19) = − r β − β , (40)where β = U / ( ~ ω ) + 1 /
2. By using the relation Γ( z )Γ(1 − z ) = π sin( πz ) , (41)we can rewrite Eq. (40) as Γ (cid:18) β + 12 (cid:19) Γ (cid:18) β (cid:19) cot (cid:16) π β (cid:17) = − r β − β , (42)For a given value of β Eq. (42) is an implicit relation determining the energy levels. Theadvantage of Eq. (42) is that the singular behavior is contained in the cotangent function.For 0 ≤ U < ~ ω/ / ≤ β <
1) the step is too small to allow for the existence ofdiscrete energy levels. The first level (the ground state) appears for β = 1 at the value E = ~ ω/
2, which is the ground state of the full-space harmonic oscillator. For 1 ≤ β < β starting from its minimumvalue ~ ω/
2. When β crosses the value 3 a second level appears at the energy E = 5 ~ ω/ β (and hence U ) the subsequent levels appear as U crosses the values E k = ~ ω (2 k + 1 /
2) ( k ∈ N ), corresponding to the ( k + 1)th even level of the oscillator. Thus,for a fixed value of β such that 2 k +1 < β < k +3 there are exactly k +1 bound states withenergies E n ( n = 0 , , . . . , k ) satisfying the inequalities ~ ω (2 n + 1 / < E n < ~ ω (2 n + 3 / k = 1 is shown in Fig. 5. Each E n is a monotonically increasing functionof U which asymptotically approaches the value ~ ω (2 n + 3 /
2) (the ( n + 1)th odd state ofthe oscillator) as U → ∞ . 10 β β − q β √ π FIG. 5: The solid and the dashed lines represent respectively the left- and the right-hand side ofEq. (42). The intersections determine the energy levels. Here β = 4 . The (unnormalized) eigenfunctions corresponding to the eigenvalue E n are u n ( x ) = F ǫ n ( αx ) e − α x ( x < J ( β n ) e − k n x ( x ≥ , (43)where the β n are the solutions of Eq. (42); ǫ n = 2 β n − ~ k n = p m ( U − E n ), and E n = ~ ωǫ n / B. The case
E > U The junction conditions on the eigenfunctions of Eq. (35) are D + E − CF ǫ (0) = 0 (44a) ik ( D − E ) − CαF ′ ǫ (0) = 0 , (44b)implying the normalized (with respect to k ) improper eigenfunctions are given by u ǫ ( x ) = 1 √ π Π( β ) F ǫ ( αx ) e − α x ( x < e − ikx + ζ ( β ) e ikx ( x ≥ , (45)11here, as usual, 2 β ≡ ǫ + 1, ~ k ≡ p m ( E − U ), 2 E ≡ ~ ωǫ , andΠ( β ) ≡ h J ( β ) + i q β − β J ( β − i − , (46) ζ ( β ) ≡ J ( β ) − i q β − β J ( β − J ( β ) + i q β − β J ( β −
1) = Γ (cid:18) − β (cid:19) − i q β − β Γ (cid:18) − β (cid:19) Γ (cid:18) − β (cid:19) + i q β − β Γ (cid:18) − β (cid:19) , (47)where we have used Eq. (A1). Note that | ζ ( β ) | = 1. As expected, the continuous part ofthe spectrum ( E > U ) is simple. IV. REFLECTION AND DELAY
To study reflection phenomenon, we consider the following superposition of continuousstates: ψ ( x, t ) = Z ∞ d k c ( k ) u ǫ ( k ) ( x ) e − i ~ E ( k ) t . (48)From Eq. (45) we have ψ ( x, t ) = 1 √ π R ∞ d k c ( k )Π( β ( k )) F ǫ ( αx ) e − α x − i ~ E ( k ) t ( x < R ∞ d k c ( k ) (cid:2) ζ ( β ( k )) e ikx + e − ikx (cid:3) e − i E ( k ) ~ t = ψ ref + ψ in ( x > . (49)We write ψ in and ψ ref in the form: ψ in ( x, t ) = 1 √ π Z + ∞ d k | c ( k ) | e − i [ kx +Ω( k ) t − γ ( k )] , (50) ψ ref ( x, t ) = 1 √ π Z + ∞ d k | c ( k ) | e i [ kx − Ω( k ) t + δ ( k )+ γ ( k )] , (51)where we have defined e iδ ( k ) ≡ ζ ( β ( k )) and Ω( k ) ≡ E ( k ) ~ = U ~ + ~ k m . (52)If c ( k ) is sufficiently regular and non-vanishing only in a small neighborhood of ˜ k , then ψ in and ψ ref represent wave packets that move according to the equations of motion x in = − dΩd k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k t + d γ d k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k = − ~ ˜ km ( t − t ) = − ˜ pm ( t − t ) , (53)for the “incoming” wave