131
770

 

 

LIBRARY
Michigan State
A University

 

 

 

This is to certify that the

thesis entitled

THE EFFECT OF GRAVITY ON QUANTUM PHASE
MEASUREMENT, WITH SPECIAL CONSIDERATION
OF THE EQUIVALENCE PRINCIPLE

presented by

Eric Grant Young

has been accepted towards fulfillment
of the requirements for

M.S. Physics

degree in

7/

 

 

0-7639 MS U I': an Afi‘iflnau'w Action/Equal Opportunity Imfitution

 

 

 

MSU

LIBRARIES
Janina-u...

 

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
Stamped below.

 

 

 

 

 

T

 

THE EFFECT OF GRAVITY ON QUANTUM PHASE MEASUREMENT,
WITH SPECIAL CONSIDERATION OF THE EQUIVALENCE
PRINCIPLE.

by

Eric Grant Young

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

HASTER OF SCIENCE

Department of Physics and Astronomy

1987

ABSTRACT

THE EFFECT OF GRAVITY ON QUANTUM PHASE MEASUREMENT,
WITH SPECIAL CONSIDERATION OF THE EQUIVALENCE
PRINCIPLE.

by
Eric Grant Young

The observed effect of gravity on quantum phase

measurements is discussed, and analyzed, in particular,

with the Feynman-Hibbs1 method of quantum mechanics.
Consideration is given to the problems these experiments

pose of the equivalence principle of general relativity.

1

I. Introduction

That a gravitational field can produce measurable

phase effects in quantum systems has been established by

experiment. Collela, et al.2, observed such effects in a
neutron_interferometer experiment. Specifically, they
observed a phase difference between the two "paths" of
the interferometer, which depends on the potential

difference between the two "paths," and the neutron's

mass. More recently, Kuhn and Schoelkopf3 have observed

phase effects in a superconducting loop. Jain, et al.4,

have considered such effects using Josephson-effect

batteries.

I begin by introducing the Feynman-Hibbs1 method of
solving quantum mechanical problems. I show that the
Feynman-Hibbs method is equivalent to solving the-
Schrodinger equation, and that low velocity problems can

satisfactorily be treated with the Feynman-Hibbs method.

Next, I analyze the Collela, et al.2, experiment using the
Feynman-Hibbs method; and demonstrate that this method
correctly reproduces their results. As a theoretical
aside, I show that, when kinetic energy is much greater
than potential energy, the integral of the Lagrangian

over time reproduces the integral of momentum over

2

distance. I briefly mention the Jain, et al.4,

experiment; and show that the Schrodinger equation for

kuhn's3 experiment leads to Airy5 functions. Kuhn's
experiment has a spatially invariant current, which poses
difficulty for general relativity. This paradox is
discussed, but not resolved. A covariant form of the
Schrodinger equation, derived from the Klein-Gordon
equation, is presented. The paper ends with a discussion
of the local/non-local paradox of general relativity.

The question of an accelerated charge, verses a charge at
rest in a gravitational field, has significance for the

equivalence principle, and has been discussed previously

by Bondi, and Gold.6 Greenberger7 has considered the
breakdown of the weak equivalence principle for quantum

systems. This has been confirmed experimentally, in

particular, by Collela, et al.2 There is concern because
the mass of a particle can be determined by measuring its
phase produced by a gravitational field. Finally, I will
show that weak gravitational fields are formally
equivalent to accelerated reference frames in special

relativity.

3
II. The Feynman-Hibbs method, and the Collela, et

al., experiment

The Feynman-Hibbs1 method is an integral method of
solving quantum mechanical problems, which is formally
equivalent to the Schrodinger equation. In practice,
those problems which are tractable with the Feynman-Hibbs
method may differ from those which are tractable with the
Schrodinger equation. The method consists of two parts:
An integral over the classical action which contributes
only to the phase; and, a complex prefactor which
provides the magnitude, and contributes to the phase.
Together, these form a kernel which is integrated to
obtain the probability of transition from one state to

another. The kernel has the form
tb
K(b,a) = expCi/hI L(x,x,t)dt] Dx(t), (l)
a t

where the differential, Dx(t), implies integration over

all possible paths between a and b. This is treated by

letting x = x + y, and using the action

S[x(t)] sc§(t) + y(t)]

