analytical solution for the de broglie wave packet description of the neutron in subcritical...

7 January 2002

Physics Letters A 292 (2002) 222–232

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Analytical solution for the de Broglie wave packet description ofthe neutron in subcritical transmission through a mirror

M. Utsuro1

Research Reactor Institute, Kyoto University, Osaka 590-0494, Japan

Received 21 March 2001; received in revised form 22 October 2001; accepted 30 November 2001

Communicated by P.R. Holland

Abstract

Theoretical solution for the de Broglie wave packet in subcritical condition for total reflection was obtained. The solutionfor the wave function shows characteristic features different from the results of the conventional plane wave analysis. Thepossibility to explain the reported anomalous reflection loss of ultracold neutrons in the storage experiment is indicated, and anestimation for the wave packet parameter from the magnitude of such a loss probability of 3× 10−5 for the beryllium bottleleads to the packet width of about 2 µm for the present solution of the de Broglie wave packet for the neutron. 2002 ElsevierScience B.V. All rights reserved.

PACS:28.20; 03.65; 03.75.B; 78.20.C

Keywords:De Broglie wave packet; Total reflection; Mirror potential; Subcritical condition; Wave mechanics; Ultracold neutrons

1. Introduction

Ultracold neutrons (UCN) with the velocity belowabout 6 m/s can be totally reflected even in the nor-mal incidence to the mirror surface of various materi-als such as beryllium, nickel, copper and others, andtherefore long time storage of UCN in a closed vesselmade of material mirrors was studied for more thanthirty years. However, recent experiments [1] indi-cated anomalously large loss probability per reflectionin the order 10−6 being significantly discrepant fromthe expected magnitude based on the conventional

E-mail address:[email protected] (M. Utsuro).1 Present address: Research Center for Nuclear Physics, Osaka

University, Ibaraki, Osaka 567-0047 Japan.

plane wave analysis on the total reflection. Further re-cent study [2] on UCN with the velocity below the lim-iting velocity for total reflectionvl of about 3 m/s forthe aluminium window revealed another phenomenaof anomalous transmission through the window withthe probability of the order 10−6, in contradiction tothe simple expectation from the plane wave theory pre-dicting essentially no transmission.

It is natural expectation that free neutrons are de-scribed by a wave packet with finite dimensions, ratherthan by an infinite plane wave. A typical approach forthe former expression is a coherent superposition ofcommon plane waves, like the Gaussian one [3], butthe distribution parameters selected to be a ‘minimumuncertainty state’ in the initial undergoes a spreadingin space as the time elapsing and no longer describesa minimum uncertainty state except in the initial.

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M. Utsuro / Physics Letters A 292 (2002) 222–232 223

Another wave packet was suggested for the neutron[4], which is described by the non-spreading, deBroglie singular and symmetrical wave packet [5]

φ0(s0; v0; r; t)(1)= C0 exp(iv0r − iω0t)

exp(−s0|r − v0t|)|r − v0t| ,

whereC0 = √s0/2π is the normalization constant,

(2)∫d3r

∣∣φ0(s0,v0, r, t)∣∣2 = 1,

s0 is a parameter, which determines the width of thepacket in the momentum space and also the inversewidth in the coordinate space,v0 is the neutronvelocity, andω0 = (v2

0 − s20)/2 by choosing the unities

with h̄ = m = 1, then becomingm/h̄ = h̄2/m = 1,thusv0 expressing also the central wave number,v2

0/2the kinetic energy. We see thatω0 is less than thekinetic energy by amounts2

0/2, which is required toresist the spreading.

The Fourier expansion of theφ0-function, Eq. (1),has the form

(3)φ0(s0,v0, r, t)=∫Φ0(p, t)e

ipr d3p,

where

(4)Φ0(p, t)=Φ0(p)exp(it

[v2

0 + s20 − 2pv

]/2)

and

(5)Φ0(p)= 4πC0/[(p − v0)

2 + s20

].

The wave packet (1) satisfies the inhomogeneousequation[i∂

∂t+ ∆

2

]φ0(s0,v0, r, t)

(6)= −2πC0eit (v2

0+s20)/2δ(r − v0t),

i.e., it satisfies the canonical Schrödinger equationeverywhere, except one point. This point is the sourceof the wave field (1), and together with this field thesingular point can be considered to be the neutron.

Another important point of wave (1) is that thereflected neutron moving in free space should satisfyagain the Eq. (6), which has the unique solution (1) forspecified conditions of symmetry and normalizability.

The usual expectation on the plane waves with thespectrum (5) should suggest that, due to the long tail

extending far away from the central wave numberv0with the component of the wave numbersp2 � 2u=v2l , whereu is the optical potential of neutron–wall

interaction, packet (3) might undergo subcritical trans-mission at every collision even for a subbarrier neutronwith the velocity below the limiting velocityvl .

