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**An Alternative Model of the**

Double Slit Interference Experiment

Double Slit Interference Experiment

W.H. Madden

November 20, 2007

ABSTRACT

This post presents an alternative model of the double-slit interference (DSI) experiment. The model is based on a novel physical interpretation of the concept of the DeBroglie wavelength.

This alternative model is shown to fulfill the essential requirements for any plausible model of the DSI experiment: It can account for the observed wave-like interference pattern, is applicable to “single quanta” type variants of the DSI experiment, and is in complete agreement with the experimentally observed—and QM predicted—results. Further, the model is entirely particle based.

The proposed model has advantages: Being strictly particle based, the wave/particle duality paradox is avoided. Non-local “action at a distance” is not required to account for the individual quanta’s sensitivity to the global condition of both slits. Also, a concrete physical interpretation is given to the DeBroglie wavelength.

The main disadvantage of this model is that this physical interpretation calls into question the fundamental assumption of continuous motion.

INTRODUCTION

In the DSI experiment, the seemingly innocent question of “what goes through the slits?” remains unresolved, even after years of debate. Despite perfect success in predicting all experimental outcomes, QM formalism offers little help in answering this question (1,2).

Numerous variations of the DSI experiment have clearly shown the gradual build-up of the wave-like interference pattern by single quanta (for example, Tonomura, et al (3). Recently, Afshar (4) has demonstrated the simultaneous presence of both wave-like and particle-like behavior for single quanta, in violation of complementarity.

These experiments, along with many others, have served to reinvigorate—but not resolve—the debate on the underlying nature of wave/particle duality. A central theme of this debate is the physical meaning of the concept of particle wavelength, as first introduced (for non-relativistic cases) by DeBroglie:

[tex]\lambda = \frac{h}{mv} \ \ \ \ \ \ Eq. \ 1[/tex]

where [tex]\lambda[/tex] is the particle wavelength, h is Planck’s constant, and m and v are the particle mass and velocity, respectively.

A NEW INTERPRETATION OF PARTICLE WAVELENGTH

For this paper, the following concrete physical interpretation of particle wavelength is adopted:

The particle wavelength is the minimum, discrete spatial distance across which the particle can travel.

For unaccelerated motion, an immediate corollary to this discrete motion interpretation (DMI) is:

Only trajectories with lengths equal to integer multiples of wavelengths are allowed:

[tex]T_{DMI} = N\lambda \ \ \ \ \ Eq. \ 2[/tex]

where N is an integer.

Consequently, the difference between any two DMI trajectories [tex]T_{a}[/tex] and [tex]T_{b}[/tex] and must also be an integer multiple of the wavelength, [tex]\lambda[/tex]:

[tex]T_{a} = N_{a}\lambda[/tex]

[tex]T_{b} = N_{b}\lambda[/tex]

[tex]T_{a}-T_{b} = (N_{a}-N_{b})\lambda[/tex]

[tex]So,[/tex]

[tex]T_{a}-T_{b} = M\lambda \ \ \ \ \ \ Eq. \ 3[/tex]

[tex]T_{b} = N_{b}\lambda[/tex]

[tex]T_{a}-T_{b} = (N_{a}-N_{b})\lambda[/tex]

[tex]So,[/tex]

[tex]T_{a}-T_{b} = M\lambda \ \ \ \ \ \ Eq. \ 3[/tex]

where M is an integer.

Applied to the DSI experiment, the DMI restriction of Equation 3 is immediately familiar as the wave model condition for an interference maxima at a given point, [tex]P_{Y}[/tex], on the screen:

[tex]P_{A}-P_{B} = M\lambda \ \ \ \ \ \ Eq. \ 4[/tex]

where [tex]P_{A}[/tex] and [tex]P_{B}[/tex] are, respectively, the path distances from slit A and slit B to the point, [tex]P_{Y}[/tex].

Unlike DMI, the wave model sets no restrictions on the allowed path lengths. In the wave model, path lengths can always be expressed in the form:

[tex]P_{A} = (N_{A}+K_{A})\lambda[/tex]

[tex]P_{B} = (N_{B}+K_{B})\lambda \ \ \ \ \ \ Eq. \ 5[/tex]

[tex]P_{B} = (N_{B}+K_{B})\lambda \ \ \ \ \ \ Eq. \ 5[/tex]

where [tex]N_{A}[/tex] and [tex]N_{B}[/tex] are integers, and [tex]K_{A}[/tex] and [tex]K_{B}[/tex] are both non-integer values between 0 and 1. So, for example, [tex]P_{A}[/tex] could be 2,000,000.3[tex]\lambda[/tex], and [tex]P_{B}[/tex] could be 1,999,998.3[tex]\lambda[/tex]. Although neither of the path lengths [tex]P_{A}[/tex] or [tex]P_{B}[/tex] is an integer multiple of the wavelength, [tex]\lambda[/tex], the difference between them, 2[tex]\lambda[/tex], is.

