Path integrals, matter waves, and the double slit

Transcription

1 University of Neraska - Lincoln DigitalCommons@University of Neraska - Lincoln Herman Batelaan Pulications Research Papers in Physics and Astronomy 05 Path integrals, matter waves, and the doule Eric R. Jones University of Neraska-Lincoln, eric.ryan.jones@huskers.unl.edu Roger Bach University of Neraska-Lincoln, roger.ach@gmail.com Herman Batelaan University of Neraska-Lincoln, hatelaan@unl.edu Follow this and additional works at: Jones, Eric R.; Bach, Roger; and Batelaan, Herman, "Path integrals, matter waves, and the doule " (05). Herman Batelaan Pulications.. This Article is rought to you for free and open access y the Research Papers in Physics and Astronomy at DigitalCommons@University of Neraska - Lincoln. It has een accepted for inclusion in Herman Batelaan Pulications y an authorized administrator of DigitalCommons@University of Neraska - Lincoln.

2 European Journal of Physics Eur. J. Phys. 36 (05) (0pp) doi:0.088/ /36/6/ Path integrals, matter waves, and the doule Eric R Jones, Roger A Bach and Herman Batelaan Department of Physics and Astronomy, University of Neraska Lincoln, Theodore P. Jorgensen Hall, Lincoln, NE 68588, USA eric.ryan.jones@huskers.unl.edu and hatelaan@unl.edu Received 6 June 05, revised 8 Septemer 05 Accepted for pulication Septemer 05 Pulished 3 Octoer 05 Astract Basic explanations of the doule diffraction phenomenon include a description of waves that emanate from two s and interfere. The locations of the interference minima and maxima are determined y the phase difference of the waves. An optical wave, which has a wavelength l and propagates a distance L, accumulates a phase of pl/ l. A matter wave, also having wavelength l that propagates the same distance L, accumulates a phase of pl / l, which is a factor of two different from the optical case. Nevertheless, in most situations, the phase difference, Dj, for interfering matter waves that propagate distances that differ y DL, is approximately pd L/ l, which is the same value computed in the optical case. The difference etween the matter and optical case hinders conceptual explanations of diffraction from two s ased on the matter optics analogy. In the following article we provide a path integral description for matter waves with a focus on conceptual explanation. A thought experiment is provided to illustrate the validity range of the approximation D j» pd L/ l. Keywords: quantum mechanics, path integrals, doule, matter optics (Some figures may appear in colour only in the online journal). Introduction The presentation of the doule typically egins with a discussion of Young s original experiments [] on the diffraction and interference of light. Demonstrations such as a ripple tank, one of Young s own inventions [], are used to reinforce the concept of wave interference and Huygens principle of the superposition of waves [3]. Figure shows circular waves that impinge on a pair of narrow s having separation d. The s ecome themselves /5/ $ IOP Pulishing Ltd Printed in the UK

3 Eur. J. Phys. 36 (05) Figure. Typical schematic of Young s two- arrangement. The condition for first order constructive interference, d sin ( q) = l, is illustrated. Shown right is recently pulished data for an electron doule interference experiment [4]. new sources of circular waves. The phase associated with a wave is the numer of wavefronts counted along a line with length L, that is, L / l, multiplied y p. Waves interfere constructively when the phase difference is an integer multiple of p. These ideas lead to the familiar condition for constructive interference D L = d sin ( q) = nl, ( ) where n indicates the diffraction order that occurs at the diffraction angle q. This analysis represented in figure, where a phase pl/ l is prescried to a path of length L, descries what we will herey refer to as the optical analogy. Even though this approach is correct for light, it is not for matter. The first prolem is that it uses an incorrect phase, pl/ l db, for a matter wave (where l db is the de Broglie wavelength). The second prolem is that the use of the optical analogy will nevertheless give the correct phase difference for most situations. In this article, the optical analogy and its limits of validity are discussed for matter waves. The analogy is also compared to a stationary phase method motivated y the path integral formalism. The path integral formalism is shown in section to give a single path phase of pl / l db, and a phase difference etween two interfering paths that is approximately pd L/ l db. The phase difference in optics, pd L/ l, is the same. The space time formulation of the path integral formalism [5] descries paths starting and ending at fixed positions, of varying length, over a fixed time interval for interfering paths. Therefore, the path integral formalism assigns different speeds (and thus wavelengths) to such different paths. This appears to e inconsistent with the idea that a doule is illuminated with a wave descried y one speed or wavelength. This apparent inconsistency will e clarified in the next sections. In sections 3 5, the relation to the optical case, wave mechanics, and timedependence is discussed, respectively, and it is demonstrated that the optical case is fundamentally different from the matter case, despite their similarities. Section 6 discusses how the uncertainty principle relates to the path integral results. In sections 7 through, a stationary phase argument completes the justification for the path selections made in section. At this point, it may appear that apart from some conceptual details, the optical analogy s validity can

