Foundations of Physics, Vol. 20, No. 1, 1990

Testing Discrete Quantum Mechanics Using Neutron Interferometry and the Superposition Principle--A Gedanken Experiment

C. W o l f 1

Received January 5, 1989

Using a neutron interferometer and the phase difference calculated from spatial discrete quantum mechanics, a test ,for discrete quantum theory may implemented by measuring the X spin polarization and its variation with position.

1. MOTIVATION FOR DISCRETE QUANTUM MECHANICAL EFFECTS

There have been numerous suggestions and attempts over the years to introduce into physics a discrete space-time lattice, both from the point of view of eliminating the divergences of quantum field theory and from the fundamental point of view of considering both space and time as discrete- like entities. (1'2) Recently, T. D. Lee has pointed out that the discreteness of time may facilitate calculations of path integrals as well as provide an avenue to explain many of the paradoxes of quantum theory. (3) There has also been an emphasis on looking for the corrections to conventional physics induced by discrete spatial effects and the shift in the spectral lines they produce. (4). In a previous note, we have discussed the effects that a discrete-time Schr6dinger equation would have on electron spin measurements and showed that a measurable shift in the resonant frequency would result due to discrete time effects. (5) In what follows, we discuss the phase difference induced in spin functions by discrete spatial effects in the neutron interference experiment, wherein one path is spin

i Department of Physics, North Adams State College, North Adams, Massachusetts 01247.

133

0015-9018/90/0100-0133506.00/0 O [990 Plenum Publishing Corporation

134 W o l f

down (d) and the other spin up. In principle, a variation of the X polariza- tion of the spin with distance along the screen can lead to a test for discrete spatial effects. (6)

2. DISCRETE SCHRODINGER EQUATION

To replace the Schr6dinger equation by a discrete spatial equation, we write

8Z ~ Z(x + Lo/2) - ;((x - Lo/2) _ AZ ( t )

8X Lo

where L o denotes the lattice spatial separation or discrete space interval:

2m A2Z = ih ~ZOI = EZ' Z = X(x) T( t)

This gives

h2 [ X(x + Lo) + X ( x - Lo) - 2X(x) I = EX(x ) 2m L~

or, setting X = e frx,

Consequently,

whence

h 2 ~ e irLO/2 - e-irL°/2 7

2h2 sin 2rL° EL 2 m - - f - =

(2)

(3)

2 . 1 r = _._+ ~oo s m - x/EL~m/2h 2

and thus, finally,

X ( X ) = C 1 e i2/LO(Sjr1.1 x~L2m/2h2)x C2 e- i2/co(sin ,/FL~/2hbx (4)

We now construct the neutron interference experiment shown in Fig. I, with a flip cord at the slit b, to flip spins originally in the + z direc- tion to the - z direction, when emerging from slit b.

Test for Discrete Quantum Mechanics 135

NEUTRON BEAM

11,>

o 1 1 /

1¢> k

Fig. 1. Neutron beam with flip coard at b.

The spin wave function is (where ~b is the phase difference between paths A and B, where we have approximated d sin O ~ dy/L in the inter- ference apparatus for L >> d, y)

~e=! ( e i~ 1 ~ > +IT >) ,A (5)

here ~b = (2/Lo) sin ~ x/EL 2 m/2h 2 (d sin O) ~ (2/Lo) sin ~ x/-E-~om/2h 2 (dy/L), where we have used the positive exponent in Eq. (4) to calculate ~b. The X polarization on the screen as a function of y is

with

h {S~.) = ~ (1, e-*~)(~ t0) (el,~) (6)

h 2~b 11 (7) = ~ [ 2 c o s

2 \I 2h2 #-/5

2 / ~ ~ (EL~mVJ2 ~ dy (8)

136 Wolf

We see in Eq. (8) that the ratio of the second term to the first term is

dO EL2m 1025L0 ' for m=mN, E=(O.O1)mNC 2 ~b 2h 2 -

( m N = neutron mass), even for Lo ~ 10 ~6 cm, AO/~ .~ I 0 - 7 In Ref. 4 a value of L o - t0 -t8 cm was quoted for significant discrete spatial effects to occur in conventional physics, and for this value A~/~ ~-10 -H, which is completely unmeasurable in the present experiments. For Lo ~ I0 -~5 cm, we have A ~ / ~ 10 -5. It would be unrealistic to set Lo ~ 10 -I3, 10 -i4 cm, since the neutron charge radius is of this order of magnitude, and hence continuum physics is known to apply at these scales. Due to this fact, we must search for a lattice size below the scale of L 0 ~ 10 -~3, 10 -14 cm for neutrons. Because a ratio of 10- 5 would be difficult to measure in polariza- tion experiments, we might entertain the idea of using the neutron polariza- tion to generate secondary effects such as e - polarization in e - , e + colli- sions. Thus, by detecting slight differences in the e - polarization measured from neutral currents in the production of hadrons and its deviation from that expected in the continuous case, we might test for the discrete space lattice.

Also, by making the neutrons relativistic, an effect might be detected, but this would also require an analysis other than the Schr6dinger equa- tion. Although neutrons have been shown to possess wave properties in the energy range 10 -7 eV up to several hundred MeV, (7) neutron interfero- metry as applied to the neutron spin (8) has only been studied in the vicinity of 0.02 eV. We would thus require higher energy neutron interferometry than is presently used, which would depend on experimental ingenuity. Nonetheless, if such an experiment could be constructed in the range of 1-10 MeV for the neutron kinetic energy, it would provide a rigorous test for discrete spatial quantum mechanics if a discrete length appeared at L o ~< 10 ~5 cm by measuring the variation of the X spin polarization with vertical distance and looking for discrete space corrections.

A C K N O W L E D G M E N T S

I would like to thank the Physics Department at Williams College and Harvard University for the use of their facilities and Javier Hasbun for his lively discussions.

Test for Discrete Quantum Mechanics 137

REFERENCES

1. P. Caldirola, Lett. Nuovo Cimento 16, 151 (1976). 2. H. S. Snyder, Phys. Rev. "71, 38 (1947). 3. T. D. Lee, Phys. Lett. B 122, 217 (1983). 4. L. Braeci, G. Fiorentini, G. Mezzorani, and P. Quarati, Phys. Left. B 133, 231 (1983). 5, C. Wolf, Phys. Lett, A 123, 208 (1987). 6, D. M. Greenberger, in Fundamental Questions in Quantum Mechanics, L.M. Roth and

A. Inomata, eds. (Gordan and Breach, New York, 1986). 7. C. G. Shull, Fiftieth Anniversary of the Neutron (Cambridge University Press, Cambridge,

1983). 8. J. G. Summhammer, H. Badurek, H. Rauch, and U. Kischko, Phys. Lett. A 90, 110 (1982).

Printed in Belgium