Boughn, Stephen P.

Rothman, Tony

English

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Haverford CollegeHaverford ScholarshipFaculty Publications Astronomy2011'Quantum interference with slits' revisitedTony RothmanStephen P. BoughnHaverford College, sboughn@haverford.eduFollow this and additional works at: http://scholarship.haverford.edu/astronomy_facpubsThis Journal Article is brought to you for free and open access by the Astronomy at Haverford Scholarship. It has been accepted for inclusion in FacultyPublications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu.Repository CitationBoughn, Stephen. "’Quantum interference with slits’ revisited” (with T. Rothman), European Journal of Physics 32, 107-113 (2011).arXiv:1009.2408v1  [quant-ph]  13 Sep 2010 “Quantum Interference with Slits” RevisitedTony Rothman∗ and Stephen Boughn†∗ Princeton University,Princeton, NJ 08544† Haverford College,Haverford PA, 09140LATEX-ed September 14, 2010AbstractMarcella has presented a straightforward technique employing the Diracformalism to calculate single- and double-slit interference patterns. He claimsthat no reference is made to classical optics or scattering theory and that hismethod therefore provides a purely quantum mechanical description of theseexperiments. He also presents his calculation as if no approximations are em-ployed. We show that he implicitly makes the same approximations foundin classical treatments of interference and that no new physics has been intro-duced. At the same time, some of the quantum mechanical arguments Marcellagives are, at best, misleading.PACS: 3.65.–w, 03.65Ca, 3.65.TA, 01.70.+wKeywords: Quantum mechanics, two-slit experiment, interference, scatteringtheoryIn a 2002 paper Marcella[1] pointed out that quantum mechanics textbooks typi-cally “finesse” discussion of the famous double-slit interference experiment by resort-ing to analogies from classical wave optics in order to obtain the standard results. Hegoes on to present a simple method of calculating a single- or double-slit interferencepattern via the Dirac formalism and claims that this provides a quantum description∗trothman@princeton.edu†sboughn@haverford.edu1Quantum Interference... 2of the experiment, moreover one that should be accessible to upper-level undergrad-uates. It is true that the usual textbook discussions of the two-slit experiment areriddled with ambiguities and omissions[2]; and this is for the reason that a consistent,quantum-mechanical treatment of the two-slit experiment does not, to our knowledge,exist. Such a treatment would necessarily introduce an interaction between the inci-dent particle beam and slits that results in a probabilistic distribution of particles onthe observation screen. The Hamiltonian governing this interaction would, in general,be quite complicated, and it is far from clear that even in principle it could resolvethe “measurement problem,” the abrupt change of a deterministic system to a prob-abilistic one at the instant of measurement. One might therefore question whetherMarcella’s approach, which includes no explicit interactions—in particular no solu-tion to a wave equation—can claim to give an adequate accounting of the two-slitexperiment. Furthermore, at first glance he makes makes virtually no assumptionswhatsoever and makes no reference to any approximations, leaving the impressionthat his result is exact. We now show that he has merely carried out the same clas-sical interference calculation that students encounter in first- or second-year physics,and with similar assumptions.slitsysourcepθFigure 1: Marcella’s setup for the one-and two-slit experiment. After passing through the slit(s),the particle is assumed to be scattered at an angle θ with a momentum p and a y-momentum ofpy = p sin θ. Note that θ is measured from the centerline.Quantum Interference... 