4 Eur. J. Phys. 36 (05) e justified y the path integral method. In section, a thought experiment is discussed for which the optical analogy predicts phase differences that disagree with the path integral method, with the purpose to illustrate that the optical analogy agrees only approximatively.. Determining the phase from the path integral Feynman developed a method [5] to construct solutions to Schrödinger s equation [6] ased on Dirac s oservations on the relationship etween the evolution of quantum states etween points in space time and the classical notion of particle trajectories [7]. In Feynman s path integral formalism, a proaility amplitude is determined from a phase, jpath, computed along a particular path connecting two space time events. The total proaility amplitude K ( ; a) for finding a particle at location x at time t, having started at location x a at the earlier time t a, is given y the sum å K ( ; a) = const exp ( i j ), ( ) all paths a path jpath where all paths connect events a and [8]. The phase accumulated along any path is given y j path = L( x, x, t) d t, ( 3) ò path where L is the Lagrangian function, which depends on the positions, x, speeds, x, and times, t, along the path. This path integral method is used to efficiently descrie experimental results for matter interferometry [9], for example the doule experiment for electrons (see the supplementary material of [4], availale online). For the case of a particle moving in free space in D, the Lagrangian is m L x, x, t x ( ) =. ( 4) In free space, the path integral accumulated phase, j = path m x x - a m ( x - xa) ( t - ta) =, ( 5) t - t t - t a a is a function of the endpoints only. Consider xa = 0 and x = x, with ta = 0 and t = t as a fixed time. When jpath p, the distance etween points in space where the phase varies y p approaches a constant value l, determined from (5) y m ( x + l) mx m xl p = ( 6) t t t where it is assumed x l. The value of l follows from (6) as l = ht/ mx, or in terms of the average speed v = x/ t as l = h/ mv. In free space, the classical momentum is p = mv, so l can e identified with the de Broglie wavelength defined l db º h/ p. By sustituting x - xa = L and the aove definitions into (5), the path integral accumulated phase is thus pl jpath =, ( 7 ) l db which followed from the discussion aove taken from Feynman and His [8] (in particular, (3.7) (3.0)). 3

5 Eur. J. Phys. 36 (05) Now consider a doule illuminated with a matter wave that is characterized y one speed. This system can e qualitatively descried y the interference of two paths, represented y the dashed lines of figure. A reasonale assumption would e that the speed, and thus the de Broglie wavelength, is the same for oth paths, so the phase difference Dj for paths of lengths L A and L B from (7) is p D j = jb - ja = ( LB - L A). ( 8) ldb This result is incompatile with the phase difference otained from the optical analogy ecause it differs y a factor of. This result is also incompatile with experiment, which agrees with the optical analogy. This is fine ecause it is indeed incorrect; the false assumption made is that the speeds along oth paths are the same. Even though this result may appear quite surprising at first, the presence of a distriution of momenta, and hence multiple speeds, is familiar in the description of wave packets (see section 4). The correct result according the path integral formalism can only e recovered y noting that interfering paths have equal durations Dt in time; they oth must egin at a and end at, as expressed in (). Because L A and L B are not equal, the consequence is that paths A and B have different speeds. The corresponding de Broglie wavelengths for paths A and B are then h t l A,B = D. ( 9) ml A,B The path length difference dl etween the two paths is taken to e small in comparison to the path lengths L A,B. For LA < L B, the de Broglie wavelength can e expanded as h t L L d L B O d d D - la -. ( 0) ml A LA L A LA Terms of order O [( d L/ L A ) ] are neglected. The phase difference etween the two paths is D j = j - B A plb dl la - L A pla p LB - L l l A ( A) A. ( ) Thus, the correct result is recovered and justified y the path integral formalism of quantum mechanics. These results can e applied in an undergraduate course utilizing the level of description indicated in (). 3. Comparison to the optical case The question may arise why the analogous situation of two diffraction for light presents no conceptual difficulty. The use of straight paths in figure for light could e justified y the application of Fermat s principle of least time [0]. These paths are called rays in the geometrical optics formulation of light propagation [ 3]. Rays are constructed from the normals of a succession of electromagnetic wave-fronts. Each ray is associated with a phase called the eikonal f that is calculated in a homogeneous medium as ò f = k dl = wd t. ( ) ray ò ray 4

6 Eur. J. Phys. 36 (05) Figure. Matter wave propagation. Three snapshots in time of the evolution of a Gaussian wave packet are shown in (a) (c). A point on the carrier wave, shown as a lue dot, moves to the right at the phase velocity v p, indicated with the lue line. The centre of the envelope, which is synchronous with the lue dot at (a), moves at the group velocity vg = vp, indicated with the lack line. A pulse that propagates a length L accumulates a phase j = kl - wt. The angular frequency is given y w = kv p, while the propagation time is given y t = L/ v G. Sustitution gives that j = pl / l db, which differs from the optical analogy of counting waves along the distance multiplied y p, indicated y the lue dot in (c). For light, this phase has the value ò k dl = kl = pl/ l along a ray. This justifies the optical analogy of counting wave-fronts along the propagation paths as in the still pictures of figure. The equality of the two integrals in () implies (note dl/ d t = c) that the dispersion relation for light is linear: w = k c. ( 3) The dispersion relation determines the group and phase velocities. For light propagating in free space, these velocities are the same. Matter wave propagation is different from light ecause it has a quadratic dispersion relation [4] and the group and phase velocities are not 5