Further, according to the usual approach decom-posing the neutron velocity vector into parallel andnormal components to a flat mirror surface, higherenergy neutrons in the total reflection region, i.e.,the neutrons with the velocity componentv⊥ < vl ,where⊥ denotes the component ofv normal to thewall surface, should show similar possibility of sub-critical transmission. The first experiment [6] with ahigh quality silicon single crystal mirror for checkingsuch possibility gave a somewhat positive result, but itseemed necessary to repeat the experiment with higherneutron flux, better spectroscopy, and with control ofscattering on surface roughness.

Based on these considerations, it will be a meaning-ful work to extend the description with the de Brogliewave packet for the neutron to such cases of the sub-critical transmission under the total reflection condi-tions. In the present Letter, a theoretical analysis tosolve the de Broglie waves inside the mirror poten-tial under the condition of total reflection is carried outwith extending the framework of wave (1), so that theneutrons inside the mirror potential should also sat-isfy an inhomogeneous wave equation correspondingto Eq. (6). The solution obtained here gives a new stateof the subcritical neutrons inside the potential not yetpresented in the wave mechanics. Further, the appli-cation of the present analytical result to the reportedmagnitude for the anomalous loss probability for UCNderives a reasonable value for the wave packet size forthe neutron.

2. Analytical solution for semi-infinite mirrorpotential

In the present section, a theoretical analysis and thesolution for the de Broglie waves inside and outsidethe mirror potential in semi-infinite geometry underthe condition of total reflection of neutrons will begiven with introducing a potential term into the waveequation (6).

224 M. Utsuro / Physics Letters A 292 (2002) 222–232

Fig. 1. Notations for wave functions in the geometry of semi-infinitemirror potentialu.

2.1. The wave equation for semi-infinite geometry

We define the wave equation for the whole spaceoutside and inside of the mirror potentialu in a semi-infinite geometry as follows under Cartesian coordi-nates withz-axis perpendicularly inward to the surfacewith the origin taking on the surface:[i∂

∂t+ ∆

2− uϑ(z)

]ψ(s,v, r, t)

= −2π{C0e

it (v20+s2

0)/2δ(r − v0t)

+C1eit (v2

1+s21)/2δ(r − v1t)

}(7)× [

1− ϑ(z)],

whereϑ(x) is a step function equal to 1 or 0 whenthe value for its variable is positive or negative, re-spectively. In the right-hand side of the equation, thefirst term in the curly bracket is the source term for thegiven condition of the incident neutron, while the sec-ond term in the curly bracket is that for the reflectedcomponent with the velocity directed backward, asshown in Fig. 1, which should be solved by the analy-sis.

2.2. The solution for the wave functions outside andinside the mirror potential

For the present geometry we put, as defined inFig. 1,

ψ(s,v, r, t)=ψ0(s0,v0, r, t)+ψ1(s1,v1, r, t)

(8)+ψBT(sB,vB, r, t),

where

(8a)ψ0(s0,v0, r, t)= φ0(s0,v0, r, t)[1− ϑ(z)

],

wave function for the incident neutron,

(8b)ψ1(s1,v1, r, t)= φ1(s1,v1, r, t)[1− ϑ(z)

],

wave function for reflected neutron,

(8c)ψBT(sB,vB, r, t)= φBT(sB,vB, r, t)ϑ(z),

wave function inside the mirror potential, and thenotationvB in ψBT is rather a temporal expressionsince the existence of real velocity for the neutroninside the potential could not be proved in the presentcase of the subcritical condition.

Decomposing the velocityvi = vi‖ + vi⊥, wherevix ≡ vi‖ and viy = 0, for i = 0, 1 and BT, thenfor the reflected neutron we definev1⊥ = −v1z, andinside the potential the normal componentvB⊥ undersubcritical condition becomesv2

B⊥/2 = v20⊥/2 − u <

0, i.e.,vB⊥ = ivBz with realvBz.Substituting Eq. (8) into Eq. (7) and performing

four-dimensional Fourier expansion process for thevariablesr andt in theψ-functions give the equation[ω+ p2

2

]Ψ0(p,ω)+

[ω+ p2

2

]Ψ1(p,ω)

+[ω+ p2

2+ u

]ΨBT(p,ω)

(9)= −2πC0

i[ω− (ωs0 − pv0)]+ 2πC1

i[ω− (ωs1 − pv1)],

where we omitted to writesi andvi in theΨ -functionsfor the simplicity for the subscripti = 0, 1 and BT,while ωsi = (v2

i + s2i )/2 for the subscripti = 0 and 1.

By making use of Eqs. (4) and (5) with Eq. (8a)and of the Fourier expansion of the step function beinggiven asΘ(pz)= δ(pz)/2+ 1/(2πipz), we obtain

Ψi(p,ω)= ∓2πCi

(10)

×[

1

i[ω+ p2/2][ω− (ωsi − pvi )]

+ 1

iqi‖

{1

(pz ∓ viz − iqi‖)1

i(ω−ωBi )

− 1

(pz ∓ viz + iqi‖)1

i(ω−ω∗Bi )

}],

where the double signs correspond to the cases for thesubscripti = 0 and 1, respectively, and similarly

ωBi = ωsi − p‖vi‖ − (viz ± iqi‖)viz,ω∗

Bi = ωsi − p‖vi‖ − (viz ∓ iqi‖)viz,

with qi‖ =√s2i + (p‖ − vi‖)2/2 by p‖ denoting the

x–y component ofp, for i = 0 and 1, respectively.


