Volume 137, number 4,5 PHYSICS LETTERS A 15 May 1989

T R A N S L A T I O N W I D T H AND T H E SCATI 'ER P R I N C I P L E

Hans MARTENS Department of Theoretical Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 2 December 1988; accepted for publication 15 March 1988 Communicated by J.P. Vigier

A recently derived "uncertainty" relation is shown to consist of two relations, viz. a scatter relation and a relation involving a "translation width". Only the latter was tested in neutron intefferometry and is generalizable to the E-t case. The confusion between the two is the result of an insufficient distinction between theory-dependent and theory-independent relations.

Let f ( x ) and g(x) be non-negative functions satisfying

1 = f ( 0 ) >.f(x)=f(--x),

i g(x) dx=l.

We can then define the following functionals [ 1 ]:

w, [ f (x) ] = the smallest value ofxo >i 0 satisfying

f(xo)<~c¢. (1)

Wp[g(x) ] = the smallest value ofxo >/0 satisfying

yo + x o / 2 t*

3yo | g(x) dx>_.p 3;0 --xo/2

(O~<a, fl~< 1 ) . (2)

Using these functionals, the following "uncertainty" relation for pos i t i on -momentum was recently de- rived [2,3]:

W~ [ I (~ol exp(ixP)I~o) I ] Wa[l(¢Jp)12 ] t> C ( a , r ) ,

(3)

with

h = l , ot~<2fl- 1,

C ( a , r ) = 2 arccos[ (1 +a-P) / /~ l •

(Note that relation (3), like all following relations for the pos i t ion-momentum case, also holds if the roles of p and x are interchanged. )

Relation (3) links an "overall width" (Wa) to a "mean peak width" (w, ) . It was claimed [ 2,3 ] to be an improvement on the more usual scatter relation:

(A2)~) (A2P) >~ ~. (4)

We shall see that this is, strictly speaking, untrue. There are, however, two other relations "der ived" from (3) in a seemingly straightforward way. One of them is an improvement on (4). To arrive at these relations, we shall first examine the measure w~ in (3). That it fails to represent some kind of "fine structure" width for the position probability distri- bution is apparent if one considers the following example:

(~olx) = c exp[ i (x/Sxl ) z_ (X/rX2)2] COS(X/~X3 )

(8x3 < 8x2).

For this state the position "fine structure" has a characteristic size 8x3, independent o f 8x~. I f 3x~ < 3x3, we, however, have

w,[ I (~lexp( ixP)I~0) I ] ~<O(rx~ ) ,

which is not related to the position distribution at all. To analyze the failure o f w, to represent the "fine

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Volume 137, number 4, 5 PHYSICS LETTERS A 15 May 1989

structure" further, three different w's are distinguished:

wax: the "translation width" of the state of the system;

wvp: the "interior width" of the Fourier transform of the momentum probability distribution;

w,: the "fine structure width" of the position prob- ability distribution. These can be expressed mathematically (in quantum mechanics) as

w~=w.[ ; (~olq)(q-xi~o)dql , (5a) - - o c

wvp=w~[l(~lexp(ixP) I~>1] , (5b)

W,=W~[i ( (o lq-x)(q l~P) ldq 1. (5c) - - o o

Of course it xs almost trivial to show that

W,- >/wFp, (6a)

wx>~w~,-, (6b)

w:~ = WFp. (6c)

Physically these relations are, however, not evident, and it is a property peculiar to quantum mechanics that they hold. Their conceptual difference results in a different operationalization:

W~: measure position probability distribution. De- termine w.

wvp: measure momentum probability distribution. Determine w.

wA¢ Suppose the preparation apparatus prepares systems in the state 1~0)(~01. Shift the preparation apparatus a distance 5x. Determine the probability that the system is still in the original state, i.e. mea- sure the observable zi= I~o) (~01. Determine w from the 5x-dependence of this probability.

Now, with the help of (6) we can separate the Hil- gevoord-Uffink relation (3) into three conceptually distinct relations:

wvpW/~[ l (pl~p) I 2] >~C( a, ,6) ,

w, WAl(pl~o) 12 ] >_, C ( a , / ~ ) ,

w~Wa[ l (pl~o > [ 2 ] >~C(a, 8) •

(7)

(8)

(9)

The first of these is a mathematical relation, not a

physical one, in the sense that it is satisfied by any probability distribution, independent of the theory from which it is derived. It is not testable. This does not hold for the relations (6), despite the triviality of their derivation.