Because the wave model allows more possible solutions for interference maxima than does the DMI model, the DMI model—as considered so far —must be incomplete.

A more sophisticated version of the DMI model should include an important uncertainty inherent in the model itself—the uncertainty in the exact position of the trajectory’s starting point, [tex]\Delta[/tex]. The term, [tex]\Delta[/tex], arises as a direct consequence of the Heisenberg Uncertainty Principle: if the trajectory start point is known to a precision smaller than the wavelength, the momentum of the particle involved—and therefore the very wavelength being considered—will be disturbed.

The starting point uncertainty, [tex]\Delta[/tex], can be physically interpreted as the distance the trajectory extends into the slit opening. [tex]\Delta[/tex] must always be between 0 and 1[tex]\lambda[/tex], and, for each event, [tex]\Delta[/tex] is definite but unknown. In a large sample of events, [tex]\Delta[/tex] will vary randomly for each event.

THE DMI MODEL OF THE SINGLE SLIT EXPERIMENT

Fig. 1 shows the DMI model of a single-slit interference experiment, including the start-point uncertainty, [tex]\Delta[/tex], and using ‘real-world’ experimental values.

In this example, the slit barrier to screen distance, R, is 1 meter; the slit width, W, is .1 millimeter, and the source wavelength, [tex]\lambda[/tex], is 500 nanometers.

In the DMI model, the basic unit of length is 1[tex]\lambda[/tex]. Expressing the length, R, in terms of the wavelength:

[tex]R = (N_{R}+K_{R})\lambda \ \ \ \ \ \ Eq. \ 6[/tex]

with [tex]N_{R}[/tex] an integer, and [tex]K_{R}[/tex] a non-integer between 0 and 1.

For the example considered here, R is exactly 1 meter. Since [tex]N_{R}[/tex]=[tex]\frac{R}{\lambda}[/tex], [tex]N_{R}[/tex] = 1 meter / 500 nanometers, and [tex]N_{R}[/tex] = 2,000,000; with the values chosen for this example, [tex]K_{R}[/tex] is 0.

Theoretically, R can be known to any precision, and can be measured from a point at the barrier not positioned on the slit, so the start point uncertainty, [tex]\Delta[/tex], does not apply to measurement of the distance, R.

On the other hand, DMI model trajectories must include the uncertainty, [tex]\Delta[/tex], and can be generally expressed as:

[tex]T_{DMI} = (N_{T}+K_{T}+\Delta)\lambda \ \ \ \ \ \ Eq. \ 7[/tex]

In keeping with the primary DMI condition, all DMI trajectories are restricted to:

[tex]T_{DMI} = N\lambda \ \ \ \ \ \ Eq. \ 8[/tex]

The only way equations 7 and 8 can be equal is if the term [tex](K_{T}+\Delta)[/tex] is also an integer. As previously defined, both [tex](K_{T}[/tex] and [tex]\Delta)[/tex] are less than 1[tex]\lambda[/tex], so their sum, [tex](K_{T}+\Delta)[/tex], must be less than 2[tex]\lambda[/tex]. To satisfy the DMI condition, [tex](K_{T}+\Delta)[/tex], must equal 1.

Returning to the example of Fig. 1, consider a single event. For simplicity, assume the uncertainty, [tex]\Delta[/tex], for this particular event is equal to 0 (a possible, although statistically unlikely, assumption).

For this example, the DMI model predicts the following discrete set of allowed trajectories:

[tex]T_{M} = \{(N\lambda)...(N+1)\lambda...(N+2)\lambda...(N+M)\lambda\} \ \ \ \ \ \ Eq. \ 9[/tex]

with N = 2,000,000, and M = {0…1…2…3…etc.}.

Each of these possible trajectories will have an associated displacement at the screen, [tex]Y_{M}[/tex], given by basic trigonometry:

[tex]Y_{M^2} = (N+M)^2\lambda^2-N^2\lambda^2 \ \ \ \ \ \ Eq. \ 10[/tex]

Simplifying Eq.10, and discarding relatively small terms, gives a close approximation for the screen displacements:

[tex]Y_M=\sqrt{(2NM)}\ \lambda \ \ \ \ \ \ Eq. \ 11[/tex]

When M = 0, [tex]Y_M[/tex] is also 0, and there is no displacement at the screen.