7 Eur. J. Phys. 36 (05) the same. The connection etween speed and phase for matter waves is discussed in the following section. 4. Determining the phase from the wave description: the motion picture Consider the motion of a one dimensional electron wave packet, illustrated in figure. This superposition of waves y ( x, t) is given y the Fourier transform of the momentum distriution f ( k - k 0 ) of the constituent waves in the group: ò ( 0) y ( x, t) = f k - k exp ( i[ kx - w ( k) t]) d k. ( 4) - For a Gaussian momentum distriution f ( k - k ) = exp[ -( k - k ) / ( Dk) ] with width Dk and a dispersion relation w ( k) = k / m, the wave packet y ( x, t) is then approximately given y ( k) k0 x t kx k t x m t y(, ) µ exp ( i[ - w( ) ]) exp - D - k0 k x m t ( k) k0 exp i 0 exp x m t = -. ( 5) - D A typical matter wave packet s width spreads and its frequency components disperse as time evolves; however, for sufficiently short times this spreading and dispersion can e neglected. The real part of (5) is illustrated in figure for three times [5, 6]. The first exponential factor, represented y a dashed grey line in figure, is a sinusoidal carrier wave travelling with the phase velocity v P w ( k) k0 º =. ( 6) k m k= k 0 The second factor is the Gaussian envelope, represented y a solid lack line, whose centre (lack dots) travels twice as fast as the sinusoidal wave at the group velocity v G w ( k) º k k= k 0 k0 = = vp. ( 7) m The group velocity is identified with the particle speed and determines the de Broglie wavelength. Suppose that the wave packet in figure travels a distance L in a time Dt. The connection etween L and Dt is determined y the motion of the centre of the Gaussian envelope as k0 L = Dt. ( 8) m Sustituting this relationship into the phase argument of the carrier wave in (5) gives an accumulated phase j of k0l pl j = =. ( 9) l 0 Thus the phase accumulated y a matter wave packet moving from one position to another follows from inspecting a time-dependent solution, and not from the time-independent part alone. Note that this result agrees with the single path accumulated phase derived in (7), ut it cannot lead to the correct phase difference etween two paths () ecause only one group 6

8 Eur. J. Phys. 36 (05) velocity is present. The results in this section could e discussed in an introductory course in modern physics or undergraduate courses in quantum mechanics. 5. The time-independent and time-dependent Schrödinger equations The fact that none of the experimental parts of a doule experiment changes over time suggests inspecting a steady state solution. Consider the time-dependent Schrödinger equation, - y Vy i y - =. ( 0) m t When the potential V in the Schrödinger equation does not depend on time, then the time-independent equation is derived from the time-dependent equation y separation of variales and division y the common factor exp( -i wt). This results in the time-independent Schrödinger equation, ( E - V) j + j = 0. ( ) m The factor ( E - V) / m can e defined as k to give the Helmholtz equation, j + k j = 0, ( ) the solutions of which also descrie the steady state solutions for optics. This well-known analogy is a defining property of the field of matter optics (see section. of [4]). Solutions to the Helmholtz equation with the same energy values (and thus k-values for free space solutions) can e summed to construct a new solution, and thus the superposition principle holds. The solutions in the case illustrated in figure have the simple form j () r = C exp ( i kr) r. The circles in figure can then e thought of as depicting the wave fronts of the real part of the circular waves ja ( ra ) and jb( rb) that are the solutions to the Helmholtz equation. The proaility to find a particle at a position x on the detection screen is then given y the Born rule, y( x, t) = ja [ ra ()] x + jb[ rb ()] x. The result is timeindependent ecause the time-dependent factor exp -iwt was factored out of the wave function. Using the lengths ra,b = LA,B = na,bldb for each dashed straight line in figure leads immediately to the condition D L = nl at maxima in the proaility distriution. It is then reasonale to question why the optical analogy is not sufficient to return to a time-dependent description of the doule experiment for matter. After all, it appears that we could recover the time-dependent description y multiplying the stationary solutions ja and j B with the factor exp( -i wt). Let us associate with the waves, for a fixed energy E, the kinematic speed as given y v = E/ m. The propagation time t along any direction is then t = r/ v. This leads to the phase kr - wt = kr/ evolving from either to the detection screen, as in (8), when the factor exp( -iwt) is added ack to the wavefunction. The phase difference at a detection point would then e ( krb - kra) / = p( rb - ra) / ldb, as in (8), which is incorrect. The correct use of time-dependent formalisms avoids this discrepancy, as shown in section. It is also reasonale to attack the argument aove as eing too simplistic. After all, the full solution of the Helmholtz equation in the diffraction prolem involves Green s functions for the prolem [7], and Green s functions at one energy for oth light and matter are the same, as they solve the same Helmholtz equation. This would give the correct spatial patterns for matter and light, and the full time-dependent results could e otained via Fourier transform. However, the time-dependent results would still not e generally correct for matter. Because matter waves are 7