Ibca(t)(§2+ 2§y + y2)...]dt
a

tb 2 2
= Scltb’a] + I [a(t)y + b(t)yy + c(t)y Jdt.
ta (2)
Substituting (2) into (1) gives
tb 2 2
K(b,a) = exp i/hI [a(t)y + b(t)yy + c(t)y Jdt x(t) X
0 t
a
iScltb,a]/h
e . (3)

The integral part of (3) is designated F(t ,ta); since

b
the end point times don't effect the integral directly, F-

must be of the form F(t) E F(tb-ta).

It is instructive to consider a particle traveling in

a potential, for which the Lagrangian is

L = 3&2 - V(x). (4)

The classical action between two points for such a

particle is

m 2
Scltb,a] 2E‘xb xa) V(xb,xa)t, , (5)

where t E t -ta, and V(x

b ,xa) s V(x ) — V(xa). The

b b

kernel K can be written

K(b,a) = J. K(b,c)K(c,a)dxc
x

C

or, changing notation,

cl

iS (t+s)/h iScl(t)/h iS l(s)/h
F(t+s)e = J F(t)e F(s)e dx
x

C

5

= F(t)F(s)J. 2

X

im
dxc EXP{EFE(xb xc)
c
+im (x -x )2 -V(x x )t -V(x x )s
255 c a b’ c c’ a

= F(t)F(s)e dxce

X
C

. im 2 1 1
lscl (t+S)/hI mxc [E +g]

Thus, with the standard form for the gaussian integral,

Zinh 1’2
F(t+s) = F(t)F(s) .
m 1 + 5

Let F(t) a f(t)[2inht/mJ—1/2, then

 

 

m 1,2 m 2inh 1’2
‘(t+5’ 2inh(t+s) = *‘t"‘5’ZTEF tsm(1/t+1/s) ’

which yields

f(t+s) = f(t)f(S), or f(t) = eat.

Take f(t) to be identically one. Then, for a particle in
a potential,

iSc
KEb,aJ = e

[b,aJ/h 1/2
1 m

firth“, -F) '
b a

(6)

 

The Feynman-Hibbs method is formally equivalent to the
Schrodinger equation as follows: Begin with the

probability of transition from one state to another

w(x2,t2-t1) = If;fi(x2,x1;t2,t1)w(x1,t1)dx1 (7)

6

Restate this in infinitesimal form,

w(x,t+s) = Foo dy nexp[cli;L[x; y,ct]]w(y,t).

the Lagrangian

(8)

where A is to be determined. Using (5),

of a particle in a potential,

m
2 .
tw) = ,1J_mdy expL‘; EFL][- ficV[x1+l,st]]w(y,t)

W‘xv
Let y = x + n; then
w‘x Us) _ firm eimn 2/2he e-ieV(x+n/2, ,t)/hw(y’t) (9)

Expand (9) to first order in e, and second order in n,

replacing eV(x+n/2,t) with eV(x,t) (the error is 0(8 )):

. 2 2
0"” éfmdn em" ’2'": [1- -1EV(x, t)][w+n;;waén2° V] (10)

 

W+£3f‘=
The leading terms give
Therefore,
A = [2i;h£]-1/? (11)

The higher terms are treated with two integrals,

. 2
leLFimn [Zhendn = 0’

and

 

. 2 .
1Emeimn [Zhenzdn = 1'18.

Equation (8) is now

2

is ihca
V+8§WE=V‘—.7W’* m V
0x

This can be manipulated to become the Schrodinger
equation

‘ 2 2
h0w__h ow

Since the Schrodinger equation applies in the low
velocity limit, and can be obtained from the Feynman-
Hibbs integral, the Feynman-Hibbs method must be able to

treat low velocity problems satisfactorily.

The Collela, et al.2, experiment is a classic neutron
interferometer experiment which firmly establishes the
effect of gravitational fields on the quantum mechanical
phase of a particle. The experiment is shown
schematically in Figure 1. The potential difference

between "paths" A and B is varied by rotating the

 

 

Figure 1

mirror

detectors
A .
showing
unnuu interference
eam
splitter B
BIT-FE?