3We focus on the single-slit experiment; the double-slit goes through in exactly thesame manner. The following is essentially Marcella’s analysis, with his experimentalarrangement shown in figure 1. Note that the angle θ is drawn from the center of thesingle slit or the midpoint between the two slits. If the system, in this case a particle,is initially in a state |ψ〉, then according to the Born postulate, the probability P thatthe particle is scattered with y-momentum py = p sin θ is given byP (py) = |〈py|ψ〉|2, (1)where the bracket represents the overlap integral between the initial and final states.In the position representation, the momentum eigenfunction for a free particle withmomentum py isφ ≡ 〈y|py〉 = 1√2pieipyy~ . (2)The scattering amplitude 〈py|ψ〉 can be calculated by inserting a complete basis ofposition states between 〈py| and |ψ〉 and integrating:〈py|ψ〉 =∫ ∞−∞〈py|y〉〈y|ψ〉dy = 1√2pi∫ ∞−∞e−ipyy~ ψ(y)dy, (3)where in the Dirac formalism the position wave function ψ(y) ≡ 〈y|ψ〉. To calculateP (py) from this expression we need to know ψ(y) at the position of the slit(s). Takingthe slit width to be a, Marcella makes the most natural assumptions:ψ(y) = 〈y|ψ〉 ={1/√a, −a/2 ≤ y ≤ a/20, elsewhere.(4)The first statement is merely that the intensity is assumed to be uniform over theslit, or that there is an equal likelihood of finding the particle anywhere within it. Inother words, we take |ψ(y)|2dy = Cdy for constant C. Integrating over the slit givesthe total probability of finding the particle P = Ca ≡ 1. Therefore C = 1/a andψ(y) = 1/√a.Quantum Interference... 4We now simply plug this expression for ψ into Eq. (3) to calculate the scatteringamplitude:〈py|ψ〉 = 1√2pia∫ a/2−a/2e−ipyy~ dy=i~py√2pia(e−ipya2~ − e ipya2~ )=2~py√2piasin(apy2~). (5)With the definition α = apy/2~, Eq. (1) gives for the probability of detecting aparticle at angle θP (y) =a2pi(sinαα)2, (6)the standard result for single-slit diffraction.Upon being presented with this procedure, one’s first reaction might well be,“Where’s the physics?” The Dirac formalism is exactly that, a formalism, and doesnot in and of itself add any physics to the problem. We see that the above techniqueincludes no wave equation to be satisfied and no interaction—in other words, nodynamics. Has a miracle occurred?In classical optics one generally treats the scalar diffraction problem by finding asolution to the Helmholtz equation:(∇2 + k2)ψ = 0, (7)of which the time-independent Schro¨dinger equation is the special case k2 = 2m(E −V )/~2, for constant E and V . The Helmholtz equation cannot be solved exactlyexcept in nearly trivial circumstances, and so one must resort to numerous approxi-mations. The usual ones for scalar diffraction theory are:1) The “freshman physics approximation,” or basic Huygens construction, whichassumes that each point within the slit acts as a source of spherical waves ψ(r) =Quantum Interference... 5eikr/r radiating only in the forward direction. Note that this wave function is asolution to the source-free Helmholtz equation;Pθ’enrθsoero rFigure 2: A source s emits a spherical wavefront whose unit normal eo is at an angle θ′ withrespect to the x-axis. At the aperture the wavefront is diffracted to an observation point P . Thenormal to the aperture is n and the unit vector from P to a point on the aperture is er.2) More sophisticated treatments include an “obliquity factor” in the Huygensconstruction; the solution is assumed to be of the form ψ ∼ f(θ, θ′)eikr/r, where θand θ′ are as shown in figure 2. To find the obliquity factor, one generally employsthe Kirchoff approximation [3, 4, 5], which in the case of an aperture in a plane screenassumes that a) the wave function ψ and ∂ψ/∂n ≡ ∇ψ · n, for n the unit normalto the aperture in the direction of propagation, are zero everywhere except in theaperture; b) in the aperture ψ and ∂ψ/∂n take on the value of the incident wave.Any text on electromagnetism or optics, including those just cited, shows that withthe Kirchoff approximation the value of ψ at the observation point P is given by theintegralψ(P ) =iAk4pi∫aeik(r+ro)rro(er − eo) · n da, (8)where A is a constant and the integration is over the aperture. As one sees fromfigure 2, the Kirchoff integral can be writtenψ(P ) = −iAk4pi∫aeik(r+ro)rro(cos θ + cos θ′) da, (9)where the obliquity factor is f(θ, θ′) = (1/2)(cos θ + cos θ′).Quantum Interference... 