Interference
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bulletSingle-slit
bulletConceptual experiment
bulletDouble-slit

Single-slit

Macroscopically a wave is an extensive disturbance in a medium (for example water serves as the medium for the ripples on its surface). When several waves come together they combine constructively or destructively, i.e. they interfere. 

When such a wave passes through a small aperture it is diffracted and spreads out into the shadow zone. In quantum mechanics the probability distribution of a particle is obtained from a wave function. This wave can diffract and interfere just  like a classical wave. 

The effect of diffraction on the probability distribution of a quantum particle can be studied by considering a wave packet that passes through a narrow slit. In the following animations the initial wave packet is taken to be 

 

with s = 7 l and k0 = ( 2p / l , 0 )

 

 

 

wpe2.gif (12741 bytes)      wpe4.gif (12568 bytes)      wpe6.gif (13098 bytes)     wpe8.gif (10597 bytes)     

(s , w) = (l / 2 , l)   (s , w) = (l , l)        (s , w) = (l , 5l)      (s , w) = (3l , l)

The animations show that 

  1. A narrow slit diffracts more strongly than a wide slit. 
  2. The length w of the slit mainly affects the probability to pass through the slit but has little influence on the diffraction pattern. 

Conceptual experiment

Consider the experiment illustrated on the right. The source at the top emits electrons that have the same energy but not the same direction. An electrical potential on the wire and plates repels the electrons. The electrons impinge on the screen (gray area) one at a time. Initially there is no interference pattern on the screen: Each electron shows up as a white dot. With the passage of time typical interference fringes appear. The electrons are most numerous at the interference maxima. The following cartoon gives an artists impression of the formation of these interference fringes.FOOTNOTE

wpeD.gif (8760 bytes)       

It turns out that in a real experiment it is impossible to follow the trajectories of the individual electrons. The systematic creation of a coherent microscopic interference pattern by non-interacting particles arriving randomly has been demonstrated in genuine experiments that use photons, electrons or atoms. This interference is one manifestation of "particle-wave duality". It is a property of a single quantum particle and does not involve an interaction between particles. 

FOOTNOTE:

From a detailed analysis of this problem it follows that the interference pattern that emerges after collecting many electrons is not the same as the interference pattern of the double slit. However, on a conceptual level there is no difference between this experiment and the experiment with a double slit.

                  

Double-slit

The numerical simulation of the two-slit experiment is shown in the next animation. Initially the quantum particle is represented by a Gaussian wave packet of width s. The dimensions of the two slits are given in the schematic diagram.

wpeF.gif (14290 bytes)       

The maxima and minima in the probability distribution are a direct proof of the presence of interference. The intensity I (q ) recorded by a detector placed far away from a source characterizes the interference pattern. The intensity I (q ) is called the angular distribution. The simulation yields an angular distribution that is in excellent agreement with the Fraunhofer diffraction pattern of a double slit. If you were to use a pencil to draw this pattern you would barely see the difference with the simulation result. 

 

The following animations illustrate the effect of changing the dimensions of the double slit. As in the case of the single slit the qualitative features of the diffraction pattern do not depend on the length w of the double slit. Remember that an experiment with a single particle does not produce an interference pattern. It is only after repeating the experiment with the single particle many times that an interference pattern emerges. 

 

 wpeC.gif (16486 bytes)      wpeE.gif (15289 bytes)      wpe10.gif (15131 bytes)      wpe12.gif (15675 bytes)   

(s , d) = (l , 2l)      (s , d) = (l , 4l)      (s , d) = (l , 6l)      (s , d) = (2l , 5l)

Imagine that you were given the double slit shown on the right. If you would use this device in your experiment would you expect that the interference pattern

  1. will be asymmetric ? 
  2. is wiped out completely ? 
  3. does not change at all ?

The answer to this question can be found by carefully looking at the following animation. It may be helpful to stop the movie and use the slide bar to look at individual frames.

wpe1.gif (16293 bytes)       


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