9 Eur. J. Phys. 36 (05) dispersive in free space, they experience the phenomenon of diffraction in time at a [8 0]. This effect, which has een realized experimentally [, ], requires careful consideration of Green s function for the time-dependent Schrödinger equation. Diffraction in time changes the time-dependence of the resulting proaility distriution. On the other hand, light is not dispersive in free space,so there is no diffraction in time for light [9]. Essential for our discussion is that we confirm that the spatial dependence of diffraction also changes for certain stationary configurations (see (57) of [0]). We give a simulation of an experimental configuration where these differences can e distinguished in section. The results of this section could e discussed in an advanced course in quantum mechanics, when students have had prior experience with partial differential equations. 6. The Heisenerg uncertainty principle and path measurements It has een estalished from section that if oth paths from the s to the screen have the same duration in time, the speeds associated with the two interfering paths will e different. These different speeds give rise to slightly different kinetic energies. Is it then possile that the actual path a particle travelled could e determined y performing an energy measurement at a particular point on the detection screen? If this would e possile, and a diffraction pattern would e present, then this would e a which-way detector and violate quantum mechanics. In the following, it is shown that this is not possile, as it violates Heisenerg s uncertainty principle. As usual, the uncertainty relation protects quantum mechanics. In order for interference to occur, the wave-fronts associated with the particles must arrive coherently at the detection screen. Thus, waves leading to interference at the point of detection can experience a variation in phase difference of no more than p; otherwise, the interference contrast averages out. Representing the variation in phase difference as df, the condition for interference is then df p. ( 3) The path integral phase difference is defined from (5) as mlb mla f = -, ( 4) t t where t = t - t a. The variation of phase difference df related to a first-order variation of the time t is given y f mlb mla df =. ( 5) t dt =- - dt t t The quantity in parentheses is the difference in kinetic energy of the paths, defined as DE. The condition for interference (3) leads to the inequality D h E dt. ( 6) The energy resolution necessary to distinguish which path a particle takes on its way from the s to the detection point must e smaller than the kinetic energy difference DE etween the two interfering paths. The necessary timing resolution on the energy measurement according to (6) would thus violate the Heisenerg uncertainty principle. Therefore, preserving phase coherence protects the inaility to distinguish through which a particle will pass on its way 8

10 Eur. J. Phys. 36 (05) to the detection point. A similar argument can e made for paths defined y fixed energy and varying arrival times. The preceding argument for the preservation of phase coherence for energy and time measurements can also e expressed in terms of momentum and position. The condition for interference is again given y (3). The variation of the phase difference df is determined y the variation in the measurement of the momentum dp as df f = d p, 7 p ( ) or, in terms of the de Broglie wavelength, p = h / l, and the phase difference (), f L df db, 8 db db db ( ) l dl p = =- D dl l where ldb = la. The variation in the momentum p is related to a variation in the wavelength as hdldb dp =-. ( 9) ldb Sustituting (8) and (9) into (3) gives the inequality D h Lp d. ( 30) The length resolution necessary to distinguish which path a particle takes on its way from the s to the detection point must e smaller than the length difference DL etween the two interfering paths. The necessary momentum resolution on the length measurement according to (6) would thus violate the Heisenerg uncertainty principle. Therefore, a which-way measurement, that also maintains a diffraction pattern, is not possile. The level of argument presented here is appropriate in any undergraduate course presented with the Heisenerg uncertainty principle. 7. Revisiting the path integral propagator in free space In section, we restricted the discussion of path integral phase differences in a qualitative description of the doule to a particular selection of two paths. However, the full path integral description for the doule calls for a summation of proaility amplitudes over all possile paths in space and time etween fixed events, not just the particular selection. In the following, section 7, we riefly review the free-space D propagator in the path integral formalism. In section 8, the free-space propagator is applied to the two-step event of crossing a at one particular time. In section 9, the sum over all crossing times is shown to converge to the choice of a particular time for each path in space. In section 0, the particular choice of time for each -crossing path is motivated y a stationary phase argument. In section, the stationary phase argument is illustrated in the two-path description of the doule, as in figure. In section, the results from the stationary phase argument and a path integral sum over times are compared to the optical analogy in a near-field arrangement where paths are summed over the entire extent of the s. The proaility amplitude for a particle to travel in free space from space time event a, denoted ( x0, t0), to event, denoted ( xn, tn), in a numer N evenly spaced time intervals, is given in (3.) of [8] as 9

