Analytic bounds on transmission probabilities

Annals of Physics 325 (2010) 1328–1339

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Annals of Physics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a o p

Analytic bounds on transmission probabilities Petarpa Boonserm a,⁎, Matt Visser b a b

Department of Mathematics, Faculty of Science Chulalongkorn University, Phayathai Rd., Pathumwan Bangkok 10330, Thailand School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, New Zealand

a r t i c l e

i n f o

Article history: Received 17 January 2009 Accepted 19 February 2010 Available online 13 April 2010 Keywords: Transmission Reflection Bogoliubov coefficients Analytic bounds

a b s t r a c t We develop some new analytic bounds on transmission probabilities (and the related reflection probabilities and Bogoliubov coefficients) for generic one-dimensional scattering problems. To do so we rewrite the Schrödinger equation for some complicated potential whose properties we are trying to investigate in terms of some simpler potential whose properties are assumed known, plus a (possibly large) “shift” in the potential. Doing so permits us to extract considerable useful information without having to exactly solve the full scattering problem. © 2010 Elsevier Inc. All rights reserved.

1. Introduction In several earlier papers [1–4], the present authors have derived a number of rigourous bounds on transmission probabilities (and reflection probabilities, and Bogoliubov coefficients) for one-dimensional scattering problems. The derivation of these bounds generally proceeds by rewriting the Schrödinger equation in terms of some equivalent system of first-order equations, and then analytically bounding the growth of certain quantities related to the net flux of particles as one sweeps across the potential. In the present article we shall obtain significantly different results, of both theoretical and practical interest. While a vast amount of effort has gone into studying the Schrödinger equation and its scattering properties [5–21], it appears that relatively little work has gone into providing general analytic bounds on the transmission probabilities, (as opposed to approximate estimates). The only known results as far as we have been able to determine are presented in [1–4]. Several quite remarkable bounds were first derived in [1], with further discussion and an alternate proof being provided in [2]. These bounds were originally used by one of the present authors as a technical step when studying a specific model for sonoluminescence [22], and since then have also been used to place limits on particle production in analogue spacetimes [23] and resonant cavities [24], to investigate qubit master equations [25], and to motivate further general investigations of one-dimensional scattering theory [26–28]. Recently, these bounds have also been

⁎ Corresponding author. E-mail addresses: [email protected] (P. Boonserm), [email protected] (M. Visser). 0003-4916/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.02.005

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applied to the greybody factors of a Schwarzschild black hole [3]. Most recently, significant extensions of the original bounds have been developed [4] by adapting the Miller–Good transformations [29]. In the current article we again return to this problem, developing a new set of techniques that are more amenable to the development of both upper and lower bounds on the transmission probabilities. For technical reasons the new techniques are also more amenable to investigating behavior “under the barrier”. The basic idea is to recast the Schrödinger equation for some complicated potential whose properties we are trying to investigate in terms of some simpler potential whose properties are assumed known, plus a (possibly large) “shift” in the potential. 2. From Schrödinger equation to system of ODEs We are interested in the scattering properties of the Schrödinger equation, 2

ψ″ðxÞ + kðxÞ ψðxÞ = 0;

ð2:1Þ

where k(x)2 = 2m[E − V(x)]/ħ2. As long as V(x) tends to finite (possibly distinct) constants V±∞ on left and right infinity, then for E N max{V+∞, V−∞} one can set up a one-dimensional scattering problem in a completely standard manner — see, for example, standard references such as [5–13]. The scattering problem is completely characterized by the transmission and reflection amplitudes (denoted t and r), although the most important aspects of the physics can be extracted from the transmission and reflection probabilities (T = |t|2 and R = |r|2). 2.1. Ansatz The idea is to try to say things about exact solutions to the ODE 2

ψ″ðxÞ + k ðxÞψðxÞ = 0;

ð2:2Þ

by comparing this ODE to some “simpler” one 2

ψ0″ðxÞ + k0 ðxÞψ0 ðxÞ = 0;