The Collela, et. al., experiment

8
interferometer about its axis. The effect of the
gravitational potential on the phase, i.e. the phase
difference between the two branches of the
interferometer, is detected as shifting of the
interference pattern. This difference is stated as

3 = sin¢, (13)

qgrav
where ¢ is the angle of rotation of the interferometer.
Ignoring ¢ , and other angles that parameterize R,

Collela, et al., obtain forfi

25% Ego = 2MgA/hv, (14)

where A is the area enclosed by the branches of the
interferometer.
This experiment can be treated exactly by the Feynman-

Hibbs method. The Lagrangian for this problem is

_ m . 2 . 2 _ '
Li - 2-[xi +yi ] mgxi, (15)

where i = 1,2 designates the two branches of the

interferometer. Similarly, the Hamiltonian is

I
II

1 2 2
1 25[pxi+pyi] + “'9":

m . 2 . 2
2(“1 4H’1 ] + ”9‘1

constant. (16)

9

Suppose that xi0 = 0, and that

_ m . 2 . 2
H1 ‘ 2["10+Y10] ‘

Suppose x

 

 

 

r3 = g.- Idt(L1 — 1",)
= 1 Jet 2m y (t)-x (t)
F 9(‘1 2 ]
2mg tb
= T X(t)t ta, X(t) 5 x1(t) -— x2(t).

 

 

Define §(T) to be the average of X(t) over the

interval T E t - t . Thus,
b a

(17)

(18)

(19)

10

Since T = y/y, Equation (20) becomes

the equation Collela, et al., obtained.

If kinetic energy is much greater than potential
energy, then the integral of the Lagrangian over time
reproduces the integral of momentum over distance. In
this limit the Lagrangian is

L 2 2T = mxz.

(Multiplying the Lagrangian by a constant in no way
changes the underlying physics.) The integral of the

Lagrangian over time is

‘7
mfo dt

= «TE-32“
«mam

fp dx,

fL dt

the integral of momentum over distance.

11
III. The Kuhn, and Schoelkopf experiment, and the Jain,

et al., experiment

Jain, et al?, have looked for gravitational effects
on charged particles. In particular, they sought
behavior in charged particles in analogy to the
gravitational redshift. Their experiment was to perform
a null measurement of the potential difference between
two Josephson-effect batteries connected in opposition,
and at different potentials. They obtained results which
are consistent with their theoretical introduction.
However, there is no apparent conclusion to draw from

this experiment about gravitational phase effects.

Kuhn, and Schoelkopf? have examined gravitational
phase effects in a superconducting loop suspended
dynamically in a gravitational field. The loop was
tightly coiled about itself so that its area would be
negligible, and connected to a superconducting quantum
interference device (SQUID) for detection. The dewar,
which housed the coil, was suspended from a tunable
pendulum so that the loop was subject to a time dependent
acceleration. A null measurement was made of the flux

modulation induced in the loop by the time dependent

12
acceleration. The induced electric field was consistent

with gravitational force balance to better than one part

in 105, and to have a relaxation rate shorter than the

acoustic time scale of the loop.

Kuhn's experiment can not be analyzed exactly; some
approximation method must be employed to analyze the

Schrodinger equation. A series expansion of the

Schrodinger equation leads to Airy functions? The

Schrodinger equation is

2 2
. 0w _ ;h 0
Letting w = f(t)Z(z), Equation (22) becomes
. 2 2
1h 0f _ 1 _h 0 _
T E - :- z-m 1 + (“92 Z - E. (23)

Equation (23) can be solved immediately for f,

f(t) = e‘Et’h. (24)

The equation to solve for Z is
z

o = -" 22 + 3E0“): - E]Z.
0x

 

°° 1:
Let 2(2) = 2 akz , then

k=o

m k 2m k k
0 == :E‘a’lu1z 4-“2 fingalhtz - E Eakz . (25)

13

Equation (25) gives the recursion relation

which is an Airy function. This solves the eigenvalue

problem exactly if carried out to all orders in n.

