1416336962.pdf

UNCERTAINTY PRINCIPLES AND INTERFERENCE PATTERNS
John F. Pr-ice
The classical Heisenberg uncertainty principle
(l) Lq/J.p ?: n/2
has been one of the key rela·tionships in quantum mechanics for over
fifty years. It does have a number of v1eaknesses, however,
particularly related to the fac·t that the standard deviations Lq and
lip only give very general information about the spreads of the
probability density func'cions of position and momentum respectively.
This paper surveys a number of recent inequali·ties which describe more
subtle relationships between posi·t.ion and momentum or, in mathematical
terms, between a func·tion and its Fourier transforrn. For example,
local uncertainty principle inequalities assert 'chat if the uncertainty
of momentum Lp is small, then not only is the uncertainty of
posit.ion Lq large, but the p:wbabili'cy of the system being localized
at any point is also small.
So as ·to add a li·t·tle more interest, I have applied in turn each
of the inequalities, starting with (l), 'co the proposition by Niels
Bohr ·that in the double-sli·t experiment you can have an interference
pattern or know the paths of the particles, but not both. In some
ways I could no·t have chosen a worse test-case since it turns out that
for this example they are all out-performed by Poisson summation.
Never'cheless it does provide an opportunity to display and contrast
some of their features. Also in the end we arrive at a rigorous
justification of Bohr's original argument which a_pparently is new.

242
I am grateful to J.B.M. Uffink and J. Hilgevoord of the
University of Amsterdam for sending me a copy of their preprint [3].
It aroused my interest in the Einstein-Bohr debate, particularly in
the area of the two-slit experiment. Also they first pointed out the
significance of the notions of w- and w-widths discussed later.
QUANTUM MECHANICAL BACKGROUND
In the Schrodinger interpretation the states of a one-dimensional
quantum mechanical system are given by complex-valued functions
The position and momentum operators are
q f H- xf and p : f I-+ -ihf'
respectively, where h denotes Planck's constant and ~ = h/2TI • The
probability density functions (pdf's) of position and momentum are
and
respectively, where
F(y) = J f(x) exp(-2Tiixy) dx
is the Fourier transform of f (The Fourier transform is extended
from L
1 (lR) to L2 (lR) in the usual manner. Also, unless stated
otherwise, all integrals are over lR.)
The expected values of position and momentum are
<q> = J xlf(x) 12 dx and = J (y/h) IF(y/h) 12 dy
and their standard deviations are
b.q = <J (x-<q>> 2 lf(x) 12 dx)~,
/:,p = (J (y-l2 h-l IF(y/h) 12 dy)~
whenever and <q> exist. (If need be, <q> will be defined as
<q> =lim J xlf(x) 12 dx ,
a~ lxl:>:a

243
and similarly for .)
It is now evident that the classical uncertain-ty principle
inequality (l) is a simple consequence of the inequality
(2) f x 2 [f(xl[ 2 dx J / [F(yJ[ 2 dy 2 (l61T2)-l (f [f(xl[ 2 dx) 2
for all f E L2 (JR) • (For a proof see [5, p.ll7].)
SINGLE SLIT EXPERIMENT
The first weakness of (l) is that for quite reasonable states f ,
~q and ~p can be infinite. For example, consider the classical
experiment in which there is a parallel stre&~ of particles passing
through a single slit as in Figure l. If the slit has width 2a, the
state function of the system (in the vertical direction) is
-1<
f = (2a) 2
1[-a,a] ,
where lE denotes the indicator function of E Hence
h
F(y) = (2a) 2 (sin 2Tiay)/2Tiay
figure l
This means that the probability fQnction of momentum is
(3) 2a [sin 27Tay/h) 2
h , 2nay/h