ð2:3Þ

for which we are assumed to know the exact solutions ψ0(x). In a manner similar to the analysis in references [1,2], will start by introducing the ansatz ψðxÞ = aðxÞψ0 ðxÞ + bðxÞψ0*ðxÞ:

ð2:4Þ

This representation is of course extremely highly redundant, since one complex number ψ(x) has been traded for two complex numbers a(x) and b(x). This redundancy allows us, without any loss of generality, to enforce one auxiliary constraint connecting a(x) and b(x). We find it particularly useful to enforce the auxiliary condition da db ψ + ψ * = 0: dx 0 dx 0

ð2:5Þ

Subject to this auxiliary constraint on the derivatives of a(x) and b(x), the derivative of ψ(x) takes on the especially simple form dψ = aψ′0 + bψ0*′: dx

ð2:6Þ

(This ansatz is largely inspired by the techniques of references [1,2], where JWKB estimates for the wavefunction were similarly used as a “basis” for formally writing down the exact solutions.)

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2.2. Probability density and probability current For the probability density we have: ρ = ψ*ψ

ð2:7Þ

= jaðxÞψ0 + bðxÞψ0 j

2

n o n o 2 2 2 2 = jaj + jbj jψ0 j + 2Re ab*ψ0 n o n o 2 2 2 = jaj + jbj ρ0 + 2Re ab*ψ0 :

ð2:8Þ ð2:9Þ ð2:10Þ

Furthermore, for the probability current: 

J = Im ψ*

 dψ dx

ð2:11Þ

i nh o = Im a*ψ0 + b*ψ0 ½aψ′0 + bψ0*′

ð2:12Þ

n o 2 2 = Im jaj ψ0*ψ′0 + jbj ψ0 ψ0*′ + ab*ψ0 ψ′0 + a*bψ0*ψ0*′

ð2:13Þ

n o 2 2 = jaj −jbj Imfψ0*ψ′0 g

ð2:14Þ

n o 2 2 = jaj −jbj J0 :

ð2:15Þ

Under the conditions we are interested in, (corresponding to a time-independent solution of the Schrödinger equation), we have ρ̇ = 0, and so ∂x J= 0. (And similarly ρ̇ = 0, so ∂x J0 = 0.) That is, Jand J0 are position-independent constants, which then put a constraint on the amplitudes |a| and |b|. Applying an appropriate boundary condition, which we can take to be a(−∞) = 1, b(−∞) = 0, we then see 2

2

jaj −jbj = 1:

ð2:16Þ

This observation justifies interpreting a(x) and b(x) as “position-dependent Bogoliubov coefficients”. Furthermore without any loss in generality we can choose the normalizations on ψ and ψ0 so as to set the net fluxes to unity: J= J0 = 1. 2.3. Second derivatives of the wavefunction We shall now rewrite the Schrödinger equation in terms of two coupled first-order differential equations for these position-dependent Bogoliubov coefficients a(x) and b(x). To do this, evaluate d2ψ/dx2 making repeated use of the auxiliary condition d2 ψ d = ðaψ′0 + bψ0*′Þ dx dx2 = a′ψ′0 + b′ψ0*′ + aψ″0 + bψ0*″ = a′ψ′0 −a′ =

ψ0 2 2 ψ *′ = ak0 ψ0 −bk0 ψ0* ψ0* 0

a′ 2 fψ *ψ′ −ψ0 ψ0*′g−k0 ½aψ0 + bψ0* ψ0* 0 0

ð2:17Þ ð2:18Þ ð2:19Þ ð2:20Þ

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=

2i J0 a′ 2 −k0 ½aψ0 + bψ0* ψ0*

ð2:21Þ

=

2ia′ 2 −k0 ½aψ0 + bψ0*: ψ0*

ð2:22Þ

Where in the last line we have finally used our normalization choice J0 = 1. This is one of the two relations we wish to establish. Now use the gauge condition to eliminate da/dx in favour of db/dx to obtain a second relation for d2ψ/dx2. This now permits us to write d2ψ/dx2 in either of the two equivalent forms d2 ψ 2ia′ 2 −k0 ½aψ0 + bψ0*; = ψ0* dx2 =−