11 Eur. J. Phys. 36 (05) Figure 3. Feynman paths for D free particle propagation. A general path (dotted line) is shown for a point particle that travels from space time point ( xa, ta) to ( x, t). The locations x that the path crosses (indicated for times separated y ) can y varied along the x-axis. The classical path is indicated with the old dark line. Feynman showed that the total amplitude for motion from a to summed over all paths (y integrating over the x-locations) is identical to the amplitude computed along the classical path alone, which is a central result from the path integral formalism [5, 8]. N N m im K ( ; a) = lim exp ( xj xj ) dx d xn. ( 3) 0 i ò ¼ò å - - ¼ - p Feynman points out that the resulting nested Gaussian integrals can e performed iteratively, leading to the result j= m K ; im ( ) exp ( i N N x N x 0) a = -. ( 3) p When the suscripts 0 and N are associated with their space time events a and, and the total time N is replaced with the time difference, t - t a, then (3) ecomes m im K ( ; ) exp x x. 33 i ( t t ) ( t t ) ( ) a = - a p - - ( ) a a This result, which is readily generalized to higher dimensions, shows that the amplitude associated with the summation over all possile paths connecting two events in free space is equal to the amplitude associated with the classical path alone, as sketched in figure 3. The formal path integral discussion of this section is suitale for graduate coursework in quantum mechanics. 8. Two-step propagator for a As the next step towards descriing the doule, consider the amplitude for an electron path intersecting a single. This path is descried y three space time events, laelled as 0

12 Eur. J. Phys. 36 (05) follows: the source, defined as the event a; the crossing, denoted y ; and the measurement at the screen,. The amplitude for such a path can e constructed as the product of the amplitude for two steps. The first step is to reach the from a, and the second step is to travel from the to [3, 4]. Applying the result of (33) to this case, one otains K( ; a) = K( ; ) K( ; a) m im exp x x i ( t t) ( t t) ( ) = - p - - m im exp x x i ( t t ) ( t t ) ( ) - a p - a - a m iml ( t t ) - a = exp, ( 34) pi ( t - t) ( t - ta) ( t - t) ( t - ta) where the sustitution ( x - xa )/ = L was made. We note that this is an approximation: an exact construction for the propagator would take into account the oundary conditions set y the walls. The integrations from - to in (3) include paths that pass through the walls; therefore, the propagator in (34) adds extraneous paths to the sum. The times t a and t defining the oundaries of this path are fixed, ut the -crossing time, t, is not. It is not a measured event in the same sense as a or and thus cannot e specified. The total amplitude to cross the, K ( ; a), is then a sum over all of the amplitudes having every possile value of t [0]. 9. Time summed amplitude for two-step propagator To otain the total amplitude to cross the, K ( ; a), consider the sum of the products K( ; ) K( ; a) of (34) for every value of t occurring etween t a and t. The result is written as K( ; a ) = K( x, t ; x, t ) K( x, t ; x, t ). ( 35) t t å t = a a a For more detail, see the derivation of (35) in appendix A. This formally estalishes the sum over intermediate times that is required from the sum over all paths given in (3). The sum in (35) over the continuous value t is proportional to the integral I ( ; a), given y t m iml t I ( ; a) = ò t dt exp. ( 36) - p i t t t t t t t t Here, ta = 0, ( x - xa )/ = L as efore, and the variale time t at the has een defined to give a symmetric integrand. This integral is derived and evaluated in greater detail in appendix B. The result, given in terms of the complementary error function, erfc ()[5], z is m I ( ; a) = erfc( i i 0) ( 37) pi p - j

13 Eur. J. Phys. 36 (05) with j º ml / t. An asymptotic expansion given y (7..3) of [5] gives, for (37), 0 Figure 4. Numerical time integration for a single electron path. (a) Real part of (36) (solid lue), integrated from -D t/ to D t/ relative to t /, showing convergence to the analytic value given in (37) (dashed line). () Complex argument of the integrated amplitude (solid red), showing convergence to p/ 4 phase shift (dashed line) from the argument of exp ij 0 (dotted lack). The amplitude proportional to exp ij0 is therefore the appropriate choice of a single amplitude to characterize the entire sum over time. m p p i 3 5i I ( ; a)» exp ( ij0) exp i. ( 38) i p j j 4j 8j 0 0 The form of (38) illustrates that the total integrated amplitude experiences a phase shift of p/ 4 from the phase j0. This phase difference is independent of the choice of path and thus the gloal phase p/ 4 can e factored out. Figure 4(a) shows the convergence of the real part of the numerical evaluation of (36) (lue curve) to the real part of the analytic result of (37) (lack dashed), for L = m and t / = - s. The numeric results are computed for variale limits of integration and plotted as a function of the total time interval eing integrated. Figure 4() gives the phase argument of the numeric results (red curve) to show the convergence of the rotation from the initial phase argument given y j0, which is defined y choice of the parameters to e p (lack dotted), to an angle of - 3p/ 4 (lack dashed). This estalishes the appropriate choice of amplitude for a path crossing one. The 0