6It is well known that specifying both ψ and ∂ψ/∂n overdetermines the system andmakes the Kirchoff approximation inconsistent; nevertheless it generally gives resultsthat are in good agreement with experiment for small angles in the short wavelengthlimit. One can remove the inconsistencies by choosing one boundary condition or theother; that is, by specifying either ψ within the aperture or ∂ψ/∂n but not both.Thus we have the other two approximations:3) Dirichlet approximation, in which ψ is taken to be the indicident wave withinthe aperture. For the case of a plane aperture, the obliquity factor becomes f =cos(θ);4) Neumann approximation, in which ∂ψ/∂n is specified within the aperture. Herefor a plane screen the obliquity factor becomes f = cos(θ′)None of these four methods lead to exact solutions of the Helmholtz equation andall differ in the limit of large scattering angles and long wavelengths. However, allfour agree in the standard slit diffraction regime of normal incidence, small scatteringangles and short wavelengths. In this limit, with the additional Fraunhofer approxi-mation that the distance to the screen is large compared to the aperture width, theKirchoff integral for an infinite slit of width a in the y-direction becomesψ(P ) = C∫ a/2−a/2e−ik sin θydy, (10)where now θ is the fixed angle between P and the center of the slit.The question is, Has Marcella made implicit use of any of this formalism? At firstglance it appears not; he does not employ the Schro¨dinger equation and makes noreference to any approximations. He is purely concerned with what happens to theparticle at the slit and makes no statement about what takes place at the distantobservation screen. This is similar to a “hidden-variable” point of view, that theparticle is emitted from the screen with a definite momentum before a measurementQuantum Interference... 7takes place. It is just this sort of interpretative statement that leads to analyses thatviolate Bell’s inequalities and is certainly not the sort of concept that should be im-planted in the mind of a student of quantum mechanics. If, however, the discussionis to take place within the framework of standard quantum theory, we have the rightto demand that the Schro¨dinger equation be satisfied, as well as quantum mechan-ics’ standard postulates. Thus, Marcella must be implicitly solving the Helmholtzequation in some limit.We can uncover this limit by a close examination of Marcella’s few assumptions.Note first that the assumption of equal intensity across the slit made in Eq. (4) isequivalent to the statement that the incoming wavefront is a plane wave. The normalvector no of a spherical wavefront impinging on a plane aperture would vary in anglewith respect to the screen normal n across the aperture; hence the relationship ofthe area elements and intensity could not be constant on the screen. This does notnecessarily imply normal incidence; however, since Marcella does not specify phases,it might as well be. That is, the intensity could either be |ψo|2 from waves normalto the aperture, or |ψ cos(θ′)|2 = |ψo|2 for non-normal wave function ψ. But since θ′must be constant, it can be absorbed into the definition of ψ. Furthermore, Marcellahas assumed that ψ = 0 outside the aperture, but has not said anything about ∂ψ/∂n.Thus he has implicitly assumed a Dirichlet boundary condition, and so the obliquityfactor should be cos(θ), which has been set to one, i.e., the paraxial approximationhas been used.At the same time, we notice that in the integration of Eq. (5), he has heldpy = p sin(θ) constant. He justifies this by saying that he is calculating the scatteringamplitude of a particle being emitted at the slit with a fixed angle θ. As mentioned,he makes no statement about what takes place at the distant observation point. Instandard quantum mechanics, however, one would assume that the wave functioncould be emitted at different angles from the top and bottom slit or integrate over θ.Quantum Interference... 8That Marcella uses a single angle (figure 1) drawn from the center of the slit, or fromthe midpoint of the double-slit configuration is permissible only in the Fraunhoferlimit. Thus, at best, he is performing the integral in Eq. (10)However, one can see that his assumptions that ψ = 1/√a and use of the plane-wave eigenstates in Eq. (5) is not consistent. All that is being done in Eq.