ð2:23Þ

2ib′ 2 −k0 ½aψ0 + bψ0*: ψ0

ð2:24Þ

2.4. SDE as a first-order system Now insert these formulae for the second derivative of the wavefunction into the Schrödinger equation written in the form 2

d ψ 2 + kðxÞ ψ = 0; dx2

ð2:25Þ

to deduce the pair of first-order ODEs: in o da ih 2 2 2 2 = + k −k0 ajψ0 j + bψ0* ; dx 2

ð2:26Þ

in o db ih 2 2 2 2 = − k −k0 aψ0 + bjψ0 j : dx 2

ð2:27Þ

It is easy to verify that this first-order system is compatible with the auxiliary condition (2.5), and that by iterating the system twice (subject to this auxiliary condition) one recovers exactly the original Schrödinger equation. We can rewrite this 1st-order system of ODEs in matrix form as h i #  2 2 "   2 i k −k0 jψ0 j2 ψ0* d a a = : ð2:28Þ dx b 2 −ψ2 −jψ j2 b 0

0

(Matrix ODEs of this general form are often referred to as Shabhat–Zakharov or Zakharov–Shabat systems [1]. This matrix ODE can be used to write down a formal solution to the SDE in terms of “path-ordered exponentials” as in references [1,2]. We choose not to adopt this route here, instead opting for a more direct computation in terms of the magnitudes and phases of a and b.) 2.5. Formal (partial) solution Define magnitudes and phases by iϕa

a = jaje

;

iϕb

b = jbje

iϕ0

ψ0 = jψ0 je

;

:

ð2:29Þ

Calculate a′ = jaj′e

iϕa

iϕa

+ ijaje

iϕa

ϕa′ = e

fjaj′ + ijajϕ′a g;

ð2:30Þ

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whence jaj′ + ijajϕ′a =

i n ih 2 2 2 −iðϕa −ϕb k −k0 jψ0 j jaj + jbje 2

+ 2ϕ0 Þ

o :

ð2:31Þ

Similarly we also have jbj′ + ijbjϕ′b = −

i n o ih 2 2 2 −iðϕb −ϕa −2ϕ0 Þ k −k0 jψ0 j jbj + jaje : 2

ð2:32Þ

Now take the real part of both these equations, whence jaj′ = +

i 1h 2 2 2 k −k0 jbjjψ0 j sinðϕa −ϕb + 2ϕ0 Þ; 2

ð2:33Þ

jbj′ = +

i 1h 2 2 2 k −k0 jajjψ0 j sinðϕa −ϕb + 2ϕ0 Þ: 2

ð2:34Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 1h 2 2 2 k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þ jaj2 −1: 2

ð2:35Þ

Therefore jaj′ = That is i jaj′ 1h 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þ; 2 2 jaj −1

ð2:36Þ

whence i n ox 1 x h 2 2 −1 2 2 = ∫x 2 k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx: cosh a x1 2 1

ð2:37Þ

Now apply the boundary conditions: At x = −∞ we have both a(−∞) = 1, and b(−∞) = 0. Therefore i 1 x h 2 2 2 ∫−∞ k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx; 2

ð2:38Þ

  i 1 x h 2 2 2 jaðxÞj = cosh ∫ k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx : 2 −∞

ð2:39Þ

−1

cosh

jaðxÞj =

and so

In particular i 1 + ∞h 2 2 2 ∫ k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx; 2 −∞

ð2:40Þ

  i 1 + ∞h 2 2 2 jað∞Þj = cosh ∫−∞ k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx : 2