Using the WKB approximation, Kuhn and Schoelkopf
obtain a current for their loop which is independent of

position; in particular

2 2 1/2
2 2 + 4m 1 Hg

i a n n 2 (26)
h

where H is the vertical height of the loop and 21 is its
length.‘ Like the weak equivalence violation paradox (see
next section), Equation (26) presents another paradox,
since general relativity predicts that the current should
vary with position as follows: We begin with the

continuity equation
j . = 0, (27)

and ignore the 1 and 2 components (potential difference

only along the z—axis ). The nearly-Newtonian metric

s
gafi 1 2mgx 0 0 O
0 1 0 0
0 0 1 0 (28)
O 0 O 1

 

 

I

is used. For the relevant components of the connectivity

14
this gives

r° = r° = mg.
90 03

Since the loop is closed, the number of charge carriers

in the loop can not vary with time; i.e.

The continuity equation (27) is now

jg 3 + mgja = 0’

D

which has the solution

3
is = e-mgx .3 . (29)
t=o
At this time, no satisfactory means of resolving the
conflict between Equations (26), and (29) is known.

A covariant form of the Schrodinger equation can be
obtained by expanding the d'Alembertian of the Klein-
Gordon equation

(a — mziwtxie“m*"‘ = o, (30)
where s is small such that 62 can be neglected. The
d'Alembertian expands as

=14”
ow _ g vavw
=1")
9 Vpovw
= gpvo a w — g“”r° o w- (31’

p v yv a

15

9“” is taken to have the nearly-Newtonian form
—(1+2§)" o o o
0 1 0 0 .
O 0 1 0
0 0 0 1

The pertinent components of the connectivity are

P, = -F , = -F , = 0,§.
100 010 001 t

Equation (31) becomes

ow = (1+2§>“<m+e>’e“’"*"‘w(x) + e“'"”"‘v’w(x)

+ (1+2§)_1V§-Vw(x)ei(m+£)tw(x). (32)

Substituting Equation (32) into equation (30) gives
[—m2 +m2 +2mc —2i>m2 -4m§e +vz +(1+2§)"V§-v]w(x) = o.

The term 46m: can be neglected, since a and 5 are
both small, and §m = -V. So that, dividing by 2m yields

a modified Schrodinger equation

_ V§(x)-Vw(x)

-1 2
£w(x) — 23V w(x) V(x)w(x) m + x

IV. The local verses non-local paradox
The question of an accelerated charge, verses a charge
at rest in a gravitational field has significance for the

equivalence principle.6 As is well known, an accelerated

charge emits electromagnetic radiation. The strong

16
equivalence principle (SEP) states that locally
gravitational fields are indistinguishable from
accelerated reference frames. Based on the SEP, one
would conclude that a particle suspended statically in a
gravitational field should radiate. That such a
suspended charge does not radiate demands attention.
Careful consideration of the problem shows that the SEP
is not tested because the two situations are
gravitationally distinguishable, except for uniform
fields of large extent where the paradox remains
unresolved. Accelerated reference frames are homogeneous
by definition to all distances. Except for uniform
fields of large extent, gravitational fields exhibit
inhomogeneities when consideration is extended to a
distance of the order of the inverse of the corresponding
acceleration. That is inhomogeneities appear in the

gravitational field, unless distances are kept small
compared to czlg. Since one must extend observation to a

distance of one over the acceleration (ca/g) to study a
radiation field, comparison of an accelerated charge to a
static charge in a gravitational field is not a proper

test of the SEP; the SEP escapes scrutiny.

One possible statement of the weak equivalence

17
principle (HEP) is that a particle's mass can not be

determined by its free motion in a gravitational field.

The HEP breaks down for quantum systems.7 The HEP is in
conflict with quantization as follows: Classically, the
HEP states that the motion of a particle may be
parameterized by its velocity, which is independent of

its mass. In the Bohr limit of quantum theory,

1p dx nh,
or,

Iv dx nh/m. (34)

Thus, the velocity is parameterized by (h/m), dependent
on mass in contradiction with the HEP. The Schrodinger
equation is similarly parameterized. Dividing the

Schrodinger equation by mass gives

 

2 .
-1 h 0 _
[ 2[fi]ox2 + ¢ ]wn — snwn, (35)
so that
w" = f(h/m), an = f(h/m), En = mf(h/m). (36)

That the HEP breaks down for quantum systems has been

established experimentally by Collela, et al.2 This

experiment establishes that the mass of a particle can be

18
determined by gravitational induced phase effects. The
SEP -- local equivalence of gravity to acceleration -- is
not contradicted by quantum theory, nor by quantum

experiment.