Relation (8) is a scatter relation, and it is this re- lation which is very useful for a discussion of the sin- gle and double slit gedanken experiments [2,3 ], un- like (4). In their discussion of these cases Hilgevoord and Uffink used states with a real and positive wave function in position representation. In those cases (3) and (8) are mathematically equivalent. Rela- tion (8) can be compared to (4), and it is indeed stronger in the sense that (4) can be derived from (8) using theory independent mathematics (al- though the lower bound will not be optimal). It is even an improvement [2] on a scatter relation by Slepian, Landau and Pollak (SLP) that is also stronger than (4) [4,5]:

w . w ~ > ~ ¢ ( a , ~ ) . ( lO)

One should however note that, reasoning along these lines, (6a) is stronger than (8). The theory-indepen- dent relation (7) is used in the derivation (6a)--, (8) in a way similar to the way Chebyshev's inequality (also theory-independent) is used in the derivation ( 1 0 ) ~ ( 4 ) [ 1-3]. Relation (6a) is already a scatter relation. The fact that w~-p ~ is used as a measure for momentum scatter may appear somewhat strange at first, but this use of parameters of the Fourier-trans- form of the momentum probability distribution is not uncommon, e.g. for angle variables [6,7 ].

We thus have a situation where a general "scatter principle", stating that position and momentum probability distributions cannot both be arbitrarily narrow, is expressed by many different "scatter re- lations" ( (6a) , (8), (10), (4) and others [8] ) dif- fering in the precise meaning they give to "narrow".

Relation (9) does not connect a functional of the position probability distribution to a functional of the momentum probability distribution, and con- sequently does not express this scatter principle. It is this type of relation which has been tested using neutron interferometry [9,10]. Of course such an experiment can also be construed as a test of (8) through the use of (6b), but since this last relation is not theory independent, such an interpretation would be much less pure. Just as in the case of (8),

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it can be argued that the physics o f (9) is already contained in (6c).

The conceptual difference between (8) and (9) clearly shows if we try to generalize them to the E - t case. Such a generalization might seem to be ap- propriate since "on the elementary level of Fourier analysis [and] in practical applications, which are likewise based on simple Fourier analysis, [there is a symmetry between pos i t ion-momentum and t ime- energy ]" [ 3 ].

Since there is no time-representation, o f the three w's, w~,, wt and wFt, only w~t makes sense. Hence (8) is not generalizable to the E- t case. Relation (9), us- ing the "life t ime" w~t, becomes

w~,Wp[ [ (E[~0) [ 2 ] ~ C ( a , p) . (11 )

We can derive this relation from (7) and wat=wvr. It can be seen to be closely related to the Mandel- s t a m - T a m m relation [ 11 ]

~ (A2~q> >/¼, (12)

with (Heisenberg picture)

z~=inf , O ( A ( t ) } / O t '

A any Hermitian operator. The analogy can be made more explicit if one takes A ( 0 ) = [~0)(~1. In that case it is not difficult to show that

I 0 a r c s i n ( 2 ( A ( t ) ) - 1 ) - i =inft I dt

WAt ~< a rccos (2a 2 - 1 ) "

In both ( 1 1 ) and (12) the time occurs only through the time dependence of the expectation value o f a Hermitian operator, i.e. as a time-shift.

Relations of the type (9), ( 11 ), (12) might be called "shift-scatter relations". The fact that in the E- t case only this kind of relation is derivable, shows that in quantum mechanics there is indeed an asym- metry between pos i t ion-momentum and t ime- energy.

The author whishes to thank W. de Muynck and J. Uffink for stimulating discussions. This work was supported by the Netherlands Organization for the Advancement of Pure Research (N.W.O.).

References

[ 1 ] J. Hilgevoord and J. Uffink, Phys. A Lett. 95 ( 1983 ) 474. [ 2 ] J. Uffink and J. Hilgevoord, Found Phys. 15 ( 1985 ) 925. [3]J. Hilgevoord and J. Uffink, in: Proc. Int. Conf. on

Microphysical reality and quantum description, Urbino (Italy), eds. A. van der Merwe, F, Selleri and G. Tarozzi (Reidel, Dordrecht, 1988 ).

[ 4 ] D. Slepian and H. Pollak, Bell Syst. Techn. J. 40 ( 1961 ) 43. [ 5 ] H. Landau and H. Pollak, Bell Syst. Techn. J. 40 ( 1961 ) 65. [6] P. Carruthers and M. Nieto, Rev. Mod. Phys. 40 (1986)

411. [ 7 ] A. Holevo, Probabilisfic and statistical aspects of quantum

theory (North-Holland, Amsterdam, 1982 ) §§ IV. 4,5. [ 8 ] H. Maassen and J. Uffink, Phys. Rev. Lett. 60 ( 1988 ) 1103. [9] H. Kaiser, S. Werner and E. George, Phys. Rev. Lett. 50

(1983) 560. [ 10] J. Uffink, Phys. Lett. A 108 (1985) 59. [ 11 ] A. Messiah, Quantum mechanics, Vol. I (North-Holland,

Amsterdam, 1962) p. 320.

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