ð2:41Þ

−1

cosh

jað∞Þj =

or equivalently

Of course this is only a formal solution since ϕa(x) and ϕb(x) are, (at least at this stage), “unknown”. But we shall argue that this formula still contains useful information. In particular, in view of the normalization conditions relating a and b, and the parity properties of cosh and sinh, we can also write

j 12 ∫

jað∞Þj = cosh

+∞ −∞

h

i 2 2 2 k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx ;

j

ð2:42Þ

P. Boonserm, M. Visser / Annals of Physics 325 (2010) 1328–1339

j 12 ∫

jbð∞Þj = sinh

+∞ −∞

h

i 2 2 2 k −k0 jψ0 j sinðϕa −ϕb + 2ϕ0 Þdx :

j

1333

ð2:43Þ

2.6. First set of bounds To determine the first elementary set of bounds on a and b is now trivial. We just note that jsinðϕa −ϕb + 2ϕ0 Þj≤1:

ð2:44Þ

Therefore   1 +∞ 2 2 2 jað∞Þj ≤ cosh ∫−∞ jk −k0 jjψ0 j dx ; 2

ð2:45Þ

  1 +∞ 2 2 2 jbð∞Þj ≤ sinh ∫−∞ jk −k0 jjψ0 j dx : 2

ð2:46Þ

What does this now tell us about the Bogoliubov coefficients? 2.7. Bogoliubov coefficients The slightly unusual thing, (compared to our earlier work in references [1,2,4]), is that now the “known” function ψ0 may also have its own Bogoliubov coefficients. Let us assume we have set our boundary conditions so that for the “known” situation ψ0 ðx ≈ −∞Þ ∼ expfikð−∞Þxg;

ð2:47Þ

ψ0 ðx ≈ +∞Þ ∼ α0 expfikð +∞Þxg + β0 expf−ikð +∞Þxg:

ð2:48Þ

and

Then the way we have set things up, for the “full” problem we still have ψðx ≈ −∞Þ ∼ expfikð−∞Þxg;

ð2:49Þ

whereas ψðx ≈ +∞Þ ∼ að∞Þψ0 ðxÞ + bð∞Þψ0*ðxÞ

ð2:50Þ

∼½α0 að∞Þ + β0*bð∞Þexpfikð +∞Þxg + ½β0 að∞Þ + α0*bð∞Þexpf−ikð +∞Þxg:

ð2:51Þ

That is, the overall Bogoliubov coefficients satisfy α = α0 að∞Þ + β0*bð∞Þ;

ð2:52Þ

β = β0 að∞Þ + α0*bð∞Þ:

ð2:53Þ

These equations relate the Bogoliubov coefficients of the “full” problem {ψ(x), k(x)} to those of the simpler “known” problem {ψ0(x), k0(x)}, plus the evolution of the a(x) and b(x) coefficients. Now observe that jαj ≤ jα0 jjað∞Þj + jβ0 jjbð∞Þj:

ð2:54Þ

But we can define jα0 j = coshΘ0 ;

jβ0 jsinhΘ0 ;

jað∞Þj = coshΘ;

jbð∞Þj = sinhΘ;

ð2:55Þ

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in terms of which jαj ≤ coshΘ0 coshΘ + sinhΘ0 sinhΘ = coshðΘ0 + ΘÞ:

ð2:56Þ

That is: since we know Θ ≤ Θbound ≡

1 +∞ 2 2 2 ∫ jk −k0 jjψ0 j dx; 2 −∞

ð2:57Þ

we can deduce   1 +∞ 2 −1 2 2 jαj ≤ cosh cosh jα0 j + ∫− ∞ jk −k0 jjψ0 j dx ; 2

ð2:58Þ

  1 +∞ 2 −1 2 2 jβj ≤ sinh sinh jβ0 j + ∫ jk −k0 jjψ0 j dx : 2 −∞

ð2:59Þ

2.8. Second set of bounds A considerably trickier inequality, now leading to a lower bound on the Bogoliubov coefficients, is obtained by considering what the phases would have to be to achieve as much destructive interference as possible. That implies jαj ≥ jα0 jjað∞Þj−jβ0 jjbð∞Þj;