244
so that /J.p = 00 Hence, given /J.p , (1) tells us nothing about the
uncertainty of position. (Of course, in this case it is trivial to
calculate directly that -~
/J.q = 3 a .)
In this and related cases, ad hoc measures are often employed to
quantify the "widths" or "spread" of the relevant distributions. For
example, the width of the momentum pdf is frequently defined as
A= h/2a , this being the wavelength of the function sin2 (2Tiay/h)
Hence
an impoverished analogue of (1).
!
h/(2.3 ),
The crux of the problem is that the weight
2
y grows too rapidly
in the definition of /J.p forcing it to be infinite. With this in
mind, and in an attempt to provide a more uniform approach, in 1982
Michael Cowling and I obtained the following generalization of (1) [4,
Theorem 5.1]. (Let t # = 2t/ (t-2) •)
THEOREM 1. Suppose p,q E [1,00] and. 8,¢ ~ 0 • There exists a
constant K such that
for all tempered distributions f with the property that f and F
are locally integrable functions if and only if
(i) 8 > 1/p# , ¢ > 1/q# and a satisfies
OR (ii) (p,8) (2,0) and a = 1 ,
OR (iii) (q,¢) = (2,0) and a = o
(If both the last cases occur, a is arbitrary.)
..

245
DOUBLE SLIT EXPERIMENT
Suppose we have the situa·tion described in Figure l but with two
slits each of width 2a with centres distance 2A apart v;here A > a
Suppose also that there is a screen situated at distance d from the
diaphragm which detects the arrivals as in Figure 2,
'=L
I , lz3j
Po 2:.11.
diaphrBJgm
Figure 2
X
detecting
screen
The state func'cion for the vertical component of the incoming
particles is
f
-~
(4a) (l + 1 )
[-A-a,-A+a] [A-a,A+al
Hence the probabili'cy density functions of position and momentum are
(4a) -l (1 + 1 ) ,
[-A-a,-A+a] [A-a,A+a]
h-1 F(y/h) 2 = (4a/h) sinc2 2nay/h cos2 2TIAy/h
respectively, where sino 8 = (sin 8)/8 for 8 t 0 and 1 for
8=0, Notice·that L:lq=(a(3A2 +a2 )/3A)~ and /J.p=oo
Denote the horizontal momentum of each of the particles by p 0
For simplici·ty assume tha'c the particles passing through the slits
leave from ·the centres of the sli·ts. This means that when a particle

246
arrives at the detecting screen we know that it has followed one of
two paths. As we shall see, the period of the interference pattern is
hd/2Ap0 so this assumption will have a negligible effect on the
pattern if hd/Ap0 >> a
The time taken for each particle to cross from the diaphragm to
the screen is md/p0 , where m is the mass of the particle. Hence
the pdf of the arrivals of the particles at the screen is
<l>(x)
(4)
(8a/'Tf) (sinc2 8a(x +A) cos2 8A(x +A)
+ sinc2 8a(x-A) cos2 8A(x-A))
where 8 = 2'TfP0/hd • We are principally interested in the cosine
terms since it is these which describe the interference pattern
characteristic of the double-slit experiment. This phenomenon was
first demonstrated by Thomas Young in 1803. In practice A >> a •
Suppose we modify the experiment by first closing one slit and
then the other. If we average the two resulting pdf's we obtain from
(3)
(5) ':l'(x) = (8a/2n) (sinc2 8a(x+A) + sinc2 8a(x-A))
Notice that the interference terms are no longer present.
By only having one slit open at a time we are imposing conditions
which enable us to know through which slit each particle passes. This
suggests the conjecture that "the interference pattern appears if and
only if we cannot determine the paths of the particles". It is
interesting to note that, as predicted by the theory, the interference
pattern has been observed even when the time interval between the
arrivals of individual particles was around 30,000 times longer than
the time for an individual particle to pass through the system [1]. A
modern variant uses two lasers instead of two slits [9].