In the substitution of Eq. (10) into Eq. (9), the firstterms in the right-hand sides of Eq. (10) fori = 0and 1 could be cancelled with the first and secondterms in the right-hand side of Eq. (9), respectively,and we obtain the solution for the inside wave functionconsisting of two components,

(11)ΨBT(p,ω)= ΨB(p,ω)+Ψ†B(p,ω),

where

ΨB(p,ω)= ω+ p2/2

ω+p2/2+ u

(12)

×[

2πC0

i(ω−ωB0)

1

iq0‖(pz − v0z − iq0‖)

− 2πC1

i(ω−ωB1)

1

iq1‖(pz + v1z − iq0‖)

],

andΨ †B(p,ω) is given by the equation replacedωB0

andωB1 in Eq. (12) toω∗B0 andω∗

B1, respectively.The inverse Fourier processes performed to Eqs.

(10) and (11) give the solutions of Eqs. (8a) and (8b)exactly and also the approximate solutions for Eq. (8c)in the sense of making use of the approximation√

2u+ (qi‖ ∓ iviz)2 ∼= vBz ∓ iviz

vBzqi‖

for i = 0 and 1, respectively, for sufficiently smallqi‖in the region dominantly contributing to the inverseFourier integration of Eq. (12).

After applying the continuity for the solutions atthe boundaryz= 0 for all the time which requires theequality s1 = s0 andv1z = v0z, i.e., v1⊥ = −v0z, thesolution for the reflected neutron becomes

ψ1(r, t)= C1eiωs0t eiv0x(x−v0xt)e−iv0z(z+v0zt)

×exp

[−s0√rxy(t)2 + (z+ v0zt)2]

√rxy(t)2 + (z+ v0zt)2

(13)(z� 0),

where r2xy(t) = (x − v0xt)

2 + y2, i.e., the projecteddistance squared on thexy-plane from the position ofthe neutron.

The similar, but a little more careful Fourier inte-gration process toΨB(p,ω) andΨ †

B(p,ω) derives theapproximate solution for Eq. (8c) as follows:

ψB(sB, r, t)

ψ†B(sB, r, t)

}=

{CB

C†B

}

× ei(ωs0−2u)t eiv0x(x−v0xt)e−vBz(z−ivBzt)

×exp

[−s0√rxy(t)2 − (v0z/vBz)2(z∓ ivBzt)2]

√rxy(t)2 − (v0z/vBz)2(z∓ ivBzt)2

(14)(z� 0),

(15)= 0 (z < 0),

where the double signs correspond to the upper andlower lines in the curly brackets, respectively. Thepresent solutions could be assured to satisfy the waveequation (7) by substituting them, then the applicableregion of solutions (14) being derived as

(16)1/s0 � rB(t)� 1/v0,

where

rB(t)=√rxy(t)2 − (v0z/vBz)2(z2 + v2

Bzt2).

The packet term in Eq. (14) written in complex for-mulas can be expressed with the phase and amplitudeterms, then becoming

ψB(sB,vB, r, t)

ψB†(sB,vB, r, t)

}=

{CB

C†B

}

× eiωBt eiv0x(x−v0xt)e−vBz(z−ivBz)t

× exp[±i{α(rxy, zB, t)

+ s0χ(rxy, zB, t)1/4 sinα(rxy , zB, t)

}](17)× exp[−s0χ(rxy, zB, t)

1/4 cosα(rxy, zB, t)]χ(rxy, zB, t)1/4

,

where

χ(rxy, zB, t)

(18)= {rxy(t)

2 − z2B + v2

0zt2}2 + 4z2

Bv20zt

2

and

(19)α(rxy, zB, t)= 1/2 arctanξ(rxy, zB, t),

with

(20)ξ(rxy, zB, t)= 2zBv0zt

rxy(t)2 − z2B + v2

0zt2,

by denoting

ωB = ωs0 − 2u, zB = zv0z/vBz.

Solution (17) for the wave function inside the po-tential is able to satisfy the continuities of the function


and also its derivative toz with the total of the incidentwaveψ0(s0,v0, r, t) and the reflected oneψ1(s1,v1,

r, t) for every time and at every point on the boundary,as studied in the next subsection.