ð2:60Þ

whence jαj ≥ coshjΘ0 −Θj:

ð2:61Þ

Therefore, using Θ ≤ Θbound, it follows that as long as Θbound b Θ0, one can deduce jαj ≥ coshfΘ0 −Θbound g:

ð2:62Þ

(If on the other hand Θbound ≥ Θ0, then one only obtains the trivial bound |α| ≥ 1.) Another way of writing these bounds is as follows   1 +∞ 2 −1 2 2 jαj ≥ cosh cosh jα0 j− ∫−∞ jk −k0 jjψ0 j dx ; 2

ð2:63Þ

  1 +∞ 2 −1 2 2 ð2:64Þ jβj ≥ sinh sinh jβ0 j− ∫ jk −k0 jjψ0 j dx ; 2 −∞ with the tacit understanding that the bound remains valid only so long as argument of the hyperbolic function is positive. 2.9. Transmission probabilities As usual, the transmission probability (barrier penetration probability) is related to the Bogoliubov coefficient by T=

1 ; jαj2

ð2:65Þ

whence 2

T ≥ sech

  1 +∞ 2 −1 2 2 cosh jα0 j + ∫−∞ jk −k0 jjψ0 j dx : 2

ð2:66Þ

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That is 2

 

1 +∞ 2 −1 −1 = 2 2 2 cosh T0 + ∫−∞ jk −k0 jjψ0 j dx ; 2

ð2:67Þ

2

 

1 +∞ 2 −1 1=2 2 2 sech T0 + ∫−∞ jk −k0 jjψ0 j dx : 2

ð2:68Þ

T ≥ sech or even T ≥ sech

Furthermore, as long as the argument of the sech is positive, we also have the upper bound 2

T ≤ sech

 

1 −1 1=2 +∞ 2 2 2 − ∫−∞ jk −k0 jjψ0 j dx : sech T0 2

ð2:69Þ

If one wishes to make the algebraic dependence on T0 clearer, by expanding the hyperbolic functions these formulae may be recast as T0 T≥ h n o n oi2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffi +∞ 2 +∞ 2 cosh 12 ∫−∞ jk −k20 jjψ0 j2 dx + 1−T0 sinh 12 ∫−∞ jk −k20 jjψ0 j2 dx

ð2:70Þ

and (as long as the numerator is positive before squaring) T0 T≤ h n o pffiffiffiffiffiffiffiffiffiffiffiffi n oi2 : ffi +∞ 2 +∞ 2 cosh 12 ∫−∞ jk −k20 jjψ0 j2 dx − 1−T0 sinh 12 ∫−∞ jk −k20 jjψ0 j2 dx

ð2:71Þ

3. Consistency check There is one special case in which we can easily compare with the previous results of references [1,2]. Take k0 = k(±∞) to be independent of position, so that our comparison problem is a free particle. In that case ψ0 =

expðik0 xÞ pffiffiffiffiffi ; k0

2

jψ0 j =

1 ; k0

J0 = 1;

α0 = 1;

β0 = 0:

ð3:1Þ

Then the bounds derived above simplify to   1 +∞ 2 2 ∫−∞ jk −k0 jdx ; jαj ≤ cosh 2k0

ð3:2Þ

  1 +∞ 2 2 jβj ≤ sinh ∫−∞ jk −k0 jdx : 2k0

ð3:3Þ

This is “Case I” of reference [1] and the “elementary bound” of reference [2], which demonstrates consistency whenever the formalisms overlap. (Note that it is not possible to obtain “Case II” of reference [1] or the “general bound” of reference [1,2] from the present analysis — this is not a problem, it is just an indication that this new bound really is a different bound that only partially overlaps with the previous results of references [1,2,4]. A second (elementary) check is to see what happens if we set ψ(x) → ψ0(x), effectively assuming that the full problem is analytically solvable. In that case T → T0, (and similarly both α → α0 and β → β0), as indeed they should.