247
The remainder of the paper is an analysis of a procedure
suggested by Einstein to disprove the above conjecture. It will also
be seen that this conjecture is, at least qualitatively, equivalent to
Heisenberg's uncertainty principle.
FIFTH SOLVAY CONFERENCE
Suppose that we have a way of keeping ~ as the pdf of the
arrivals of the particles but still knowing which slits the particles
passed through. In this case the pdf of position of the particles at
the diaphragm gives
-~
tJ.q = 3 a ,
the standard deviation for the single slit. experiment. In other words,
the uncertainty of position has been reduced, and substantially if
A >> a • Since the form of <P is a consequence of the dist:t·ibution
of momentum at the diaphra~n, this means that the uncertainty of
position has been reduced without changing the pdf of momentum.
Although this does not defeat the uncertainty principle inequality (1)
as it stands (since in this case tJ.p = co ) it certainly undermines the
spirit of the general principle.
The above argument was appreciated by ZUbert Einstein as early as
October 1927 for at that time he presented it to the Fifth Physical
Conference of the Solvay Institute in Brussels. FurtheL-more, he put
forward a method, a mind-experiment, for resolving the ambiguity of
the slits. He suggested that a very delicat;e mechanism be attached to
the screen that v1as capable of measuring the vertical impulses or
kicks of the arriving particles. (Actually his suggestion was to
attach it to the diaphragm but the resulting argument is the same.)
It is to be so sensitive that it can detect the difference between the

248
momentum of a particle coming from the top slit and one coming from
the bottom. In other words, by virtue of this mechanism we can detect
the paths of the particles.
As was so often the case during this period, it was Niels Bohr
who supplied the counterargument. Its general thrust is that if we
can measure these minute impulses so precisely, the uncertainty of
momentum of the screen must be very small. But then, by (1), the
uncertainty of its positron must be large, so large, in fact, that it
would obliterate the interference pattern. (This is just one of a
series of mind-experiments put forward by Einstein in an attempt to
locate weaknesses in the quantum theory. For a fascinating account,
see Bohr [3]. Another highly-readable introduction to some of these
experiments is contained in [7].)
We shall see, however, that this conclusion cannot be inferred
from (1) since ~q is too general a measure of spread. Other
inequalities will then be brought to bear on the problem with the
final conclusion that Bohr was correct, the interference pattern would
be lost.
MOMENTUM AND PROBABILITY
Suppose that the state function of the screen in the vertical
direction is g • Denote the pdf of momentum by
where G is the Fourier transform of g • Suppose that a particle
coming from the bottom slit hits the detecting screen at a height x •
(The height-measuring scale is assumed to be rigidly fixed with
respect to the diaphragm. It is separate from the screen.) The
reading of the strength of the impulse should be (x+A)p0/d since

249
the time taken to crass from ·the diaphragm is md/p0 , m being the
mass of an individual particle. But if it is nearer to (x -A) p0 /d
tha·t is, if the reading is less ·than Xf>0 /d , then it will be
interpreted as coming from the top slit. Thus the probability of
incorrectly de-termining ·tha·t this particle comes from the top slit is
fxp0/d _ f-Ap0/d
ll(Z ·-(X +A)p0 /d) dz : ll(Z) dz •
-CO -00
Similarly, if the reading exceeds xp0/d the particle will be judged
as coming from the bottom slit: the probability of ·this being in
error is
r](z) dz •
(This decision rule is plausible given t.hat vile do no·t have any further
information on n . It always favours ·the correct conclusion when n
is even with n(x) decreasing for positive x .) Thus the probability
of corJ~ec·tly interpreting the readinsr as to the path of the particle is
This means t-J1at correct judqements are made wi·th probability one if
<md only if
(6)
From now on we assume that this is the case. Also 6q and 6p will
denote the uncertainties of position and momentum of the screen. (We
assuine <ci> and exist.)
DETECTED PATTERN
We now calculate ·the pdf of arrivals at the screen in the case
that ·the pdf of the position of the screen is I g 12 . Suppose the

