Comment on "Longitudinal Coherence in Neutron Interferometry"

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<ul><li><p>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 </p><p>Comment on '^Longitudinal Coherence in Neutron Interferometry" </p><p>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 prop-agation, i.e., that the coherence length A is constant in time. The authors furthermore state that A is propor-tional 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 </p><p>I / (Ax) I = \ Jy/* (x,0)y/{x -^ Ax,0)dx \ </p><p>has decayed for the first time to exp( y ) =0.61 of its maximal value ( = 1, and Ax = 0 ) . </p><p>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 ar-bitrary normalized function (t)(y,z,t) and let the in-tegrals in the definitions of 0 , 0 elsewhere a ~^'^ for X E [xo,xo + ^ ] , 0 elsewhere (2 / ; r2) i /4^^p(_^2/^2) </p><p>( 2 / ; r a ) ' / 2 ( l + x V ' ) - ' ( 2 ; r ) - / M ^ r ^ / 2 e x p ( - x V a ? ) </p><p>+ ^ 2 ' ^ / ' e x p [ - ( x - ^ ) 7 ^ ^ 1 } , A ^ax.ai </p><p>all </p><p>(1 - i / V e ) a </p><p>a l{e-\yl^a &gt; min(ai ,^2) &lt; [-21n(2/V^-</p><p>xmin(3[i,fl2) -1)1 1/2 </p><p>all </p><p>al^Vi </p><p>all {nl^yl^a All </p><p>1 </p><p>1.36 </p><p>2 2.78 &gt; 2 m i n ( a i , a 2 ) / ^ &lt; 3.5 min((3i,a2)/^ </p><p>2274 1987 The American Physical Society </p></li></ul>