A. Waves

Fig. 1: The diffraction of water waves

Fig. 2: The distribution φ of wave in the x-direction for two times t1 und t2.

Two water waves appear in figures 1 and 2 . These two waves interfere and are detected by the detector D which gives a reading of the light intensity I.  If we close one of the slits, we will have only one wave. The sine-like wave that is produced should be recognizable to you. The distance between two wave crests (or two minimums) is called the wavelength l. If we choose any point x on this wave, this point vibrates as φ from t like a sine wave with the vibration frequency v.
Detector D registers the Wave intensity I which is given by I = |φ|².
Next we open both slits and watch the waves φ₁ and φ₂ overlap in time. Both waves have the same frequency. The overlapping of the waves causes a transitional shift which is described by the wave constant j (see figure 2). We have:
 

A more precise equation for the intensity is                I = I₁ + I₂ + 2(I₁I₂)^½.cos j. One can see that intensity also oscillates with cos j. These oscillations are especially strong for I₁ = I₂ = I₀, since then all the intensity swings between 0 and 4I₀:   I = 2I₀·(1+cos j). We only have fourfold intensity in the most extreme case. This is the case for each point x, however having different phases in either point x.
 

B. Balls

Fig. 3: The experiment with balls.

When we conduct the "Interference" experiment with balls, we can find the probability P₁ of each ball passing through slit 1 whether or not we keep slit 2 closed. We also can find the probability P₂ of each ball passing through slit 2 if we keep slit 1 closed. If the two slits are opened one can find the total probability which is equal to:
 

When the slits are situated far away from each other then there is no total probability like the previous case and we will have the individual probabilities P₁ and P₂. This scenario is displayed on the right in a more crude form:
 
 

C. Electrons
 

Fig. 4: We will always have P12¹ P1+ P2, even in the case when we can only detect one electron at a time.

In a similar experiment with electrons we learn that the total probability P₁₂ is not equal to the sum of particular probabilities. This is the case even when we detect one electron at a time:
 

The interference pattern which we observe on our screen corresponds to the behavior of  wave interference which we can easily describe. Next we can describe the result by applying two complex numbers φ₁ and φ₂. The square of absolute φ₁ value gives the probability P₁ whether slit 1 is opened: P₁ = |φ₁|². Similar to previous statement one will have for slit 2 (when slit 1 is closed): P₂ = |φ₂|². And if both slits are opened however we will have in this case: 
 

In order to describe our observations correctly we must use the mean of complex numbers in our calculations. In general, what will happen when we add a single electron that has passed through slit 1 or 2 ? 

To get a result, we must carry out an experiment in which electrons are observed together with the light source:
 

Fig. 5: Experiment with electrons that are passing through one slit which is lighted by a lamp.

The observed electron distribution on the screen matches exactly with the alternately closed slit. We have
 

Finally: If we carry out an experiment to determine the way an electron moves we can determine (with complete certainty) whether an electron took one path or another. The interference pattern disappears, however. If we aren't certain of the path of the electron then we can't say that it has taken by path 1 or 2. Consequently, we can't even be certain that an electron has passed through either slit. 

What will happen if an experimenter who is waiting for a particle to pass through the two slits decides to obtain measurements (and moreover the two slits are situated far away from each other)?

In this experiment the laser beam is directed on the beamsplitter and the split laser beams are collected by a detector which records the interference pattern. We can direct part of initial beam on the Pockels Cell. We can also direct photons on the auxiliary detector when applying different voltages. When we have zero voltage everything stays the same as it was before. The random-signal generator switches on the cell after the beam has passed the beamsplitter, but before it reaches the main detector. The photon can seemingly take either one way or the other. If the Pockels cell is turned off then we will see the interference pattern on the main detector. 

This double-slit experiment shows that the two paths can be located as far away from each other as possible. The split beams can travel infinitely far in opposite directions before they combine with each other to produce an interference pattern.  

Efforts with atoms
 

Fig. 6: The double slit experiment features atoms passing through system of diagrams and a laser beam. Then atoms lose energy by falling to a lower energy level and producing photons which are stored in a cavity. Because this process doesn't influence atomic motion, we won't take the uncertainty principle into account here. Thus, this experiment shows that it is not possible to determine the path of atoms from the interference pattern.

Auf diesem Webangebot gilt die Datenschutzerklärung der TU Braunschweig mit Ausnahme der Abschnitte VI, VII und VIII.