250
screen is displaced a distru1ce u The probability of an arrival
exceeding X is r ljl(s) ds Hence, in general,
x-u
I 2
<[_u
prob(arrival 2: x) lg(u) I if>( s) ds) du
I lg(u) 12 ( [ ljl(s- u) ds) du
Hence the pdf of arrivals on the detecting screen is
(7)
It is emphasised ·that this pdf is with respec·t to a scale on the
screen. The problem is to show that under ·the above assumption (6),
ljl equals, or is at. least close to, '¥ as defined in (5) •
g
I. CLASSICAL UNCERTAINTY INEQUALITY
Under assumpt.ion (6)
- 2 ,; I
'
2 2
z n(z) dz < (Ap0/d)
and hence from (1)
(8) t:,q > hd/4TrAp0 .
The usual response to this is tha·t since ·the right side of this
:i.nequali·ty is of the same order as the period hd/2Ap0 of the
interference pattern, ·then the pa·ttern will be obliterated. (See, for
example, [3], [2], [7] and [8].) But standard deviations give us
meagre informa-tion about the fine de-tails. For example, it could be
that, as depicted in Figure 3, n consists of two peaks far from the
origin but is very small elsewhere. In this way (8) can be satisfied
and if the distance between the peaks is a multiple of the period

251
+---multiple of hd/2Ap
0
figure 3
A
J
I
.-~ ... ~ ~
hd/2Ap0 , then there would be no essential effect on the interference
pattern. Something stronger is needed to uphold Bohr's conclusion.
II. LOCAL UNCERTAINTY PRINCIPLE INEQUALITIES
Local uncertainty principles assert that .if the uncertainty of
momentum .is small, then not only is the uncertainty of position large,
but the probability of being localized at any point is also small. A
number of inequalities supporting this principle are developed and
applied in Faris [6]. Recently I have extended two of these:
THEOREM 2 [10]. Suppose that 1 $ t $ oo and 8 ~ o.
(i) Let (t,8) satisfy 1/t# < 8 < 1/t' (1uhere t# = 2t/(t-2)
and t' = t/(t-1) ) or (t,8) = (l,Ol or (2,0) There exists a
constant K such that
<f IF(y) 12
h #
II IX - b 18f II t
dy) 2 $ Km(E) 8-l/t
E
for aU f E L2 (IR) and measurable E _so IR.
(ii) If 8 ~ 1/t' [except for (t,8) = (1,0) ] or 8 $ 1/t#
[except for (t,8) = (2,0) ], no such inequality is possible.
THEOREM 3 [ll]. Suppose E EO IR is measurable and e > • Then

252
for all f E L2 (lR) and b E lR where
and K1m(E) is the smallest possible constant. I.f 6 ~ ~ no such
inequality is possible.
Proof. In the one-dimensional situation the most important inequality
is that of Theorem 3 with 6 ~ 1 • Fortunately, as communicated to me
by Henry Landau, it has a very simple proof. Standard completeness
arguments show that it is enough to establish the inequality for
f E S , the Schwartz space of rapidly decreasing func<tions. Given
y E lR,
fy
• -co
and hence
(FF' + F'F)
- f
co
(FF' + F'F)
y
IIFF' I
(The las-t inequality must be strict since f E S •) Thus
f IF(y) J 2 dy ~ m(E) IIFII: < 21Tm(E) lifll 2 lltfl 2 ,
E
as required. The full proofs of Theorems 2 and 3 are given in [10]
and [11] respectively.
The fact that Kf!(E) is -the best constant in Theorem 3 can be
shown in the following way. Define
where a(6) e;r <l/26J r (l-l/2!:l l .
Simple calculations show that

253
[lglli (e<.(6) 2/8) B(l+l/28, l-1/28),
8 2 2
llltl gll 2 = (a(8) /6) B(l+l/28, l-1/26) 0
Define gn = ng(n •) for Then is an approximate
identity with
L1Gnl2
( IG 12
II 11 2-1/8
11 1t I8 g 11 ; 18
Kl 0
)E n
gn 2 n -
As n -+ oo
'
Gn (y) _,_ l- for each y so that
LIG 12 + m(E) by the
n
dominated convergence theorem, demonstrating the sharpness of the
constant K1m(E) (Of course vle can have equality in Theorem 3 in a
trivial sense by supposing that E has infinite measure.)
Discussion. In general, local uncertainty principles are stronger
than global ones. For example, it follows from Theorem 3 with 8 = l
that
IIFII~
-2 2
< 4Tib llfll 2 llxfll 2 + b llyFII 2
Se'cting
gives
•;hich has the same general fomt as (3).
Theorem 3 wi·th 8 = 1 applied to the double-slit experiment under
ass~mption (6) gives
where g is the state function of the screen. Hence, if g is as