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4. Keeping the phases? We can extract a little more information by taking the imaginary parts of Eqs. (2.31) and (2.32) to obtain: ϕ′a =

  i 1h 2 jbj 2 2 k −k0 jψ0 j 1 + cosðϕa −ϕb + 2ϕ0 Þ ; 2 ja j

ð4:1Þ

  i 1h 2 jaj 2 2 k −k0 jψ0 j 1 + cosðϕb −ϕa −2ϕ0 Þ : 2 jbj

ð4:2Þ

ϕ′b = − Subtracting

( ) h i jaj2 + jbj2 2 2 2 cosðϕa −ϕb + 2ϕ0 Þ : ðϕa −ϕb Þ′ = k −k0 jψ0 j 1 + 2jajjbj

ð4:3Þ

This is now a differential equation that only depends on the difference in the phases — the overall average phase (ϕa + ϕb)/2 has completely decoupled. (Likewise, in determining the transmission and reflection probabilities, this average phase also neatly decouples). To see how far we can push this observation, let us now define a “nett” phase Δ = ϕa −ϕb + 2ϕ0 :

ð4:4Þ

Furthermore, as per the previous subsections, we retain the definitions jaj = coshΘ;

jbj = sinhΘ:

ð4:5Þ

Then Eq. (2.39) becomes   i 1 x h 2 2 2 ∫ k −k0 jψ0 j sinðΔðxÞÞdx : ΘðxÞ = 2 −∞

ð4:6Þ

while the “nett” phase satisfies

ΔðxÞ′ =

nh

k

2

2 −k0

h i i o k2 −k20 ψ20 2 jψ0 j + 2ϕ0′ + cos½ΔðxÞ: tanh½2ΘðxÞ

ð4:7Þ

We can even substitute for Θ(x) and thus rewrite this as a single integro-differential equation for Δ(x):

ΔðxÞ′ =

nh

k

2

2 −k0

i o 2 jψ0 j + 2ϕ0′ +

h

tanh ∫x

−∞

i k2 −k20 jψ0 j2

cos½ΔðxÞ:  2 k −k20 jψ0 j2 sin½ΔðxÞdx

ð4:8Þ

This equation is completely equivalent to the original Schrödinger equation we started from. Unfortunately further manipulations seem intractable, and it does not appear practicable to push these observations any further. 5. Application: small shift in the potential Let us now consider the situation V ðxÞ = V0 ðxÞ + δV ðxÞ; for  “sufficiently small”.

ð5:1Þ

P. Boonserm, M. Visser / Annals of Physics 325 (2010) 1328–1339

1337

5.1. First-order changes To be consistent with previous notation let us define 2

2

k = k0 + 

  2mδV 2 ≡k0 + δv: 2 h

Using Eq. (2.39) we obtain the preliminary estimates

2 jaðxÞj = 1 + O  ;

ð5:2Þ

ð5:3Þ

and similarly jbðxÞj = OðÞ:

ð5:4Þ

It is now useful to change variables by introducing some explicit phases so as to define   i i h 2 2 2 a = aexp ˜ + ∫ k −k0 jψ0 jdx ; 2

ð5:5Þ

  i i h 2 2 2 ˜ b = bexp − ∫ k −k0 jψ0 jdx : 2

ð5:6Þ

Doing so modifies the system of differential Eqs. (2.26) and (2.27) so that it becomes