14 Eur. J. Phys. 36 (05) physical meaning of j0 will now e discussed. The level of difficulty of this, the previous section, and accompanying appendices A and B is more appropriate for advanced graduate coursework. 0. Stationary phase for the two step propagator The phase j accumulated along a path crossing a is determined from (5) to e m L L j = t + t - t, ( 39) where t = t - t a, L is the path length from source to, and L is the length from to screen. When t is varied y dt, the phase can e expanded as a power series in dt as j j j = j0 + d t + ( d t) +, ( 40) t t where j0 is associated with a particular choice of t. The first-order term of (40) is written out j t m L = ( t - t ) L - t. ( 4) The factor j/ t = 0 when L /( t - t) = L / t : that is, when the speeds along the path are equal efore and after the. The phase j will then experience no first-order variation from j0 when t is chosen y this condition. We then say that the phase is stationary for this choice of path, and the value of the stationary phase is j0. As shown in section 9, thisphase characterizes the amplitude arising from the sum of choosing all values of t ; therefore, it is the appropriate choice for a single path. The phase in terms of the de Broglie wavelengths along this single path is now estalished pl / ldb + pl / ldb = pl path / ldb. The argument presented here should e appropriate for advanced undergraduates, in what could e a simplified discussion of which paths to choose in the path integral formalism.. Stationary phase in the doule Figure 5(a) shows two interfering paths (green and red) in a space time diagram for the doule. The times t and t, when paths and intersect the s, respectively, take any value etween the initial time t initial and final time t final. The proaility distriution at the screen is shown to the right of the screen as an intensity plot. In figure 5(), a phasor diagram of the complex amplitudes for the varying times t and t is shown. The highlighted paths in figures 5(a) and () are the paths of stationary phase. In figure 5(c), the phases corresponding to the amplitudes in () are given as a function of time to illustrate the stationary phase ehaviour. Notice that the stationary phase time for path, indicated y the largest red dot in figure 5(a), occurs after the stationary phase time for path. The reason is that the length of path (that is, the length of the dashed line in the x y plane) is shorter than the length of path. As the initial time and final times are the same for oth paths, the speeds of the paths are different. The equal length of the part of oth paths etween the source and s explains the difference in the stationary phase times for this example. For some other path integral 3

15 Eur. J. Phys. 36 (05) Figure 5. Path integral illustration for destructive interference in a doule. (a) The times t and t at which the paths intersect the s take any value etween the initial time t initial and final time t final. The resulting proaility distriution at the screen is the square of the sum of the amplitudes for all of the times t and t. () Shown is a phasor diagram for complex amplitudes associated with the intermediate times for (red) and (green). The highlighted paths in () are the paths of stationary phase shown in (a). (c) The phases corresponding to the amplitudes in () are shown as a function of intermediate time to illustrate the stationary phase ehaviour. calculations, the crossing times are chosen to e identical for all paths [4, 3, 4], while for the optical analogy, the times are the same for paths of the same length from source to.. Phase matters Does the optical analogy and the path integral method make the same predictions? In other words: does the phase of a single path matter? After all, real experiments are only sensitive 4

16 Eur. J. Phys. 36 (05) Figure 6. Feynman paths and proaility distriution from all paths in a near-field two arrangement. The source is positioned in line with one of the s and the detection point. The separation d, propagation length L, and speed are chosen to highlight the discrepancy etween the predictions of the two methods. The normalized proaility distriution functions at the screen are computed with the path integral stationary phases (lue), the time-summed amplitudes (green points), and the optical analogy (dashed red). to phase differences, which were shown to agree for the optical analogy and the path integral formalism in (). This agreement is not always the case. Consider now the doule arrangement in figure 6, where the electron source and oservation point are in-line with one of the s. Computing the phases of the drawn paths y (5) (dashed lines) leads to a phase difference D md jpath integral =. ( 4) D t This result does not depend on the length L in this configuration. If instead we use the optical analogy, we compute the phase difference md md p D j L d L optical = + - l () v D t D tl db ( ) 4, ( 43) where the time and speed etween the two methods are connected y v = L/ Dt. The phase difference in (43) now depends on L. The phase difference thus depends on the choice of method used for single path phases. When d / 4L ~ p Dt/ md, (43) conflicts with (4), and thus the choice of method matters. To est exemplify this conflict, let us now choose the experimental conditions so that the common term of (4) and (43) is set to an integer multiple of p, and the second term of (43) set to p. Now, (4) predicts constructive interference in line with the, while (43) predicts destructive interference. For electron diffraction in the symmetric doule arrangement of figure 6, a separation of 73 nm with widths of 63 nm can e chosen. Note that such a 5