We can study here shortly one of the importantcharacteristic features on solution (14) for the deBroglie wave packet inside the potential under sub-critical condition. That is the singularity of Eq. (14) at

t = 0, then the denominator reduces to√r(0)2 − z2

B,

i.e., the wave function becomes singular atx2 + y2 =z2

B = (zv0z/vBz)2, where is the region outside the cor-

rect applicability of these equations as restricted byEq. (16). Since the regions outside the applicability ofthe present solutions seem to indicate the central re-gion of the packet, the present singular region insidethe potential under the subcritical condition should besuggesting that the tunnelling wave packet distributestoward deviated direction from the normal direction tothe surface. The predicted region locates along a circu-lar cone with the vertical anglev0z/vBz dependent onthe velocityv0z of the incident neutron and the poten-tial u. This is one of the essentially different features ofthe present packet from the plane waves, in which nosuch localized region onx–y coordinates is predictedconcerning the position of the tunnelling particle in-side the potential but the tunnelling waves distributehomogeneously onx–y plane with the same attenua-tion coefficientvBz towards normal direction ofz fromthe surface.

2.3. Reflectivity and transmission probability

The continuity of the wave functions at the bound-ary z= 0, i.e.,

(21)ψ0(x, y,0, t)+ψ1(x, y,0, t)=ψBT(x, y,0, t),

for whole the time and the values forx andy requiresthe next equality to be satisfied:

(22)C0 +C1 = CB +C†B.

Further, the similar continuity of the derivative tozof these wave functions requires the next simultaneousequations to be satisfied:

(23)iv0z(C0 −C1)= −vBz(CB +C

†B

),

(24)iv0z(C0 −C1)=(v0z

vBz

)2

vBz(CB −C

†B

).

These simultaneous equations (22)–(24) give thesolutions for the coefficientsC1, CB, andC†

B as fol-lows:

(25)C1

C0= 1− iβ

1+ iβ,

(26)CB

C0= 1− β2

1+ iβ,

(27)C

†B

C0= 1− iβ,

whereβ = vBz/v0z.Eq. (25) is the same result with the plane wave

analysis for the semi-infinite geometry in the subcrit-ical condition, giving the reflectivity|C1/C0|2 = 1,i.e., total reflection in the present condition of the sub-critical incident angles.

On the other hand, concerning the property of thewave function inside the potential, we should studyfurther rewriting Eq. (17) by making use of Eqs. (26)and (27), as

ψBT(sB,vB, r, t)

=ψB(sB,vB, r, t)+ψ†B(sB,vB, r, t)

=DBTe−vBzzeiωBt eiv0x(x−v0xt)eiv

2Bzt

(28)×ΛB(s0, rxy, zB, t),

where

DBT = C0

(29)

×2[cosγ (s0, rxy, zB, t)− iβ2 sinγ (s0, rxy, zB, t)]1+ iβ

,

ΛB(s0, rxy, zB, t)

(30)= exp[−s0χ(rxy, zB, t)1/4 cosα(rxy, zB, t)]

χ(rxy, zB, t)1/4,

and

γ (s0, rxy, zB, t)= α(rxy, zB, t)

(31)+ s0χ(rxy, zB, t)1/4 sinα(rxy , zB, t).

Eq. (28) consists of theβ-dependent coefficientpart and the attenuation factore−vBzz = e−βv0zz, whichdefine the magnitude for the solutionψBT inside thepotential.


Fig. 2. Notations for wave functions in the geometry of the mirror potentialu with finite thickness.

3. Analytical solution for a mirror potential withfinite thickness

The results derived in the previous section indicatedthe possibility that the central region of the wavepacket for the tunnelling neutron under the subcriticalcondition locates along a definite direction deviatedfrom the normal direction to the surface. Therefore,it should be one of the interesting tasks to study on thecase of a mirror potential with finite thickness and tolook for where the tunnelling neutron should emergeon the backside surface of the potential and what isthe magnitude of the tunnelling probability.

3.1. The wave equation for the geometry with a finitethickness layer

In the present case of the mirror potential withfinite thicknessl, the inside wave functionψBT shouldsatisfy the condition to be zero forz > l, i.e., Eq. (8c)in the previous section should be replaced to

ψBT(sB,vB, r, t; l)=ψBT(sB,vB, r, t)

[1− ϑ(z− l)

](32)= φBT(sB,vB, r, t)ϑ(z)

[1− ϑ(z− l)

],

then for the continuity of the total wave function atthe backside boundary surface, the tunnelling wavefunctionψ ′

0T should exist outside the potential in thebackside space. As similar as Eq. (8a), the tunnellingwave functionψ ′

0T should be expressed as

(32)ψ ′0T(s

′0,v

′0, r, t)= φ′

0T(s′0,v

′0, r, t)ϑ(z− l),

which should further induce an additional boundaryeffectψ ′

B0 inside the potential. In this way, successiveboundary effect terms are induced by the front andbackside surfaces accumulating as shown in Fig. 2.