packet, and x ref = dΩd k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k t − d γ d k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k − d δ d k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k = ˜ pm (cid:20) ( t − t ) − m ˜ p d δ d k (cid:12)(cid:12)(cid:12)(cid:12) k =˜ k (cid:21) , (54)12 ββ -0.500.511.522.5 1 2 3 4 5 6 7 8 9 10 ββ -0.500.511.522.5 1 2 3 4 5 6 7 8 9 10 ββ -0.500.511.522.5 1 2 3 4 5 6 7 8 9 10 ββ -0.500.511.522.5 1 2 3 4 5 6 7 8 9 10 ββ -0.500.511.522.5 1 2 3 4 5 6 7 8 9 10 ββ FIG. 6: Plots of the delay time τ (in units of T /
2, where T = p κ/m is the period associated tothe harmonic oscillator) versus the “energy” β of the incoming wave packet for six values of thestep height β . (a) β = 1 .
5, (b) β = 2, (c) β = 2 .
5, (d) β = 3 .
5, (e) β = 4, and (f) β = 4 . for the reflected “outgoing” one.The solution represents a particle of well defined momentum ˜ p = ~ ˜ k which approachesthe origin from the right, interacts with the harmonic potential (at t = t ), and is totallyreflected. The phase shift results in a delay in the time the wave packet bounces back, whichis caused by the interaction with the confining harmonic barrier. Because the phase shift δ depends only on k through β , we can write the delay as τ ( ˜ β ) = 1 ω d δ d β (cid:12)(cid:12)(cid:12)(cid:12) β = ˜ β , (55)where ˜ β = β (˜ k ).We prove that lim β →∞ δ ′ ( β ) = π, (56)lim β → β δ ′ ( β ) = + ∞ β ∈ S k ∈ N (2 k, k + 1) −∞ β ∈ S k ∈ N (2 k + 1 , k + 2)0 β ∈ N = 1 , , . . . , (57)13here the prime denotes the derivative with respect to β . To this purpose, note that byemploying Eq. (41) δ ′ ( β ) can be cast in the form: δ ′ ( β ) = 12 √ β − β sin ( βπ ) (cid:20) β − β + Ψ (cid:18) β (cid:19) − Ψ (cid:18) β + 12 (cid:19) + 2 π sin ( βπ ) (cid:21) ( β − β ) Γ ( β/ β/ / √ (cid:18) βπ (cid:19) + Γ ( β/ / √
2Γ ( β/
2) cos (cid:18) βπ (cid:19) , (58)where Ψ is the Digamma function (that is, the logarithmic derivative of the Gammafunction). In Fig. 6 we plot τ versus β for different values of β . Note the resonanceslocated at β ≃
3, 5, 7, 9, . . . , corresponding to the formation of metastable states at the re-spective energies E ≃ ~ ω/
2, 9 ~ ω/
2, 13 ~ ω/
2, 17 ~ ω/ . . . . These states have lifetimes whichdecrease as the corresponding energies increase and move farther away from the thresholdenergy U . Conversely, as U increases, the lifetime of the resonance closest to the height ofthe step becomes progressively longer and then infinite when the resonance turns into thenext bound state. This behavior is evident in Fig. 6, in which the first three plots correspondto values of β for which there is only one bound state. In the successive three plots theresonance at β = 3 has disappeared, having turned into the second bound state.It is simple (using steepest descent or Stirling’s formula, for example) to show thatΓ( z + 1 / z ) = √ z (cid:20) O (cid:18) z (cid:19)(cid:21) , (59)for z ≫
1. By using one of the integral formulas for the Digamma function, we can alsoshow that lim z →∞ (cid:20) Ψ ( z ) − Ψ (cid:18) z + 12 (cid:19)(cid:21) = 0 . (60)Thanks to Eqs. (59) and (60) it is straightforward to derive Eqs. (56) and (57). In particularEq. (56) implies that lim β →∞ τ ( β ) = πω = T . (61)The wave packet undergoes half an oscillation during the interaction with the harmonicpotential before being reflected, which results in a delay of half a period compared withthe reflection from a perfect mirror (that is, when the confining barrier is an infinite wall).Thus, as expected, the high energy limit reproduces the classical behavior.14 . CONCLUSIONS The main features of the discrete part of the spectrum can be summarized as follows.For sufficiently small U (the height of the step) there is no discrete spectrum. When U increases and approaches the value ~ ω/ E ≃ ~ ω/