(5) is to takethe Fourier transform of ψ(y) = 1/√a. If we transform from position to momentumspace and back again, we should end up where we started: ψ′(y) = ψ(y) = 1/√a.Thus,ψ′(y) =1√2pi∫ km−kmφ(k)eikydk, (11)where we have let the limits of integration be finite and whereφ(k)1√a∫ a/2−a/2e−iky′dy′. (12)Hence,ψ′(y) =12pi√a∫ km−km∫ a/2−a/2eik(y−y′)dy′dk,=12pi√a∫ a/2−a/2dy′∫ km−kmeik(y−y′)dk. (13)If km →∞, then the last factor becomes 2piδ(y − y′) and the result ψ′(y) = 1/√a asrequired. However, in Marcella’s problem py = ~ky = p sin(θ) and there is a maximumvalue of k. In that caseψ′(y) =1pi√a∫ a/2−a/2sin(km(y − y′))(y − y′) dy′, (14)which is nonanalytic and must be approximated for a given value of y. For km(y −y′) << 1 one gets Marcella’s result only when, interestingly, λ = 2a, but in that casethe assumption that km(y − y′) << 1 cannot always be met.In fact Marcella’s approximations are no better than the freshman physics ap-proach, which ignores all boundary conditions. Consider figure 3, in which we divideup a slit of width a into 2N Huygens sources, N above the central source and NQuantum Interference... 9θa θ∆= (a/2)sinFigure 3: Consider a slit of width a and divide it into 2N sources of Huygens wavelets, N above thecentral source and N below. For parallel rays the phase shift with respect to the central source willbe ±k(a/2) sin θ(n/N) for the n-th source. The phase shift at the end of the slit is ∆ = ±k(a/2) sin θbelow. Assume that the screen is at an infinite distance so that the rays are parallel.At the observation point P on the screen the wave function is merely the addition ofall the Huygens wavelets. Thusψ(P ) = ψo(P )N∑n=−Nei(ωt+δn), (15)where the nth phase shift with respect to the central source is δn = k(a/2) sin(θ)n/N .Write δn = αnsinθ with αn ≡ k(a/2)n/N . Notice that αn plays the role of k ormomentum, while θ is the position variable. The spacing between adjacent n’s is∆α = (ka/2N) = constant. Multiplying and dividing Eq. (15) by ∆α givesψ(P ) =ψo(P )∆αN∑n=−Neiαn sin θ∆α, (16)where we have absorbed the maximum value (say) of the rapidly varying function eiωtinto the definition of ψo. Absorb the first ∆α into the definition of ψo as well. Then,Quantum Interference... 10in the limit of large N ,ψ(P ) → ψo(P )∫ ka/2−ka/2eiα sin θdα,= ψo(P )2 sin((ka/2) sin θ)sin θ(17)which is exactly the standard answer. We thus see that the freshman physics con-struction of adding together Huygens wavelets is really a Fourier transform, whichis exactly what Marcella has introduced by chopping his wave function at the edgeof the slits. Although he is going from position to momentum space, by writing theresult Eq. (6) in terms of θ, we have the same position-space result just obtained.In sum, Marcella does make the valid point that quantum interference should betreated as a quantum phenomenon and quantum texts ought not immediately redirectthe discussion to classical wave optics. But a more reasonable way to do this would beto simply show that the Schro¨dinger equation reduces to the Helmholtz equation, thusreducing the problem to one of classical scalar scattering with its concomitant approx-imations. This would also provide the opportunity of discussing relevant boundaryconditions and to point out the difficulty of specifying them precisely in both thequantum and electromagnetic cases. As it stands, while Marcella’s procedure is use-ful in giving students practice with the Dirac formalism, it has introduced no quantumphysics into the problem other than setting p = ~k, and has implicitly made all theassumptions that show this is indeed a problem of classical optics. That his resultis the same as the one obtained by the simplest Huygens construction is merely areflection of the fact that he has implicitly made the lowest-order approximations,where all methods converge to the same result.References[1] Marcella Thomas V 2002 Eur. J. Phys. 23 615-621Quantum Interference... 11[2] Sudarshan E C G and Rothman T 1991 Am J. Phys. 59 592-595[3] Elmore William C and Heald Mark A 1969 Physics of Waves (New York; McGraw-Hill)chaps. 9-10[4] Marion Jerry B 1972 Classical Electromagnetic Radiation (New York; Academic Press)chap. 12[5] Jackson J D 1975 Classical Electrodynamics (New York; Wiley) chap. 9

\u27Quantum interference with slits\u27 revisited

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\u27Quantum interference with slits\u27 revisited

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