According to these considerations on successiveboundary effects resulting successive tunnelling and

reflected components, the source terms in the right-hand side of the wave equation should consist ofsuccessive reflected sources on the front surface andalso of transmitted sources from the backside surface,and then the wave equation becomes[i∂

∂t+ ∆

2− uϑ(z; l)

]ψ(s,v, r, t)

= −2π

[{C0e

it (v20+s2

0)/2δ(r − v0t)

+n∑i=1

Cieit (v2

i +s2i )/2δ(r − vi t)

}[1− ϑ(z)

]

+{C′

0eit (v′2

0 +s ′20 )/2δ(r − v′0t)

(33)

+n∑i=2

C′i eit (v′2

i +s ′2i )/2δ(r − v′i t)

}ϑ(z− l)

],

where u is again the mirror potential,l the mirrorthickness, and hereϑ(z; l)= ϑ(z)[1 − ϑ(z− l)] withthe same notation forϑ(x) as in the previous section,and the wave functionψ is now decomposed to

ψ(s,v, r, t)=ψ0(s0,v0, r, t)+ψ1(s0,v1, r, t)

+ψBT(s0,vB, r, t; l)+ψ ′0T(s

′0,v

′0, r, t)

+∞∑i=2

ψiT(si ,vi , r, t)+∞∑i=2

ψBiT(sBi ,vBi , r, t; l)

(34)

+∞∑i=2

ψ ′iT(s

′i ,v

′i , r, t)+

∞∑i=2

ψ ′BiT(s

′Bi ,v

′Bi , r, t; l).

In the right-hand side of the wave equation (33),the first term in the first curly bracket is the sourceterm for the given condition of the incident neutronand the remaining terms in that curly bracket expressthose for the reflected components with the final ve-locity directed backward, after thei-times interactions


with the front surface, while the first term in the secondcurly bracket is the source term for the direct transmit-ted component and the remaining terms in that curlybracket correspond transmitted components with thefinal velocity directed forward after also thei-times in-teractions with the front surface. Thus, the subscriptsi in these terms express the total orders of reflectionsand transmissions at the front boundary of the mirror.All these terms except the initial condition of the inci-dent neutron should be solved by the analysis.

3.2. The wave functions outside and inside the mirror

We can obtain the solutions for each componentsin Eq. (34) by substitutingψ into the wave equation(33), with the Fourier expansion method as used in theprevious section, and successively equating the same-order terms for the boundary effects. For example,the Fourier expansion of Eq. (32) with making useof the solution in the previous section, i.e., Eqs. (11)and (14), gives the result of the approximate solutions

ΨB(p,ω; l)=ΨB(p,ω)⊗[1− e−ipzlΘ(pz)

](35)=ΨB(p,ω)+ΨBB(p,ω; l),

where⊗ denotes the convolution onpz, and

ΨBB(p,ω; l)= e−ipzl

[2πC0e

−q0Bl (iq0B + v0z + iq0‖)2iq0Bi(ω−ωB0)iq0‖(pz − iq0B)

(36)− 2πC1e−q1Bl (iq1B − v0z + iq1‖)

2iv1Bi(ω−ωB1)iq1‖(pz − iq1B)

],

with

qiB =√

2u+ (qi‖ ∓ iviz)2 ∼= vBz ∓ iviz

vBzqi‖

for i = 0 and 1, respectively, with the approximationin the same sense as used in the previous section. Thesimilar equation forΨ †

BB(p,ω; l) can also be derived

fromΨ†B(p,ω) in the same way.

The second term in the right-hand side in Eq. (35)comes from the backside boundary effect, and it gives,with the substitution of Eq. (35) into Fourier expansionof Eq. (34) and further into Fourier expanded equation(33), the solution for the direct component of thetunnelling wave as

(37)Ψ ′

0(p,ω)

Ψ′†0 (p,ω)

}= ω+ p2/2+ u

ω+ p2/2

{ΨBB(p,ω; l),Ψ

†BB(p,ω; l).

The inverse Fourier processes for Eq. (37) givethe solutions for the wave functionsΨ ′

0 andΨ ′†0 , but

a remark for the present situation should be notifiedthat there is no incoming neutron outside the potentialin the backside space, thus the coefficients for thesolutions with the singularity fort � 0 in the spacez > l should be taken to be 0. In this way, we obtainfrom the inverse Fourier process the solution for thedirect component of the tunnelling wave as follows:

ψ ′0T(s

′0,v

′0, r, t)=ψ ′

0(s′0,v

′0, r, t)

(38)+ψ′†0 (s

′0,v

′0, r, t),

where

ψ ′0(s

′0, r, t)

ψ′†0 (s

′0, r, t)

}=

{C′

0C

′†0

}exp[−vBzl]

× eiωs0t eiv0x(x−v0xt)eiv0z(z−l−v0zt)

×exp

[−s′0√rxy(t)2 + (

z− l − v0zt ∓ iv0zvBzl)2 ]

√rxy(t)2 + (

z− l − v0zt ∓ iv0zvBzl)2

(39)(z� l).

It is obvious that the present solution for the tun-nelling wave packetψ ′

0T is able to satisfy the continu-ity conditions at the backside boundaryz= l with theinside waves (14) for all the time with equatings′0 = s0

and the coefficientC′0 andC′†

0 equating to satisfy thecontinuity conditions. The present solution has the sin-gularity att = 0 emerging on the cut circle of the conedescribed in the previous section, and developing fort � 0 at

z= l + v0zt,

(40)rxy(t)2 =

(v0z

vBzl

)2

,

which is the indication of the existence of the tun-nelling wave packet for the neutron in the backsidespace outside of the potential.