2. This resonance converts into a bound state when U reaches the value ~ ω/ x <
0, to the eigenfunction of theground state of the free harmonic oscillator and is flat otherwise. By further increasingthe height of the step the ground state energy increases monotonically with U and, as U → ∞ , approaches asymptotically from below the first odd level 3 ~ ω/ E k = ~ ω (2 k + 1 / k ∈ N ) whenever U crosses the value E k . In the limit of infinite U (leading to the half-space oscillator), the energy levels become the odd levels of the oscillator itself, as expectedfrom the symmetry of the problem. Loosely speaking, the levels are born as “even” and,upon increasing the height of the step, end up as “odd” (see Fig. 5). This behavior is notpeculiar to the step problem associated with the harmonic oscillator, but is typical of thecorresponding step variant of every symmetric confining potential.The continuous spectrum is simple and extends from U to ∞ . A wave packet comingfrom infinity collides with the confining harmonic branch and is thereby entirely reflected.The interaction with the potential results in a delay of the reflected packet which, as iswell known for problems of this kind, is proportional to the derivative of the phase shiftof the plane wave component evaluated at ˜ β = β (˜ k ). This delay can be interpreted as theinteraction time with the harmonic barrier. When the confining part of the potential isinfinite ( U ( x ) = + ∞ at x <
0) the delay vanishes, and the reflection on a perfect mirror isinstantaneous. The desirable feature of our example is that we can derive an exact analyticexpression for the delay [Eq. (58)] as a function of the step height and of the peak energyof the incoming packet.Although these characteristics are typical of the step variants of all symmetric confiningpotentials, the harmonic oscillator potential is “more equal” than the others. It is the onlyanalytic (except possibly at x = 0), convex or concave locally bounded symmetric andconfining potential that gives rise to classical isochronous oscillations and thus to evenlyspaced energy levels. In our step variant of the problem we recover both these features15n the limit U → ∞ , when the potential reduces to the half-space harmonic oscillator. Thus,it is possible that the harmonic potential is the only confining barrier that displays a constantnonvanishing interaction time in the limit of high energies. For steeper barriers we expect theinteraction time τ to vanish at high energies, while for milder potentials we expect the delayto become infinite in this limit, in accordance with the corresponding classical behavior.Similarly, we expect that, as U → ∞ , the spacing between two neighboring discrete levelsapproaches infinity in the former case and zero in the latter.In a forthcoming paper we corroborate this conjecture by analyzing two examples usingthe integral representation method. This method can be employed to analyze a wide classof “step-something” potentials in which the harmonic part is replaced by another type ofbarrier. We encourage readers to investigate, for example, the step-linear (sl) and the step-exponential (se) potentials U sl ( x ) = U ( x ≥ − M x ( x <
0) (62) U se ( x ) = U ( x ≥ M e − x/σ ( x < , (63)where M , σ , and U are real positive constants. Acknowledgments
We are grateful to Carlo Garoni for having inspired the topic that we have analyzed inthis paper.
