V O L U M E 58, N U M B E R 21 P H Y S I C A L R E V I E W L E T T E R S 25 M A Y 1987

Comment on '^Longitudinal Coherence in Neutron Interferometry"

In their recent paper Klein, Opat, and Hamilton^ have demonstrated that the longitudinal coherence length Ac of de Broglie wave packets remains unchanged even though the length a^ of the packets increases upon prop­agation, i.e., that the coherence length A is constant in time. The authors furthermore state that A is propor­tional to the minimum length CJX(O), which is true for the Gaussian wave packets considered by them. We have examined the generality of this statement and have calculated the factor of proportionality for various forms of de Broglie wave packets. CTX(O) was defined by cr;c(0) = J ( x — (x))^ I v/(x,0) I '^dx, with normalized V/(x,0), and A^ was defined as that value of Ax where the function

I / (Ax) I = \ Jy/* (x,0)y/{x -^ Ax,0)dx \

has decayed for the first time to exp(— y ) =0.61 of its maximal value ( = 1, and Ax = 0 ) .

The results are summarized in Table I. Any of the functions v^(x,0) may be replaced by y/(x,0)cxp(ikox) without altering the results. The real (or imaginary) parts are then wave trains having the forms of Table I as envelopes. In order to obtain three-dimensional wave packets we may multiply any function i//(x,0) by an ar­bitrary normalized function (t)(y,z,t) and let the in­tegrals in the definitions of <jjc(0) and A go over all three dimensions. We obtain again the results of Table I.

Table I shows that usually there is proportionality, and the proportionality factor is between 1 and 3. However, there exist cases where this does not hold, that is, where A and cJxiO) do not depend on the same combination of parameters that are used in the mathematical description

of the wave packet. One example is the wave packet consisting of two well separated peaks. There, A is close to the width of the narrower of the two peaks, whereas GxiO) is half the distance between the peaks. Thus, if we double OxiO), A remains unchanged.

One might suspect that this disproportionality is due to the somewhat different manner in which the widths were defined [Gaussian width and exp(— y ) width, re­spectively]. It is not difficult, however, to convince one­self that the disproportionality does not depend on the details in the definition of the width. Thus, not only do A (a constant) and cjx (a function of time) behave differently under time evolution, as shown by Klein, Opat, and Hamilton,^ but they are, in general, complete­ly different quantities because of the intrinsically different properties of the wave function y/ix.O) and the autocorrelation function / (Ax) , as pointed out by Hil-gevoord and Uffink.^

Arthur Jabs Federal University of Paraiba Joao Pessoa, Brazil

Romi ldo R a m o s Federal University of Mato Grosso Cuiaba, Brazil

Received 24 February 1987 PACS numbers: 03.65.Bz, 42 .50 . -p

• A . G . Klein, G. I. Opat, and W. A. Hamilton, Phys. Rev. Lett. 50, 563 (1983).

2j. Hilgevoord and J. B. M. Uffink, Phys. Lett. 95A, 474 (1983), and Eur. J. Phys. 6, 165 (1985); J. B. M. Uffink, Phys. Lett. 108A, 59 (1985); J. B. M. Uffink and J. Hilgevoord, Phys. Lett. 105A, 176 (1984), and Found. Phys. 15, 925 (1985).

TABLE I. Coherence length Ac and minimum length OxiO) for various forms of the wave packet y/(x,0).

Name V/(x,0) Ac aAO) Ac/oAO)

Exponential

Rectangular

Gaussian Lorentzian Two well

separated peaks

(2/a)^^^Q\p{-x/a) f o r x > 0 , 0 elsewhere a ~^'^ for X E [xo,xo + ^ ] , 0 elsewhere (2 / ; r«2) i /4^^p(_^2/^2)

( 2 / ; r a ) ' / 2 ( l + x V « ' ) - ' ( 2 ; r ) - » / M ^ r ^ / 2 e x p ( - x V a ? )

+ ^ 2 ' ^ / ' e x p [ - ( x - ^ ) 7 ^ ^ 1 } , A ^ax.ai

all

(1 - i / V e ) a

a l{e-\yl^a > min(ai ,^2) < [-21n(2/V^-

xmin(£3[i,fl2) -1)1 1/2

all

al^Vi

all {nl^yl^a All

1

1.36

2 2.78 > 2 m i n ( a i , a 2 ) / ^ < 3.5 min((3i,a2)/^

2274 © 1987 The American Physical Society