Furthermore, as similar as in the previous section,the backside boundary effect derives the next-ordercomponents of the inside wave function as follows:

ψ ′B0T(s0, r, t; l)=ψ ′

B0(s0, r, t; l)(41)+ψ

′†B0(s0, r, t; l),


where

ψ ′B0(s0, r, t; l)

ψ′†B0(s0, r, t; l)

}=

{C′

B0C

′†B0

}

× e−vBz(2l−z−ivBzt)ei(ωs0−2u)t eiv0x(x−v0xt)

×exp

[−s0√rxy(t)2 − (v0z/vBz)2(2l − z∓ ivBzt)2]

√rxy(t)2 − (v0z/vBz)2(2l − z∓ ivBzt)2

(42)(0 � z� l),

(43)= 0 (z > l, z < 0).

From these solutions, we will look for the relationsamong their coefficients to be satisfied for the continu-ity conditions on the backside surface ofz= l.

3.3. The reflectivity and the transmission probability

The continuity of the total wave function atz = l

can be constructed from Eqs. (14), (39) and (42), andby making use of Eqs. (26) and (27), which requires

C′0

C0= 1− β2

1+ iβ+ C′

B0

C0,

(44)C

′†0

C0= 1− iβ + C

′†B0

C0.

The continuity of the derivative toz of the wavefunction produces two independent sets of require-ments since the every wave functions consist of twoparts, i.e., the pilot wave part in the same form withthe plane wave formula and the packet distribution partcharacterizing the de Broglie waves. These require-ments are reduced to the following equality sets, re-spectively:

C′0

C0= iβ

[1− β2

1+ iβ− C′

B0

C0

],

(45)C

′†0

C0= iβ

[1− iβ − C

′†B0

C0

]and

C′0

C0= 1

iβ

[1− β2

1+ iβ− C′

B0

C0

],

(46)C

′†0

C0= − 1

iβ

[1− iβ − C

′†B0

C0

].

The requirements on these coefficients to satisfyEqs. (45) and (46) simultaneously and exactly are

contradictory for a realβ , and therefore we look forthe approximate solution to satisfy Eqs. (44) and (45)only, i.e., we permit some discontinuities at the back-side boundary in the derivatives on the packet parts,then the results are obtained as

(47)C′

0

C0= 2iβ(1− β2)

(1+ iβ)2,

(48)C

′†0

C0= 2iβ(1− iβ)

1+ iβ,

(49)C′

B0

C0= − (1− β2)

(1+ iβ)2(1− iβ),

(50)C

′†B0

C0= − (1− iβ)2

(1+ iβ).

As similar as in the previous section, Eq. (39) canbe simplified by making use of Eqs. (47) and (48) be-coming

ψ ′0T(s0, r, t)

=D′0Te

−vBzleiωs0t eiv0x(x−v0xt)eiv0z(z−l−v0zt)

(51)×Λ′0(s0, rxy, z, t; l),

where

D′0T = C0 ×

(52)

4iβ[cosγ ′(s0, rxy, z, t; l)− iβ2 sinγ ′(s0, rxy, z, t; l)](1+ iβ)2

,

Λ′0(s0, rxy, z, t; l)

(53)

= exp[−s0χ ′(rxy, z, t; l)1/4 cosα′(rxy, z, t; l)]χ ′(rxy, z, t; l)1/4 ,

with

γ ′(s0, rxy, z, t; l)= α′(rxy, z, t; l)

(54)+ s0χ′(rxy, z, t; l)1/4 sinα′(rxy, z, t; l),

χ ′(rxy, z, t; l)

={rxy(t)

2 + (z− l − v0zt)2 − v2

0zl2

v2Bz

}2

(55)+ 4(z− l − v0zt)2v

20zl

2

v2Bz

,


(56)α′(rxy, z, t; l)= 1/2 arctanξ ′(rxy, z, t; l),and

ξ ′(rxy, z, t; l)

(57)=2(z− l − v0zt)

v0zvBzl

rxy(t)2 + (z− l − v0zt)2 − v20z

v2Bzl2.

On the other hand, the latter solutions, Eqs. (49)and (50), are related toCB andC†

B by a common ratio

(58)C′

B0C

′†B0

}= −

[1− iβ

1+ iβ

]{CB,

C†B.

Therefore, we can extend the results of the presentanalysis, which were derived for only a finite order ofthe wave function components, towards the accumula-tion of the infinite number of the orders by multiplyinga factor

(59)F = 1

1− [1−iβ1+iβ

]2e−2vBzl

,

i.e., the coefficientD′0TF for the total accumulated

wave function for the tunnelling neutron becomes

D′0TF = C0 ×

(60)

4iβ[cosγ ′(s0, rxy, z, t; l)− iβ2 sinγ ′(s0, rxy, z, t; l)](1+ iβ)2 − (1− iβ)2e−2vBzl

,

the magnitude of which comparing to that from theplane wave analysis will be discussed in the next sec-tion.