17 Eur. J. Phys. 36 (05) doule has een demonstrated recently for electron diffraction in [4]. In contrast to [4], now the source and screen are placed at the much closer distances of 3.37 μm from the s. When Dt is fixed for the path integral method y choosing the electron speed to e 0 7 ms over the straight path, the difference of p is set etween the predictions of the two methods. The diffraction patterns are computed with oth methods and shown on the right of figure 6. At a location on the detection screen that is in line with the source (at y = 36.5 nm), the path integral method (solid lue) predicts a constructive maximum, while the optical analogy (dashed red) predicts a minimum. A time sum of the form of (35) performed for 6000 points per and five points on the oservation screen over intervals of s centred on the stationary phase time of each path from the source to the screen points (green points) agrees with the lue curve computed with the stationary phase times alone. This configuration is experimentally challenging to realize. Nevertheless, near field interferometry for matter waves does exist and may e pushed towards this regime [6, 7]. In conclusion, phase difference predictions from the optical analogy and the path integral formalism will not agree in some near-field conditions. While the gloal phase of a single path does not matter, to otain correct phase differences, the single path phases must e handled appropriately. This result, which follows from the stationary phase argument of section 0, could e presented in an advanced undergraduate course, ut the technical details of justifying the stationary path results for the simulation would again e a topic for an advanced graduate course. 3. Summary and conclusions The optical analogy can give excellent approximate phase differences in most situations and thus leads to the correct prediction of the positions of interference extrema. This method is justified y considering stationary solutions to the Schrödinger equation. The conceptual trap is that a student may infer from the correct phase difference, pd L/ l, that the phase of a single path is given y pl/ l (as would e correct for optical waves). The path integral description of quantum mechanics gives the correct phase difference pd L/ l etween paths, the correct phase pl/ l accumulated over time along a single path, and justifies drawing paths in space to compute phases. The path integral method (and the time-dependent Schrödinger equation) gives the exact phase difference in all situations. It is therefore an appropriate method to use in conceptual discussions of matter wave diffraction. In some physics textooks, oth paths and waves are omitted from the description of matter wave diffraction. Instead, the discussion refers ack to water waves or Young s experiment for light waves and quotes the condition for interference or phase differences y analogy [8 3]. This presentation is correct to otain phase differences, ut it ignores the differences in propagation, that is, the time dependent ehaviour, etween light and matter waves. Some physics textooks [30 3], as well as some advanced undergraduate and graduate texts, will draw attention to group and phase velocities in sections unrelated to the doule description [5, 33, 34]. It is interesting to contemplate at what level and in what manner the conceptual difficulty discussed in this paper could e addressed. For example, it could follow a discussion of the group and phase velocities of a matter wave packet. The results from the path integral formalism could thus e presented at the undergraduate level [35 37] to elucidate the idea of a path. 6

18 Eur. J. Phys. 36 (05) Acknowledgments The authors gratefully acknowledge discussions with Ron Capelletti, Sam Werner, Mike Snow, Bradley Shadwick, Ilya Farikant, and Mathieu Beau. ERJ, RAB, and HB gratefully acknowledge funding from the NSF (grant numers and ). ERJ also gratefully acknowledges funding from the DOE GAANN programme numer Appendix A. Time sum derivation In the following, the propagator is derived for a single crossing as descried in section 9. In (3), each of the positions x k represent the range of positions a point could have along a path at the kth time step of the sum. Requiring that a path intersects the at x in the kth time step is defined as a multiplication of a term d ( x - x k ) to the integrand. This intersection happens at any time step from the first up to the last, so a factor c N- å = d( x - x ) ( A.) k= k must e included in (3) to descrie all of the alternative times a path can intersect the. Sustituting (A.) into (3) gives the total amplitude to travel from a to as N N m im K ( ; a) = lim exp x x dx dx i ( j j ) N 0 ò ¼ò c å - - ¼ - p j= N- N N m im = lim å exp ( xj xj ) 0 k i ò ¼ò å - - = p j= d ( x - x ) dx ¼d x. ( A.) k N - Performing the N - integrations in (A.) leads to the total sum N- å K( ; a ) = lim K x, t ; x, t + k K x, t + k ; x, t. Î 0 k= ( a ) ( a a a) ( A.3) The sustitution t º t + k is made into (A.3) to otain the final result a K( ; a ) = lim K( x, t ; x, t ) K( x, t ; x, t ) 0 t t t - = t + = K( x, t ; x, t ) K( x, t ; x, t ). ( A.4) t å = t a å a a a a a Appendix B. Evaluating the integral of the full time sum The time sum derived in (A.4) over the continuous value t must e handled carefully near the singular points occurring at t = t and t = t a = 0, so we convert the sum to an integral prior to performing the limit 0 to otain 7