i h i d a˜ ih 2 2 ˜ 2 2 2 2 = + k −k0 bψ 0* exp −i∫ k −k0 ψ0 dx ; dx 2

ð5:7Þ

i h i

d b˜ ih 2 2 2 2 2 2 = − k −k0 aψ ˜ 0 exp + i∫ k −k0 jψ0 jdx : dx 2

ð5:8Þ

The advantage of doing this is that in the current situation we can now estimate

d a˜ 2 =O  ; dx

ð5:9Þ





d b˜ i 2 2 3 = − δvðxÞψ0 ðxÞexp + i∫δvjψ0 jdx + O  : dx 2

ð5:10Þ

Integrating



i + ∞ 2 2 3 b˜ ð∞Þ = − ∫−∞ δvðxÞψ0 ðxÞexp + i∫δvjψ0 jdx dx + O  : 2

ð5:11Þ

This is not the standard Born approximation, though it can be viewed as an instance of the so-called “distorted Born wave approximation”. In terms of the absolute values we definitely have jb˜ ð∞Þj = jbð∞Þj ≤



 +∞ 2 3 ∫−∞ jδvðxÞjjψ0 ðxÞjdx + O  : 2

ð5:12Þ

5.2. Particle production When it comes to considering particle production we note that

2 β = β0 að∞Þ + α0*bð∞Þ = β0 + α0*bð∞Þ + O  ;

ð5:13Þ

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so the change in the number of particles produced is

2 2 δjβ j = Ref2α0*β0 bð∞Þg + O  :

ð5:14Þ

In particular jδNj ≤ jα0 jjβ0 j∫

+∞

−∞



2 2 jδvðxÞjjψ0 ðxÞjdx + O  ;

ð5:15Þ

which we can also write as

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ∞ 2 2 jδNj ≤  N0 ðN0 + 1Þ∫−∞ jδvðxÞjjψ0 ðxÞjdx + O  :

ð5:16Þ

Note that one will only get an order  change in the particle production if the “known” problem {ψ0, k0} already results in nonzero particle production. 5.3. Transmission probability To see how a small shift in the potential affects the transmission probability we note T=

1 1 1 = =

 : jαj2 jα0 að∞Þ + β0* bð∞Þj2 jα0 + β0* bð∞Þ + O 2 j2

ð5:17Þ

But then T=

1 n o

 ; jα0 j2 j1 + β*0 bð∞Þ =α0 + O 2 j2

ð5:18Þ

implying     β *bð∞Þ 2 : +O  T = T0 1−2Re 0 α0

ð5:19Þ

So the change in the transmission probability is     β *bð∞Þ 2 : +O  δT = −T0 2Re 0 α0

ð5:20Þ

Taking absolute values one obtains δT ≤ T0



pffiffiffiffiffiffiffiffiffiffiffiffiffi + ∞ 2 2 1−T0 ∫ jδvðxÞjjψ0 ðxÞjdx + O  : −∞

ð5:21Þ

Note that one will only get an order  change in the transmission probability if the “known” problem {ψ0, k0} already results in nonzero transmission (and nonzero reflection). 6. Discussion What are the advantages of the particular bounds derived in this article? • They are very simple to derive — the algebra is a lot less complicated than some of the other approaches that have been developed [1–4]. (And a lot less complicated than some of the blind alleys we have explored.) • Under suitable circumstances the procedure of this article yields both upper and lower bounds. Obtaining both upper and lower bounds is in general very difficult to do — see in particular the attempts in [2].

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• All of the other bounds we have developed [1–4] needed some condition on the phase of the wavefunction, (some condition similar to φ′ ≠ 0), which had the ultimate effect of making it difficult to make statements about tunnelling “under the barrier”. There is no such requirement in the present analysis. (The closest analogue is that we need J0 ≠ 0, which we normalize without loss of generality to J0 = 1.) In particular this means that there should be no particular difficulty in applying the bound in the classically forbidden region — the “art” will lie in finding a suitable form for ψ0 which is simple enough to carry out exact computations while still providing useful information. In closing, we reiterate the fact that generic one-dimensional scattering problems, which have been extensively studied for close to a century, nevertheless still lead to interesting features and novel results. Acknowledgments This research was supported by the Marsden Fund administered by the Royal Society of New Zealand. PB was additionally supported by a scholarship from the Royal Government of Thailand. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

[23]

[24] [25] [26] [27] [28] [29]

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