4. Possibility of causing the UCN anomaly

Now we can study on the possibility of the presentsolutions for the de Broglie wave packet becomingthe causes of the UCN anomalous loss [1] and theanomalous wall transmission [2] during the storageexperiments reported. These UCN anomalies are theanomalously large reflection loss probability [1] com-paring to the theoretical expectation from the planewave analysis for the absorption and inelastic scatter-ing losses, and the anomalously high wall transmission[2] of the subbarrier UCN detected after penetratingthrough a thin foil.

A simple estimation for the effects of the absorp-tion and inelastic scattering on the reflection lossescan be carried out by introducing the complex poten-tial for the mirror wallu = u0 − iu1, the imaginarypart of which we now try to use was given in thetextbooks [7,8] with their notations asu1 = h̄2Nk×(σa + σie)/2m, whereN is the atomic density,k theneutron wave number, andσa andσie are the absorp-tion and inelastic scattering cross sections, respec-tively. We notice by substituting the complex poten-tial u into our Fourier inversion process for Eq. (12)or Eq. (37), i.e., in the more exact sense, into the re-

lation q0B =√

2u+ (q0‖ ∓ iv0z)2, that, if the magni-tude of the imaginary potential becomes significant,i.e., u1 ∼ v0zs0, as expected to be probable for suffi-ciently smallv0z, which corresponds to small grazingangles to the mirror surface, the correct constructionof the wave packet will be obstructed and the reflec-tion loss should occur. The present assumption that thereported reflection loss [1] had occurred due to sucha situation leads to the numerical estimation for thepacket parameters0 with its physical unit as

(61)s0 ∼ u1

v0z∼ Nvn(σan+ σien)

πvlPloss

∼= 0.5 µm−1,

where isotropic UCN flux being assumed,Plossdenot-ing the reflection loss probability reported as 3×10−5,the loss cross sectionσan + σien was taken as about13 mb normalized tovn of thermal neutrons for theberyllium bottle [1] with the limiting wave numberfor total reflection of 0.11 nm−1, and here our unitiesof h̄2/m = 1 explained in Section 1 were used. Thepresent magnitude estimated fors0 gives the packetwidth of 1/s0 ∼ 2 µm, which belongs to the accept-able range for the magnitude of the packet width.

Another hypothesis for the loss process will bepossible since both of the anomaly observations cor-respond to the particle reaction of the neutron, thenstudying the present task with based on Eqs. (51)–(60).That is, for the UCN losses due to the absorption andinelastic scattering reactions in the wall, such a situa-tion is supposed as, after the subcritical transmissionthrough the thicknessl, the particle reaction occurswith the nucleus at the positionz� l. In this case also,a noticeable situation occurs for small grazing anglesto the mirror surface, which gives a large value forβ ,i.e.,vBz ∼= vl � v0z, thenα′ approaches to zero,χ ′1/4


to√rxy(t)2 + (z− l − v0zt)2, andΛ′

0 to

exp[−s0√rxy(t)2 + (z− l − v0zt)2

]√rxy(t)2 + (z− l − v0zt)2

.

In this situation, the magnitude for the coefficientD′

0TF from Eq. (60) can be estimated as

(62)|D′0TF|2 ∼ 16C2

0[s0l(z− l − v0zt)]2r2xy + (z− l − v0zt)2

.

A very rough estimation for the probabilityP ′0TF

for such subbarrier neutron tunnelling through thepotential reduces to

(63)P ′0TF ∼ 16

3

(s0

vl

)2[vlle

−vl l]2,

for the isotropic packetΛ′0 and with assumings0 � vl .

The present result (63) gives the largest value for thetunnelling probability around the region ofl ∼= 1/vl as

(64)P ′0TFmax ∼ 0.7

(s0

vl

)2

.

Such possibility for the tunnelling of UCN withthe velocity componentv0z � vl will induce absorp-tion or up-scattering inside the wall, and thus lostfrom the storage device. If we roughly estimate theobserved anomalous loss probability for the presentsituation to be holding, for sayv0z < vl/2, then withthe loss probabilityPloss= 3 × 10−5 substituted intoP ′

0TF max∼ 2Ploss, we obtain the packet parameters0with its physical unit as

(65)s0 ∼ 9× 10−3vl ∼= 1 µm−1,

for the beryllium bottle with the limiting wave numberof 0.11 nm−1, which gives the packet width of about1 µm, also belonging to the acceptable range for themagnitude of the packet width.

On the other hand, we could not present a reason-able estimation for the case of the anomalous trans-mission probability through nearly 20 µm thick alu-minium foil, if we suppose the phenomena beingcaused by a direct transmission process due to tun-nelling, but we suppose it should be some multiplestage phenomena, such as via some energy-gainingprocess for UCN, then succeedingly resulting immedi-ate transmission of such UCN. Such a chance of ener-gy-gaining scattering is plausible as UCN could tunnel

through the potential with the probability of the order10−5 as estimated above.