depicted in Figure 3, the bases of the triangles would have to exceed
This does no·t help much since hd/Ap0 >> 1 in practice
so that the period of the interference pattern hd/2Ap0 greatly
>,
exceeds (hd/2TIAp0) 2 •
We can do better with Theorem 2. Taking t 2 , e ~ and E
the base of one of the triangles,
"" (JE
Jg2)~ k k
(~) 2 "' ~ Km(E) 4
11y4 G12
;!,; k
~ Km(E) 4 (Ap0/hdl"
since supp G s [-Ap0/hd, Ap0/hd] . This implies
4 -1
rn (E) <: (4K ) hd/Ap0
In other words, the lengths of the bases of the triangles in Figure 3
areof the same order as the period of the interference pattern so, at
least, the pattern would be markedly reduced. (One estimate for K
-k
is 1+2(1-26) 2 when t ~ 2[10].)
')
However, this whole approach becomes less useful if g ·· is
made up of many low peaks separated by integer multiples of the wave-
length of the interference pattern.
III. CONSTANCY VERSUS CONCENTRATION
The essence of the family of inequalities in ·this section is that
the more a function is concentrated, the less variable is its
transform. Suppose that f is a one-dimensional state. Given
a E: (0,1) , define
while for S E (0,1) define
w = w (F) ~ min{v
13
a} '
S} .

255
Uffink and Hilgevoord [14] and myself [12] independently obtained
lower bounds for the product Ww ••• with the physicists getting the
sharper result! Work is under way for a paper containing all the
details but the central result is:
THEOREM 4. Suppose a,S E (0,1) • Then
for aZZ states f provided 2a > S+l . Further, if 2a ~ S+l • there
is no positive constant K BO that Wa(f) WS(F) ~ K for all states,
Returning to the double-slit experiment, replace f by G ,
where g is the state function of the screen. By letting a + 1 and
observing that
Supp(h-ljG(•/h) j2 J S [ A /d A /d]
- - Po ' Po '
we arrive at
w6(g) ~ ( (1- Sl /2) ~ hd/TIAp0
for S E (0,1) • This means, for example, that g cannot have
support in disjoint intervals (En):=l where
(i) m(En) ~ Ahd/7TAp0 for all n with 0 < A < 1//2 ,
(ii) the distance between adjacent pairs of intervals exceeds
For if we had such a fm1ction, letting
2
i3 "' l- 2A
In particular, the pdf of position jg 1 2 cannot be supported in
intervals of length Ahd/TIAp0 with centres hd/Ap0 , the period of
the interference pattern, apart.
Ho'VJever, it; is clear that we are still a long way from
0 '
establishing that the interference pattern is destroyed. This will be
done in the next section using Poisson suromation.

256
IV. POISSON SUMMATION
As before, g is the state function of the screen and we assume
that supp n s [-Ap0/d, Ap0/d] , where n is the corresponding pdf of
momentum. Hence
(9) supp G s [-Ap0/hd, Ap0/hd] .
Define g# by
~ lgl 2 (x +nhd/2Ap0) •
nEZ
Considering g# as a function on
# 1
[O, hd/2Ap0 1 , g E L [O, hd/2Ap0 1
since I 2 1
g I E L (JR) • Its Fourier coefficients are:
fhod/2Apo
(2Ap0/hdl
(2Apo/hdl J lgl2<xl
In view of (9) it follows that
exp(-4Tiixk Ap0/hd) dx
Furthermore, lgl 2"
is continuous since lgl 2 E L1 so that it is 0
Hence ck = 0 for all integers k ~ 0 with the
consequence that
(almost everywhere)
Hence, modulo the period of the interference pattern, hd/2Ap0 g is
a constant almost everywhere. In general terms this means that g
kills off the interference terms
2
cos (2TIAp0/hd) (x +A) and
2
cos (2TIAp0/hd) (x -A) announced in (4).
A stronger statement of this phenomenon is obtained by letting
a + 0 • As explained above, the pdf of arrivals at the screen is
given by (7), namely Suppose that the rate of

