4 The two-​​slit experiment revisited

The two-​​slit exper­i­ment with elec­trons fea­tured two alter­na­tives:
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  • The elec­tron went through the left slit (L).
  • The elec­tron went through the right slit (R).

Under the con­di­tions stip­u­lated by Rule A, as we have seen, the prob­a­bility of detec­tion at the posi­tion D of a detector is the sum of two prob­a­bil­i­ties, pL = |AL|2 and pR = |AR|2. This is con­sis­tent with the view that an elec­tron detected at D went through either L or R.

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two-slit experiment

Setup of the two-​​slit experiment

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Let us try to under­stand what hap­pened, under the con­di­tions stip­u­lated by Rule B. Let us assume, to begin with, that

  1. each elec­tron goes through a par­tic­ular slit (either L or R), and
  2. the behavior of elec­trons that go through a given slit does not depend on whether the other slit is open or shut.

If the first assump­tion is true and both slits are open, the dis­tri­b­u­tion of hits across the back­drop is given by

n(x) = nL(x) + nR(x),

where nL(x) and nR(x) are the respec­tive dis­tri­b­u­tions of hits from elec­trons that went through L and elec­trons that went through R. If the second assump­tion is true, then we can observe nL(x) by keeping the right slit shut, and we can observe nR(x) by keeping the left slit shut. What we observe if the right slit is shut is the dotted hump on the left side of Figure 1.4.2 (repro­duced here), and what we observe when the left slit is shut is the dotted hump on the right side of this figure. If both assump­tions are true, we thus expect to observe the sum of these two humps.

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Plot A

Figure 1.4.2 The prob­a­bility of detec­tion according to Rule A (the solid curve) is the sum of two prob­a­bility dis­tri­b­u­tions (the dotted curves), one for elec­trons that went through L and one for elec­trons that went through R.

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But this is what we observe under the con­di­tions stip­u­lated by Rule A. What we observe under the con­di­tions stip­u­lated by Rule B is plotted in Figure 1.4.3. At least one of the two assump­tions is there­fore false.

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Plot B

Figure 1.4.3 The prob­a­bility of detec­tion according to Rule B.

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Bohmian mechanics

According to an inter­pre­ta­tional strategy pro­posed by David Bohm,[1] only the second assump­tion is false: all elec­trons follow well-​​defined paths, which wiggle in a pecu­liar manner and cluster at the back­drop so as to pro­duce the observed dis­tri­b­u­tion of hits.

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Bohmian trajectories

Figure 3.4.1 Bohmian trajectories

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What causes the wig­gles? Bohmians explain this by positing the exis­tence of a “pilot wave” that guides the elec­trons by exerting on them a force. If both slits are open, this passes through both slits; the sec­ondary waves ema­nating from the slits inter­fere, and the result is that the elec­trons are guided along wiggly paths.

According to the Bohmians, the reason why elec­trons emerging from the same source or slit arrive in dif­ferent places is that they start out in slightly dif­ferent direc­tions and/​or with slightly dif­ferent speeds. If we had pre­cise knowl­edge of these ini­tial values, we would be in a posi­tion to pre­dict each electron’s future motion with clas­sical pre­ci­sion. Since quantum mechanics decrees that such knowl­edge cannot be had, Bohmians must declare that in spite of the fact that well-​​defined elec­tron paths and exact ini­tial values exist, they are hidden from us. What they don’t say is why they are hidden from us, even though there is a per­fectly simple answer to this ques­tion: they are hidden from us because they do not exist.

Bohmian mechanics is an extreme instan­ti­a­tion of the prin­ciple of evo­lu­tion. It not only posits a wave func­tion that evolves between mea­sure­ments but also attrib­utes to it the reality of a clas­sical force that acts on clas­sical par­ti­cles, in blithe dis­re­gard of the fact that the pilot wave asso­ci­ated with a phys­ical system with N degrees of freedom prop­a­gates in an N-​​dimensional con­fig­u­ra­tion space, which can be iden­ti­fied with phys­ical space only in the spa­cial case that N=3. (Another unpalat­able fea­ture of Bohmian mechanics is that on this theory energy and momentum and spin and every par­ticle prop­erty other than posi­tion are con­tex­tual.[2])

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The meaning of “both”

Our inter­pre­ta­tional strategy leads us to con­clude that it is the first assump­tion which is wrong. Under the con­di­tions stip­u­lated by Rule B, each elec­tron does in some sense pass through both slits. But in what sense? Saying that an elec­tron went through both slits cannot be equiv­a­lent to saying that the elec­tron went through L and that it went through R. To ascer­tain the truth of a con­junc­tion we must indi­vid­u­ally ascer­tain the truths of its com­po­nents, yet we never find (i) that an elec­tron launched at G and detected at D has taken the left slit and (ii) that the same elec­tron has taken the right slit.

Nor can saying that an elec­tron went through both slits mean that a part of the elec­tron went through L while another part went through R. In point of fact, the ques­tion of parts does not arise. Anal­o­gous exper­i­ments have been per­formed with C60 mol­e­cules using a grating with slits 50 nanome­ters wide and spaced 100 nanome­ters apart.[3] The sixty carbon nuclei of C60 are arranged like the cor­ners of an old-​​fashioned soccer ball having a diam­eter of just 0.7 nanome­ters. We do not pic­ture parts of such a mol­e­cule as get­ting sep­a­rated by many times 100 nanome­ters and then reassemble into a ball less than a nanometer across.

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Buckminsterfullerene

Figure 3.4.2. A Buck­min­ster­fullerene (C60)

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Saying that an elec­tron went through both slits can only mean that it went through L&R — the two elon­gated cutouts in the slit plate con­sid­ered as an undifferentiated/​undivided whole. When­ever Rule B applies, the dis­tinc­tion we make between L and R is a dis­tinc­tion that has no reality as far as the elec­tron is con­cerned. The dis­tinc­tion between “the elec­tron went through L” and “the elec­tron went through R” is a dis­tinc­tion that “Nature does not make” — it cor­re­sponds to nothing in the actual world. The posi­tion at which the elec­tron passed the slit plate is the entire undif­fer­en­ti­ated region L&R. It is not any part or seg­ment of L&R, let alone a point some­where in L&R.

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1. [↑] Bohm, D. (1952). A sug­gested inter­pre­ta­tion of quantum theory in terms of hidden vari­ables. Phys­ical Review 85, 166–193.

2. [↑] Albert, D.Z. (1992). Quantum Mechanics and Expe­ri­ence, Har­vard Uni­ver­sity Press, Chapter 7.

3. [↑] Arndt, M., Nairz, O., Vos–Andreae, J., Keller, C., van der Zouw, G., and Zeilinger, A. (1999). Wave-​​particle duality of C60 mol­e­cules. Nature 401, 680–682.