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9 November 1998

Physics Letters A 248 (1998) 114-I 16

PHYSICS LETTERS A

Quantum phase shift of spatially confined de Broglie waves in a gravitational field

Yu.N. Pokotilovski ’ Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia

Received 26 September 1997; revised manuscript received 17 August 1998; accepted for publication 18 August 1998

Communicated by P.R. Holland

Abstract

A small change in momentum in the direction of motion and a corresponding change in the phase of the wave function follow the introduction of a transverse spatial constriction. It is shown that gravity significantly alters this phase shift. @ 1998 Elsevier Science B.V.

PAW 03.65.B~ Keywords: De Broglie wave; Wave function; Quantum phase shift; Spatial confinement; Gravitation

Recently, a planned experiment was described [ 1 ]

to measure the phase shift of a slow neutron mov- ing through a narrow channel with impenetrable walls.

The experiment is based on the geometrical exam-

ple of quantum nonlocality, shown in Refs. [ 2,3] : the

phase shift in the neutron wave function arises due

to a change in its environment (boundary conditions)

when the neutron enters a narrow channel. The resulting phase shift can be calculated via en-

ergy arguments. The energy E of a neutron with the mass m moving in the z-direction in the spatially

(x, y) restricted geometry (a tube with a square cross section with side a, see Fig. la) is

E = p2/2m + (nz + n$) ~2fi2/2ma2. (1)

For the case n, = ny = 1, the change of the incident momentum po due to the introduction of constriction is

1 E-mail: pokot@nf.jinr.ru.

Ap = po - p = m( 1 - Jl - v2/i2/mEa2)

M (di/a)2/p. (2)

The corresponding phase shift at the channel length L is

A40 = LAP/~.. (3)

However, perfect crystal neutron interferometers are usually oriented horizontally. Therefore, for a neu- tron travelling through the channel in the horizontal z-direction, an additional shift of the energy levels oc- curs due to the presence of a gravitational field (in

the vertical y-direction) and the reflecting horizontal, inner lower surface of the horizontally oriented chan- nel [4].

The more realistic potential V(r) = V, (n) + V2( y ) + V3 ( z ) for the neutron in the square horizontal channel is

v,(x) =O, fora>x>O,

0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved.

PIi SO375-9601(98)00638-O

Yu.N. Pokotilovski/Physics Otters A 248 (1998) 114-116 115

Fig. I, (a) Geometry of the constriction for the neutron wave.

(b) Coordinates for the problem of a massive quantum particle

on the reflecting inclined plane.

v,(x) = vo, for x 6 0 and x 2 a, (4)

V2(Y) = %Y * fora>y>O,

b(Y) = WY + v, 7 fory<Oandy>a, (5)

Vs(2) = const, (6)

where VO = (h2/2m)4rrNb is the potential of the re-

flecting walls of the channel, N is the atomic density, b isthe coherent scattering length of the wall substance, and g is acceleration of gravity.

Table 1 shows the results of computations for the

lowest energy levels for a neutron in the potentials relevant to this problem.

It follows from the above that ( 1) transforms into

E = pzj2m + n~rr2fi2i2/2ma2 + Ee . (7)

For the case where n, = i = 1, for the above example, the corresponding phase shift will be

Efi A@= I

+r2fi2/ma2 + l/2

> A40 M 1.9Aqo. (8)

In this experiment, both the nonlocal quantum me- chanical effect and the influence of gravity on the quantum phase may be demonstrated simultaneously. To separate these two effects, measurements with dif- ferently sized constriction are possible. Another pos-

sible way to demonstrate the influence of gravitation on the neutron phase shift in the constricted geometry

is to use channels not of square cross section but for example a system of thin plates (sheets) of a proper material, separated by thin gaps. It is possible to show that rotation of such a sandwich system around the horizontal axis will produce a change of the gravita-

tional part of the phase shift. The effect may be seen

from what follows. We consider the simple two-dimensional problem of

the particle in an uniform field with boundary condi-

tion of a perfectly reflecting inclined plane (Fig. 1 b) . Gravitational field is directed along the y’-axis. The

Schrbdinger equation is given by

-& n’rl, + (Mgy’ - E)+ = 0. (9)

This problem is separable in the coordinates (x, y)

turned by the angle cy, in which the y-axis is normal

to the inclined plane surface,

x’=xcosff-ysina, y’=xsincu+ycos(~,

-&&+[Mg(ycosa+xsino)-E]I/,=O.

