4 Two slits

According to a Physics World poll con­ducted in 2002,[1] the most beau­tiful exper­i­ment in physics is the two-​​slit exper­i­ment with elec­trons. According to Richard Feynman,[2] this classic gedanken exper­i­ment “has in it the heart of quantum mechanics” and “is impos­sible, absolutely impos­sible, to explain in any clas­sical way.” The setup con­sists of an elec­tron gun G, a plate with two slits L and R equidis­tant from G, and a screen at which elec­trons are detected.

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two-slit experiment

Figure 1.4.1 Setup of the two-​​slit experiment

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To apply the rules of the game, we need to iden­tify the ini­tial mea­sure­ment (on the basis of whose out­come prob­a­bil­i­ties are assigned), the final mea­sure­ment (to the pos­sible out­comes of which prob­a­bil­i­ties are assigned), and a set of alter­na­tives. The ini­tial mea­sure­ment indi­cates that an elec­tron has been launched at G. The final mea­sure­ment indi­cates the posi­tion — along an axis across the back­drop — at which the elec­tron is detected. (If we assume that G is the only source of free elec­trons, then the detec­tion of an elec­tron behind the slit plate also indi­cates the electron’s launch at G.) A single inter­me­diate mea­sure­ment, if made, indi­cates the slit through which the elec­tron went. Thus there are two alternatives:

  • The elec­tron went through the left slit (L).
  • The elec­tron went through the right slit (R).

The cor­re­sponding ampli­tudes are AL and AR. The event whose prob­a­bility we wish to cal­cu­late is the detec­tion of the elec­tron by a detector D sit­u­ated some­where at the back­drop. (As we will use the letter G for both the gun and its posi­tion, so we will use the letter D for both the detector and its posi­tion.) Here is what we need to know in order to be able to per­form this cal­cu­la­tion (→ proofs):

  • AL is the product of two com­plex num­bers, which we will refer to as prop­a­ga­tors, and for which we will use the sym­bols <D|L> and <L|G>. Thus AL = <D|L> <L|G>. Like­wise AR = <D|R> <R|G>.
  • The mag­ni­tude of the prop­a­gator <B|A> — that is, of the ampli­tude asso­ci­ated with a particle’s prop­a­ga­tion from A to B — is inverse pro­por­tional to the dis­tance between A and B.
  • The phase of <B|A> is pro­por­tional to this distance.

According to Rule A, the prob­a­bility with which an elec­tron launched at G is found by a detector sit­u­ated at D thus is

pA(D|G) = |<D|L><L|G>|2 + |<D|R><R|G>|2.

Here is how pA(D|G) depends on the hor­i­zontal posi­tion of the detector at the screen:

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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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Cal­cu­lated according to Rule B, the prob­a­bility with which an elec­tron launched at G is found by a detector sit­u­ated at D is

pB(D|G) = |<D|L><L|G> + <D|R><R|G>|2.

Here is how this depends on the hor­i­zontal posi­tion of the detector at the screen:

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Plot B

Figure 1.4.3 The prob­a­bility of detec­tion according to Rule B.

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Strictly speaking these graphs are plots of prob­a­bility den­si­ties (in one dimen­sion) — that is, they plot a prob­a­bility per unit length. One has to pick an interval on the hor­i­zontal axis and draw (or imagine) two ver­tical lines at the interval’s end points to obtain the prob­a­bility with which an elec­tron is detected within the interval. This prob­a­bility is given by the area enclosed by the plot, the hor­i­zontal axis, and the two ver­tical lines. (What then is the area between the entire hor­i­zontal axis and the entire plot? Since the prob­a­bility of detecting the elec­tron any­where along the axis equal 1, this area also equals 1.)

Two remark­able facts are worth noting. The first is that near the (local) minima of Fig. 1.4.3 the prob­a­bility of detec­tion is less if both slits are open than if either of the slits is shut, as a com­par­ison with Fig. 1.4.2 reveals. The second is that the cen­tral max­imum of Fig. 1.4.3 is twice as high as the max­imum of Fig. 1.4.2. The first fact is gen­er­ally attrib­uted to “destruc­tive inter­fer­ence,” the second to “con­struc­tive inter­fer­ence.” This ter­mi­nology is likely to cause much con­fu­sion. Except in the imag­i­nary world of clas­sical physics, inter­fer­ence is not a phys­ical process. When­ever we speak of con­struc­tive inter­fer­ence, all we mean is that a prob­a­bility cal­cu­lated according to Rule B is greater than the cor­re­sponding prob­a­bility cal­cu­lated according to Rule A. And when­ever we speak of destruc­tive inter­fer­ence, all we mean is that a prob­a­bility cal­cu­lated according to Rule B is less than the cor­re­sponding prob­a­bility cal­cu­lated according to Rule A.

Here is how the inter­fer­ence pat­tern plotted in Fig. 1.4.3 builds up over time[3]:

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Build-up of interference pattern

Figure 1.4.4 Build-​​up of an inter­fer­ence pat­tern. The number of detected elec­trons is 100 (b), 3000 (c), 20000 (d), 70000 (e).

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Now sup­pose that the con­di­tions stip­u­lated by Rule B are met: there is nothing — no event, no state of affairs, any­where, any­time — from which the slits taken by the elec­trons can be inferred. Fur­ther sup­pose that, nevertheless,

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

If assump­tion (1) is true, then the dis­tri­b­u­tion of hits across the back­drop, when both slits are open, 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 assump­tion (2) 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 when the right slit is shut is the left dotted hump in Fig. 1.4.2, and what we observe when the left slit is shut is the right dotted hump. If both assump­tions are true, we thus expect to observe the sum of these two humps. But this is what we observe under the con­di­tions stip­u­lated by Rule A. At least one of the two assump­tions is there­fore false.

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1. [↑] Crease, R.P. (2002). The most beau­tiful exper­i­ment, Physics World, Sep­tember, 19–20.

2. [↑] Feynman, R.P., Leighton, R.B., and Sands, M. (1965). The Feynman Lec­tures in Physics Volume 3, Sec­tion 1–1, Addison–Wesley.

3. [↑] Tono­mura, A., Endo, J., Mat­suda, T., and Kawasaki, T. (1989) Demon­stra­tion of single-​​electron buildup of an inter­fer­ence pat­tern, Amer­ican Journal of Physics 57 (2), 117–120.