Solving for the next few displacements gives:

[tex]Y_{1} \sim 1\ millimeter[/tex]

[tex]Y_{2} \sim 1.4142\ mm.[/tex]

[tex]Y_{3} \sim 1.7320\ mm.[/tex]

In general, the allowed displacements for this single event (where [tex]\Delta[/tex] is 0) are approximated by [tex]Y_M[/tex] = [tex]\sqrt{M}\ mm.[/tex]

Obviously, this pattern is not what is observed at the screen, but these results are for a single event, with a single value—zero—of [tex]\Delta[/tex].

The actual pattern observed will be the composite of a large sample of events, with the variable, [tex]\Delta[/tex], different each time.

To include the uncertainty, [tex]\Delta[/tex], Eq.10 must be re-written:

[tex]Y_{M^2} = (N+M)^2\lambda^2-(N+\Delta)^2\lambda^2 \ \ \ \ \ \ Eq. \ 12[/tex]

After simplification, and again disregarding small terms, a close approximation for [tex]Y_M[/tex] is:

[tex]Y_{M} = \sqrt{2N(M-\Delta)} \ \lambda\ \ \ \ \ \ Eq. \ 13[/tex]

As [tex]\Delta[/tex], varies from 0 to approaching 1, Eq.13 varies from [tex]\sqrt{2NM}[/tex] to [tex]\sqrt{2N(M-1)}[/tex]; for smaller values of M, this variation leads to a significant change in the pattern on the screen.

For example, consider the trajectory when M = 1. Its displacement at the screen is 1 mm when [tex]\Delta[/tex], is 0[tex]\lambda[/tex], but approaches 0 mm as [tex]\Delta[/tex] approaches 1[tex]\lambda[/tex].

As [tex]\Delta[/tex], varies across a large sample of events, the cumulative distribution of points on the screen becomes almost continuous, in agreement with what is actually observed.

THE DMI MODEL OF THE DOUBLE SLIT EXPERIMENT

Applying the DMI model to the double-slit interference experiment now becomes straightforward.

Figure (2a) shows the DSI experiment, with the same values for R, W, and [tex]\lambda[/tex], as in the single-slit example just discussed. The slit separation, d, is .2 mm. For clarity, the slit barrier side of the experiment has been greatly exaggerated.

Recall the wave-model condition for a bright interference fringe at a point, [tex]P_Y[/tex], on the screen:

[tex]P_{A}-P_{B}=M\lambda[/tex] and

[tex](N_{A}+K_{A})\lambda-(N_{B}+K_{B})\lambda=M\lambda \ \ \ \ \ \ Eq. \ 14[/tex]

where the left hand terms are the expressions for the wave-model path lengths.

In the DMI model, these expressions are simply replaced by ones which include the start point uncertainty, [tex]\Delta[/tex]:

[tex]T_{a} = (N_{a}+K_{a}+\Delta)\lambda[/tex]

[tex]T_{b} = (N_{b}+K_{b}+\Delta)\lambda\ \ \ \ \ \ Eq. \ 15[/tex]

It is important to note that, for each event, the uncertainty, [tex]\Delta[/tex], is the same for both slits; this is because the uncertainty applies to the start point of a single particle trajectory—that is, a single event—which must originate at only 1 of the 2 possible slits.

Substitution of the DMI trajectories (Eq.15) into the wave-model interference condition (Eq.14) has no effect; since the variable [tex]\Delta[/tex] is the same for both trajectories [tex]T_{a}[/tex] and [tex]T_{b}[/tex], it simply cancels out.

The underlying geometrical framework of both the wave and DMI models is seen to be identical, and predictions from one model must carry over to the other. The observed outcomes, if not the underlying physical basis, of both models are the same.

Fig 2b shows the interference pattern resulting from the DSI experiment of Fig. 2a. The pattern is described by the conventional wave-model equation:

[tex]Y_{M}=\frac{(M \lambda R)}{d}\ \ \ \ \ \ Eq. \ 16[/tex]

where, again, M = {0…1…2…3…etc.}, R = 1 m, [tex]\lambda[/tex] = 500 nm, and

d = .2 mm.

Substituting the values {0,1,2,3…etc.} for M gives the displacements for the first few interference maxima:

[tex]Y_{0} \sim 0[/tex]

[tex]Y_{1} \sim 2.5\ millimeters[/tex]

[tex]Y_{2} \sim 5.0\ mm[/tex]

[tex]Y_{3} \sim 7.5\ mm[/tex]...,etc.

This result, as predicted by both the wave and DMI models, agrees exactly with that which is experimentally observed.

By incorporating both the discrete motion interpretation of particle wavelength and the uncertainty in the trajectory start point, the DMI model of the DSI experiment is shown to have the following features:

It can account for the observed wave-like interference pattern, although it is entirely particle-based, and it is applicable to individual events—that is, to single particles.

