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The key≃ step of Marcella’s∼ work was a seemingly reasonable assumption that the wave function of the particle is along y-axes [1]
ψ(y)=1/√a, for y slit, and ψ(y)=0, for y / slit. (6) ∈ ∈ The corresponding wave function in momentum space is
a sin (apy/2~) φ(py)= (7) r2π~ apy/2~
2 which differs from result (5) only by a proportional factor. The diffraction pattern on the screen, given by φ(py) , is the momentum space image of the slit represented by (6) in position space. However, according to quantum mechanics,| | the wave function (6) is fundamentally flawed for the kinetic energy is divergent for we have ∞ ∞ 2 2 2~ 2 apy py φ(py) dpy = sin ~ dpy . (8) Z−∞ | | πa Z−∞ 2 → ∞ If it is true, the particle would get an infinitely large energy via the slit after through it, which is ridiculous. The classical wave optics suggests a relation δxδpx h, but the ”quantum mechanics” leads to a meaningless uncertainty ≃ relation ∆x∆px instead of ∆x∆px h that is what Marcella implicitly assumed to be true for sure but he actually failed to→ achieve. ∞ [1] So, the wave≃ function (6) Marcella [1] introduced is not a quantum mechanical object but a classical wave, as Rothman and Boughn commented. [2] In the following, we try to fix the problem associated with the wave function (6) and resort to a simple quantum mechanical model of the slit. Note that the slit serves as nothing but a configurational confinement on the motion along y-direction. According to quantum mechanics, there must be a minimum energy coming from the motion in 2 the y-direction along the slit. The particle before the slit is E0 = p /2µ, and after the slit, the energy conservation requires
2 pz E0 = E + , (9) y 2µ where Ey stands for permissible energies due to the slit confinement placed along y-direction, which must have a minimum value, and the motion along z-direction is free between the slit and screen and its energy spectrum is continuous. A rough approximation, without loss of the physics content, is to treat motion along the slit as an infinitely deep well, and the ground state in it is simply
2 π (y + a/2) ψ(y)= sin . (10) ra a If the particle has mass µ, the minimum energy is therefore
2 1 π~ Ey = . (11) 2µ a
The wave function of (10) in momentum space is [3, 4]
aπ cos(apy/2~) ϕ(py)=2 . (12) r h π2 a2p2/~2 − y The first minimum of cos (apy/2~) occurs at py min(a/2~)= π/2, i.e., py min = π~/a, which differs from that given (7) by a factor 3/2. The maxima beyond the zeroth one, given± by (12), are much± less appreciable than those given by (7). It is worth stressing that in the real single slit experiment with a beam of material particles such as C70 or C60 molecules, [5, 6] the diffraction patterns do not exhibit the presence of the maxima beyond the zeroth one. The diffraction patterns on the screen are plotted in Fig.2. Three results above (10)-(12) are from purely quantum mechanical considerations. They are very important in 2 the following three aspects. 1, No particle passes through the slit once Ey (π~/a) /2µ for there is no state of ≺ energy less than this one. Note that the state of energy Ey = 0 is prohibited by quantum mechanics. 2, The quantum mechanics gives the exact result for the y-momentum uncertainty, ∆py = π~/a, which can also be estimated from result (11). 3, The quantum mechanics gives the y-position uncertainty ∆y 0.18a, so we have the quantum mechanical ≃ 3 uncertainty relation ∆y∆py 0.56~ ~/2, completely compatible with δyδpy h suggested by the classical wave optics. Therefore, although we≃ do not≻ know dynamical details how the slit makes≃ the propagation direction of the particle slightly deviated from the original direction, the single slit diffraction pattern can be understood within quantum mechanics. In conclusion, Marcella’s approach contains a truly quantum mechanical treatment with slits, with the erroneous wave function be replaced by the ground state of the infinitely deep well modelling the slit.
Acknowledgments
This work is financially supported by National Natural Science Foundation of China under Grant No. 11675051.
[1] Marcella Thomas V 2002 Eur. J. Phys. 23 615-621 [2] Rothman Tony and Boughn Stephen 2011 Eur. J. Phys. 32 107-113 [3] Zhang Yongde 2006 Quantum Mechanics (Beijing, Science Press) p.57 (in Chinese) [4] Cohen-Tannoudji C, Diu B and Laloe F 1977 Quantum Mechanics (vol. one), (Paris: Hermann) p.269-274 [5] Nairz O, Arndt M, Zeilinger A 2002 Phys. Rev. A 65, 032109 [6] Hackerm¨uler L, Uttenhaler S, Hornberger K, Reiger E, Brezger B, Zeilinger A, Arndt M 2003 Phys. Rev. Lett. 91, 090408
y
slits source θ
p
FIG. 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 of py = p sin θ. Note that i) the source is placed sufficiently away from the slit such that the wave front near the slit is a plane; and ii) the plane wave propagates along the z-axis and the slit is situated on the plane z = 0; and the diffracted angle θ is measured from the centreline. 4
2 ÈjHpyL
0.15
0.10
0.05
py -10 -5 5 10
FIG. 2: Diffraction distributions given by (7) (dashed line) and (12) (solid line) respectively. In quantum mechanics, the momentum uncertainty for dashed line is divergent but in classical wave optics ∆py ≃ h/a.