( 10)

Representing 3 = X(x)Y( y), dividing by XY, and separating variables we obtain two equations,

-&Y”+ [Mgycoscu-(E-y)]Y=O, (11)

and

-&X”+ (Mgxsina+y)X=O, (12)

with y as the constant of separation. Introducing

2M2g cos a/n= = 1 /P,

2M( E - y)/fi= = h/l= (13)

forEq. (ll),and

2M2g sin a/h2 = 1 /s3,

2My/fi2 = ,u/s= (14)

for Eq. (12), we have

d2yld[= - 6Y = 0, (15)

where & = y/l - h, with boundary conditions Y(c =

-A) = 0, Y(.$ -+ oo) --+ 0, and

d2X/d12 - CX = 0, (16)

where 5 = x/s + p, and boundary condition X( [ -+

co)-+O.

116 Yu.N. Pokotilovski/Physics Letters A 248 (1998) 114-116

Table 1 Neutron energies (in lo- I3 eV) for the first four levels in a one-dimensional square well of size a, gravitational field with a reflecting horizontal surface, and the potential (5) (a = 20 pm, VO = 100 neV)

Level number i

1 2 3 4

potential well without gravity 5.09 20.4 45.8 81.5 gravity with a reflecting horizontal surface 14.0 24.5 33.05 40.6 gravity in the potential well (Ef ) 14.4 30.0 54.0 87.3

The solutions of these equations which are evanes- cent at infinity are Airy functions, and Dirichlet bound- ary conditions for (15) give the spectrum of station- ary states. The total energy E is not quantized in this situation, but the part of the energy (E - y) corre- sponding to the motion normal to the surface of the inclined plane is quantized in accordance to ( 13) and ( 15). The energy levels for the neutron in the gravi- tational field (for the case when LY = 0) may be seen in the third row of Table 1.

The solution of Eq. (16) describes a wave moving in a uniform field with changing energy y.

Recently, a planned experiment on the measurement of energy quantization of neutrons in the vicinity of a perfectly reflecting horizontal plane in the presence of a gravitational field was announced [5]. One of the main experimental problems is very small energy (and respectively space) separation of energy levels, N several microns. In Ref. [ 61 it was proposed to use the vertical gradient of the magnetic field to compen- sate the gravitational force and in this way to increase the separation of levels for one of the neutron spin

components. It is seen from ( 13) that rotation of the plane around

the horizontal axis changes the energies of the quan- tum levels for motion in the direction along the y-axis and respectively the phase shift of the neutron wave moving in the sandwich type constriction.

References

[ 1 ] S. Griffin, A. Cimmino, A. KLein, G. Opat, B. Allman, S. Werner, Advance in neutron optics and related research facilities, in: Proc. Int. Symp. Neutron Optics in Kumatori ‘96, 19-21 March 1996, J. Phys. Sot. Jpn 65A (1996) 71.

[2] J.-M. Levy-Leblond, Phys. L-ett. A 125 (1987) 441. [3] D.M. Greenberger, Physica B 151 (1988) 374. [4] S. Flugge, Practical Quantum Mechanics 1, Problem 40

(Springer, Berlin, Heidelberg, New York, 1971) [Russian Edition: Zadachi po kvantovoy mekhanike, t. I (Mir, Moscow, 1974) p. 1121.

[ 51 V.V. Nesvizhevsky, Preprint ILL 96NE14T. On an experiment related to the oservation of quantum states of a neutron in a gravitational held (ILL, Grenoble, November 1996).

[6] V.I. Lushchikov, AI. Frank, JETP L&t. 28 (1978) 559.