257
particles passing through the slits is
density distribution of arrivals at the screen. In other words, the
expected number of arrivals in a set E is J a-14>
E g
Working
within s• , the space of tempered distributions, we show that
(10)
-1
lim a 4>g = 2p0/hd (= lim
a+O+ a+O+
for all states g satisfying (6) The effect of letting a tend to
0 is to remove the influence of the sine-terms in 4> since
sine (21fp0/hd)a (x ±A) + 1 as a+ 0 Thus
-1 2 2
a 4> + (2p0/hd) (cos (21TAp0/hd) a(x+ A) + cos (21TAp0/hd) (x -A))
as a+ 0 • Next, since
-1 2
F (o-b + 200 + obl = 4 cos 1fbx , where 0c
is the point measure at c , the Fourier transform of the preceding
limit is
Hence
F-l((lim a-14>)A lgi2A)
1 2A
(p0/hdl F- (cos 21fAy(o_2A /hd + 200 +o2A /hd> lgl > •
Po Po
0 for lxl ~ 2Ap0/hd and lgi 2A(O) = 1 so that
lim a-14>g = (p0/hdl F-1 (2o0>
a+O
as asserted. Thus in the limit as a + 0 , we see that Bohr was
correct when he asserted that knowledge of the paths of the particles
precludes the appearance of an interference.pattern.
REFERENCES
[1] L.B. Biberman, N. Sushkin and V. Fabrikant, 'Diffraction of
individually proceeding electrons', Doklady Acad. Nauk SSSR 66
(1949), 185-186.

258
[2] D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J.,
1951).
[3] N. Bohr, 'Discussions with Einstein on epistemiological problems
in atomic physics', in Albert Eins-tein: Philosopher-Scientist,
ed. P.A. Schilpp (Library of Living Philosophers, Evanston, Ill.,
1949) ' 201-241.
[4] M.G. Covling and J.F. Price, 'Bandwidth versus time
concentration: the Heisenberg-Pauli-Weyl inequality', SIAM J.
Math. AnaL 15 (1984), 151-165.
[5] H. Dym and H.P. McKean, Fourier Series and Integrals (Academic
Press, New York, 1972).
[6] W.G. Faz-is, 'Inequalities and uncertainty principles', J. Math.
Phys. 19 (1978) , 461-466.
[7] R.P. Feynman, R.B. Leighton and M. Sands, The Feynman Lectures
on Physics: Quantum Mechanics (Addison-wesley, 1965).
[8] A. Messiah, Quantum Mechanics, Vol. I (North-Holland, Amsterdam,
1965) 0
[9] R.L. Pfleegor and L. Mandel, 'Interference effec'cs a_t the single
photon level', Phys. Lett. A (1967), 76fr767.
[10] J.F. Price, 'Inequalities and local uncertainty principles',
J. Math. Phys. 24 (1983), 1711-1714.
[11] J.F. Price, 'Sharp local uncertainty inequalities' (submitted).
[12] J·.F. Price, 'Posit:ion versus momentum', Phys. Lett. A.}i2.5.(1984) ,343-345.
[13] J .B.M. Uffink and ,J. Hilgevoord, 'Uncertainty principle and
uncertainty relations" (submitted).
[14] J.B.M. Uffink and J. Hilgevoord, 'New bounds for the uncertainty
principle', Phys. Lett. A (to appear).
School of Mathematics
University of New South Wales
P_o. Box 1
Kensington NSW 2033
AUSTRALIA

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