5. Concluding remarks

The wave functions for the de Broglie wave packetunder the total reflection conditions were studied in astraightforward manner starting from the proposed in-homogeneous wave equation for the packet, and thesolutions inside and outside the reflection potentialwere derived for the geometries of the semi-infiniteand the finite thickness mirror potential. The wavefunctions for the neutron inside and tunnelling to thebackside of the potential showed the very character-istic features drastically different from the solution ofthe conventional plane wave analysis.

The solution for the wave function of the tunnellingneutron indicated the possibility of significantly largeprobability for inducing particle reactions such as ab-sorption and inelastic scattering by the wall nucleusfor the typical case of the very small grazing anglesof the neutrons to the mirror surface, and therefore arough numerical estimation was performed in compar-ison with the observed anomalous loss probability ofUCN in storage experiments. The result gave the valuefor the packet width parameter of about 2 µm for theloss probability of 3× 10−5 for the beryllium bottlewith the limiting wave number 0.11 nm−1 and the losscross section of 13 mb.

Further study for possible direct experimental ver-ifications on our present analytical results on the deBroglie wave packet is now under planning to be car-ried out by making use of UCN and also of very coldneutrons.

At the last, some general remarks should be added,since, as shown in the beginning of the Letter, thepresent concept is going to provide a corpuscle de-scription on a quantum system with a modificationof the Schrödinger equation at the one point intro-duced as the source of the wave. On the other hand,the usual Schrödinger equation explains a tremendousamount of empirical data in a very wide range ofphysics. Therefore, how the present approach couldbe extended to such general descriptions of physics isthe important task remaining, in which exact consis-tency with every experimental observations should berequired.


Among various directions for corpuscle descrip-tions on atomic systems proposed in the past, a no-ticeable concept will be the reformulation of theSchrödinger equation initiated by Bohm [9], and itsthorough exposition has recently been done as the deBroglie–Bohm framework of causal interpretation ofquantum mechanics on particle motion [10]. It ex-plains the stability of matter, such as the stable tra-jectory of an electron in a hydrogen-like atom, withappearance of the quantum force. For our present ap-proach and the solution being able to sustain such gen-eralizations, it must answer at least the same require-ment on the stability of the atom.

Another instructive results are indicated in elec-tronic dynamics studies based on recent advances inlaser technology and Rydberg atom spectroscopy [11],which has brought the concept of the wave packet trav-elling a Kepler orbit of an electron in an atom.

From these points of view, in addition to the directverifications with neutrons mentioned above, a lot oftasks remain to be studied on the present approach forrequired consistency with every experimental observa-tions as well as for theoretical generality in quantummechanics.

Acknowledgements

The author thanks Dr. P. Geltenbort at the InstitutLaue-Langevin on much valuable collaborations withhim in the UCN and very cold neutron optics research

which promoted so much the present study. He alsoappreciates valuable discussions with Dr. M. Hino,KURRI, based on his detailed knowledge of neutronoptics and wave mechanics.

Further, the author appreciates instructive discus-sions with Prof. K. Asahi, Tokyo Institute for Tech-nology, on the framework of the present analysis.

It should be especially noted that the present workwas stimulated by the collaboration of the author withDr. V.K. Ignatovich, Frank Laboratory of NeutronPhysics, JINR, on the UCN anomaly problem duringhis stay at KURRI.

References

[1] V.P. Alfimenkov et al., JETP Lett. 55 (1992) 92.[2] A.V. Strelkov, V.V. Nesvizhevsky, P. Geltenbort et al., Nucl.

Instrum. Methods Phys. Res. A 440 (2000) 695.[3] See, for example, J. Byrne, Neutrons, Nuclei and Matter,

Institute of Physics Publishing, Bristol, 1994.[4] V.K. Ignatovich, M. Utsuro, Phys. Lett. A 225 (1997) 195.[5] L. de Broglie, Non-Linear Wave Mechanics: A Causal Inter-

pretation, Elsevier, Amsterdam, 1960.[6] M. Utsuro, V.K. Ignatovich, Phys. Lett. A 246 (1998) 7.[7] L. Koester, A. Steyerl, Neutron Physics, Springer Tracts in

Modern Physics, Vol. 80, Springer, Berlin, 1977, p. 93.[8] V.K. Ignatovich, The Physics of Ultracold Neutrons, Clarendon

Press, Oxford, 1990, p. 8.[9] D. Bohm, Phys. Rev. 85 (1952) 166, 180.

[10] P.R. Holland, The Quantum Theory of Motion, CambridgeUniversity Press, Cambridge, 1993.

[11] J.A. Yeazell, C.R. Stroud, J. Phys. Rev. Lett. 60 (1988) 1494.

analytical solution for the de broglie wave packet description of the neutron in subcritical...

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