No “wave collapse” is required in the DMI model, and the apparent non-locality (that is, the instantaneous change in outcome dependent upon the state of both slits) is caused not by any interaction between the particle and the slits, but by the presence or absence of possible DMI trajectories. The non-local aspects of the DSI experiment are seen to be geometrical—and not physical—in nature.

VARIATIONS OF THE DSI EXPERIMENT

These include delayed-choice experiments and single quanta “self-interference” experiments. A very brief discussion of each follows:

1. Delayed Choice: in these types of experiments, the decision as to what type of detector to employ is made after the quanta (wave or particle) has already passed through the slit barrier (5). If two narrow field of view detectors are used, with each focused on one slit or the other, the detector outputs can be analyzed to determine which slit the quanta passed through. If, on the other hand, an extended detector (the phosphorescent screen) is used, an interference pattern is observed. The choice of detector appears to alter the nature (wave-like or particle-like) of the quanta involved, after it has already passed the slit barrier. In a sense, the choice of detector seems to influence the past.

2. Self Interference: in this variant, one photon at a time is directed through a beam splitter, with a 50% chance of passing through, and 50% chance of reflection. See Fig. 3. If the photon follows path A, it goes directly to the screen; if it follows path B, it is first reflected by a mirror before proceeding to the screen. The position of the mirror is adjustable along path B. As the mirror is moved, the intensity at the screen varies. Although only 1 photon at a time is considered, an interference-like effect occurs at the screen, as though the single photon were somehow interfering with itself (6).

The DMI model deals readily with both these variants. Delayed-choice experiments are essentially the DSI experiment posed in more dramatic terms, and, in DMI, the choice of detector is irrelevant; in all cases, a single particle-like quanta passes through one slit or the other; the interference pattern arises as a statistical consequence of the permitted DMI trajectories. Only those quanta following DMI trajectories are detected as point-like flashes on the screen; those which do not are not detected at all.

For the self-interference experiment, the same reasoning applies. The presence of the beam splitter provides for 2 separate paths to the screen; in effect, the beam splitter acts as a double slit, and the photon simply follows one of the available paths at random. When at least one of these paths is a DMI permitted trajectory, the photon can be detected at the screen. When both paths are DMI trajectories, the probability of detection is that much greater. If one of the path lengths is varied (by changing the position of the mirror very slightly), the intensity at the point on the screen will vary as well.

CONCLUSIONS

Returning to the original question—what goes through the slits? —the DMI model furnishes an entirely particle-based answer, which can also account for the observed wave-like interference pattern. DMI does so by adopting a concrete physical interpretation of the particle wavelength, and this interpretation then restricts the allowed trajectories the particle can follow to the screen. Furthermore, the DMI model can be applied to individual particles.

If the second slit is opened, a new set of possible DMI trajectories appears instantaneously, but again, for each event, the particle merely follows one of these at random. For a large sample of events, the probability of detecting the particle within a given region on the screen depends on the number of allowed DMI trajectories with endpoints in that region.

Non-locality is very much present in the DMI model, but it is of a geometrical—not physical—nature. It is the DMI trajectories which can interact non-locally, and the experimental setup can be manipulated to alter these instantaneously.

The central tenet of DMI—that the DeBroglie wavelength has a concrete physical interpretation—serves to clearly frame what Feynman (7) once called “the only mystery” of QM, but does not resolve it. In this paper, discrete motion is treated as a first principle, and no explanation is offered here for the mechanism of particle “jumps”, if such a mechanism exists at all.

Obviously, DMI has far-reaching implications, extending to the foundations of both quantum and classical mechanics, and these must be investigated more thoroughly. Some immediate tests of the validity of DMI could be found in applications to the “electron in a box” experiment, quantum tunneling, and development of a DMI model of the hydrogen atom.

References

- Bell, J.S., “Six Possible Worlds of Quantum Mechanics”, Chapter 20, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (1987)
- Zeilinger, A., “On the Interpretation and Philosophical Foundation of Quantum Mechanics”, Vastakohtien Todellisins, Helsinki University Press (1996)
- Tonomura, et al. “Demonstration of Single-Electron Buildup of an Interference Pattern,” Amer. Journal of Physics, vol.57, p.117 (1989)
- Afshar, Shahriar S.; Flores, Eduardo; McDonald, Keith F., et al ,“Paradox in Wave Particle Duality”, Foundations of Physics, vol. 37, p. 295 (2007)
- Helmuth, P., et al, “Delayed Choice Experiments in Quantum Interference”, Phys. Review A, vol. 35, p. 2532 (1987)
- Granger, P., Adler, G., Aspect, A, “Experimental Evidence for a Photon Anti-Correlation Effect on a Beam Splitter”, Europhys. Letts., vol. 1, p. 173 (1986)
- Feynman, R.P., “The Feynman Lectures on Physics”, vol. III, Addison-Wesley (1965)