19 Eur. J. Phys. 36 (05) t- I ( ; a) = lim dt 0 ò m iml t exp, ( B.) pi t ( t - t) t ( t - t) ( ) where ( x - xa )/ = L as efore. Next, x = t/ t - / is sustituted to otain - t I x m ml ; lim d exp i ( a) = ò t p i ( + x)( - x) t ( + x)( - x) ( B.) Equation (B.) is simplified y the definition of the stationary phase as j 0 º ml / t as in section 9. The next sustitution to e performed is x = sin ( q). This trigonometric sustitution eliminates the square root, as dx/ - x = d q, and we otain m sin- - t ij 0 I ( ; ) lim a = d exp 0 i sin ò q p - cos ( q) - + t ( ) m sin- - lim t = d exp ( i exp i tan 0 i sin 0) 0 ( ) ò q j j q p -, ( B.3) - + t where the identity sec ( q) = + tan ( q) is used in order to factor out a term exp i j0. Next, we sustitute u = tan ( q) and otain m tan sin- t u I ; lim exp i du exp i - ( 0 ) ( ) ( 0 i 0) j a = j p ò - + u tan sin - + t. ( B.4) The limits of integration are symmetric and now tend to as 0, so they are redefined as tan sin- ( - / t ) = R and tan sin - (- + / t ) = -R, with the limit R. Finally, we extend the integrand into the complex plane y performing the sustitution t =-i ij0 u to otain -i ij0 R m exp (- t ) I ( ; a) = lim exp ( ij0) i ij0 dt. R i ò i i R p - j0 (- ) ( ij0) - t ( B.5) The integrand is analytic everywhere in the complex plane except for first-order poles at i j0, therefore the path of integration, which lies on the line t = R exp ( i3p/ 4 ), can e rotated to lie entirely on the real axis. The integrand s even symmetry then permits m i I ( ; a) = exp ( ij ) i p ij 0 0 ò 0 dt exp (- t ) ( ij0) - t. ( B.6) The term in square rackets of (B.6) has the form of the complex-valued function w ()given z, in (7..4) of [5], as iz t wz dt exp (- ) () = ò. ( B.7) 0 p z - t This function can e readily evaluated from the definitions given in (7..) and (7..3) of [5], as 8

20 Eur. J. Phys. 36 (05) wz () = exp (- z) erfc( -i z), ( B.8) where erfc ()is z the complementary error function. Sustituting (B.7) and (B.8) into (B.6),we otain the result, m I ( ; a) = erfc( i i 0), ( B.9) pi p - j which is (37). References [] Young T 804 The akerian lecture: experiments and calculations relative to physical optics Phil. Trans. R. Soc. A 94 6 [] Young T 807 A Course of Lectures on Natural Philosophy and the Mechanical Arts vol (London: William Savage) pp 464 8, 776 7, [3] Feynman R P, Leighton R B and Sands M L 965 The Feynman Lectures on Physics vol 3 (Reading, MA: Addison-Wesley) pp 6 [4] Bach R, Pope D, Liou S Y and Batelaan H 03 Controlled doule- electron diffraction New J. Phys [5] Feynman R P 948 Space time approach to non-relativistic quantum mechanics Rev. Mod. Phys [6] Schrödinger E 98 Collected Papers on Wave Mechanics 3rd edn (New York: Chelsea) pp 0 4 [7] Dirac P A M 945 On the analogy etween classical and quantum mechanics Rev. Mod. Phys [8] Feynman R P and His A R 005 Quantum Mechanics and Path Integrals emended edn ed D F Styer (Mineola, NY: Dover) pp 6 47 [9] Cronin A D, Schmiedmayer J and Pritchard D E 009 Optics and interferometry with atoms and molecules Rev. Mod. Phys [0] Sara A I 98 Theories of Light from Descartes to Newton (New York: Camridge University Press) pp [] Feynman R P 006 QED: the strange theory of light and matter (Princeton, NJ: Princeton University Press) pp [] Born M and Wolf E 993 Principles of Optics 6th edn (Tarrytown, NY: Pergamon) pp 09 3 [3] Landau L D and Lifshitz E M 975 The Classical Theory of Fields 4th edn (Burlington, MA: Elsevier) pp 40 3 [4] Adams C S, Sigel M and Mlynek J 994 At. Opt. Phys. Rep [5] Griffiths D J 005 Introduction to Quantum Mechanics nd edn (Upper Saddle River, NJ: Pearson Prentice Hall) pp [6] Bohm D 989 Quantum Theory (Mineola, NY: Dover) pp [7] Jackson J D 999 Classical Electrodynamics 3rd edn (Hooken, NJ: Wiley) pp [8] Moshinsky M 95 Diffraction in time Phys. Rev [9] Brukner Č and Zeilinger A 997 Diffraction of matter waves in space and time Phys. Rev. A [0] Beau M and Dorlas T C 04 Three-dimensional quantum diffraction and diffraction in time Int. J. Theor. Phys [] Szriftgiser P, Guéry-Odelin D, Andt M and Daliard J 996 Atomic wave diffraction and interference using temporal s Phys. Rev. Lett [] Hils Th F J, Gähler R, Gläser W, Golu R, Haicht K and Wille P 998 Matter-wave optics in the time domain: results of a cold-neutron experiment Phys. Rev. A [3] Barut A O and Basri S 99 Path integrals and quantum interference Am. J. Phys [4] Beau M 0 Feynman path integral approach to electron diffraction for one and two s: analytical results Eur. J. Phys [5] Aramowitz M and Stegun I A 965 Handook of Mathematical Functions with Formulas, Graphs, and Mathematical Tales 9th edn (Mineola, NY: Dover) pp 97 8 [6] McMorran B J and Cronin A D 009 An electron talot interferometer New J. Phys [7] Bach R, Gronniger G and Batelaan H 03 A charged particle Talot-Lau interferometer and magnetic field sensing Appl. Phys. Lett

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