Is the ability to determine the mass of a particle
from its gravitationally produced phase worrisome? Hhile
this is clearly a departure from classical thinking, as
long as the mass dependence is confined to the phase, and
so far it is, I claim that this is not problematic. The
HEP is a classical concept, which does not survive in the
quantum domain; while the phase is a strictly quantum
mechanical concept. That the non-classical phase
violates the HEP should be cause for thought but not
worry. However, if the amplitude, from which classical
variables, e.g. velocity, can be obtained by integration,
should be shown to violate the HEP, there might be

extreme concern.

Heak gravitational fields may be equivalent to
accelerated reference frames in special relativity; i.e.
the metrics for the two have the same form. This is
important, from the geometric perspective of general
relativity, since this perspective holds that, locally,

gravitational fields are indistinguishable from the

19
Minkowskian space of special relativity, and the
criterion for locality is the same criterion that gives a

weak field. Begin with the condition distance << 9",

local coordinates t“ , and basis vectors Ek' for the

locally defined hypersurface? The observer is located

at P(r), which is displaced from the origin by 2(1).

The typical point in this hyperplane is

5 = (k e_;(r) + 2(1). (37)

t is identified with 7, and the coordinates of a point

are found by solving

I :*'[ek.[t°']]“ + 2"[:°']. (38)
. The equations obtained are

° [9“. :"]sh[g:°'].

(.1. ("We”), ' ‘

x = (2'.

x”

for

x
II

x3 = :3 . (39)

Thus, the metric for an accelerated reference frame is
ds? = n dxpdxv
pv

2 2 2 2 Z

=-[1 + at“) (m («"1 («=1 m

20
The coefficient of the first term is

2

1 + 29:" + [96']-

0

Since we are restricted to {1 << g_‘, or gt1 << 1, (gt1

)2

can be neglected. Further, gt1 is identifiable as the

gravitational potential 5. The metric (40) becomes
2

de.2 = —[1 + 25] [dz°']2+[d:"]z+[d:2'] +[dta'] (41)

The metric (41) is the standard metric for nearly—

2

Newtonian gravitational fields.

V. Conclusion

Two conclusions can be drawn from this discussion: 1)
Theory and experiment are in agreement. 2) The
application of the equivalence principle leads to
paradoxes which remain to be explained. The status of
the strong equivalence principle for a charged particle
suspended in a gravitational field which is uniform to
large extent remains uncertain. The position independent
current observed by Kuhn, and Schoelkopf is an unresolved
paradox for general relativity. That gravitational phase
effects are mass dependent in contradiction with the weak

equivalence principle is not worrisome, but awaits an

21

adequate explanation.

Acknowledgements

I have benefitted priciplely from J. Kuhn's

supervison. H. Repko also provided helpful conversation.

22

References

1R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path

Integrals (McGraw-Hill, New York, 1965)

 

2R. Collela, A.H. Overhauser, and S.A. Herner, Phys. Rev.

Lett. 34, 1472 (1975)

3J.R. Kuhn and R. Schoelkopf, preprint

4A.K. Jain, J.E. Lukens, and J.-S. Tsai, Phys. Rev.

Lett. 58, 1165 (1987)

5C.M. Bender and S.A. Drszag, Advanced Mathematical

Methods for Scientists and Engineers (McGraw-Hill, New
York, 1978), p. 51

6H. Bondi and T. Gold, source unknown, but believed to be

Proc. Roy. Soc. (1955)

7D.M. Greenberger, Rev. Mod. Phys. 55, 887 (1983)

BC.H. Misner, K.S. Thorne, and J.A. Hheeler, Gravitation

(H.H. Freeman, San Francisco, 1973), p. 172

   

"'llifillllfilllllllllljlllslllllllljllls