Appendix A: Calculation of J ( β ) In this appendix we prove Eq. (37). According to the definition given in Sec. III we have J ( β ) ≡ F (2) ǫ (0) = Z Γ d t e − t t − β , (A1)where 2 β ≡ ǫ + 1. Assume β ∈ C . For | β | ≤ R , the integrand function in Eq. (A1) isbi-continuous and holomorphic with respect to β in any compact disc. Furthermore, itsabsolute value is bounded by a summable positive function: | e − t t − β | ≤ e − ℜ ( t ) | t | R . These16 Im(t) Re(t) Im(u) Re(u)Γ ′ Im(u) Re(u)Γ ′ FIG. 7: The path Γ and its transformed one Γ ′ after the change of variable u = t . When we takeinto account the presence of the cut, we must choose 0 ≤ arg( u ) < π on the first sheet (dashedline), and 2 π ≤ arg( u ) < π on the second one (solid line), that is, we choose the positive squareroot on the first sheet, and the negative one on the second. properties imply that the integral in Eq. (A1) is uniformly convergent. Therefore, J ( β ) isan entire function. If we change the variable in Eq. (A1) to u = t , we must cut the complexplane and define t = √ u on its complete Riemann surface, which is composed of two sheets.The new path Γ ′ is shown in Fig. 7.On both sheets of the u -plane, there is an integral along a straight line and an integral ona semi-loop. It is easy to show that the integral on the semi-loop vanishes when shrunk toa point, provided that β ∈ ( −∞ , J ( β ) = − Z ∞ d u e − u u − β +12 + 12 e i π (cid:18) − β +12 (cid:19) Z ∞ d u e − u u − β +12 , (A2)which we can write as J ( β ) = sin( πβ ) ie iπβ Γ (cid:18) − β (cid:19) . (A3)Because the poles of the Gamma function in Eq. (A3) are cancelled by the zeroes of thesine, the right-hand side is an entire function so that Eq. (A3) holds on the entire complexplane by analytic continuation. Appendix B: Taking limits under the integral sign
For convenience we give here a very useful elementary theorem which we have used inthe paper:
Theorem 1: The dominated convergence theorem . Let f k : R → R be summableon an interval I , that is, R I f k < ∞ , ∀ k ∈ N . Moreover, let f k converge almost everywhere17o a function f ∞ : R → R . Suppose that there exists a positive I -summable function g : R → R + that dominates every f k (that is, | f k ( x ) | ≤ g ( x ) ∀ k ∈ N ). It follows that f ∞ is I -summable and that we can take the limit under the integral, that is,lim k →∞ Z I d x f k ( x ) = Z I d x lim k →∞ f k ( x ) = Z I d x f ∞ ( x ) . (B1)It is noteworthy that, in our case, we do not need to invoke Lebesgue integration and thedominated convergence theorem, and our calculations are based on Riemannian integration.Even though Riemann integration theory lacks theorems regulating the interchange betweenlimit and integration operations, the following theorem is sufficient for our purposes: Theorem 2 . Let f n be a sequence of functions defined on [ a, ∞ ) and Riemann integrableon [ a, b ] for all b > a . Assume that (i) f n ( x ) → f ∞ ( x ) almost everywhere in [ a, ∞ ), f ∞ beingRiemann integrable on every finite interval. (ii) There exists a positive function g definedon [ a, ∞ ) such that R ∞ a g is convergent and | f n ( x ) | ≤ g ( x ) for all n . Then R ∞ a f n → R ∞ a f ∞ .Note that, in this theorem, the integrability of the limiting function is part of the hypothesis,whereas in the dominated convergence theorem it is a consequence of the theorem itself. A. Consortini and B. R. Frieden, “Quantum-mechanical solutions for the simple harmonic os-cillator in a box,” Il Nuovo Cimento (B2), 153–164 (1976). W. N. Mei and Y. C. Lee, “Harmonic oscillator with potential barriers,” J. Phys. A: Math. Gen. , 1623–1632 (1983). J. L. Marin and S. A. Cruz, “On the harmonic oscillator inside an infinite potential well,” Am.J. Phys. , 1134 (1988). G. Barton, A. J. Bray, and A. J. McKane, “The influence of distant boundaries on quantummechanical energy levels,” Am. J. Phys. , 751–755 (1990). V. G. Gueorguiev, A. R. P. Rau, and J. P. Draayer, “Confined one-dimensional harmonicoscillator as a two-mode system,” Am. J. Phys. (5), 394–403 (2006). H. E. Montgomery, Jr., N. A. Aquino, and K. D. Sen, “Degeneracy of confined D-dimensionalharmonic oscillator,” Int. J. Quantum Chemistry , 798–806 (2007). R. W. Robinett, “Visualizing the collapse and revival of wave packets in the infinite square wellusing expectation values,” Am. J. Phys. (5), 410–420 (2000). R. W. Robinett, “Wave packet revivals and quasirevivals in one-dimensional power law poten-tials,” J. Math. Phys. , 1801–1813 (2000). J. D. Chalk, “Tunneling through a truncated harmonic oscillator potential barrier,” Am. J.Phys. (2), 147–151 (1990). P. Caldirola, R. Cirelli, and G. M. Prosperi,
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