This Quantum World/Implications and applications/How fuzzy positions get fuzzier

How fuzzy positions get fuzzierEdit

We will calculate the rate at which the fuzziness of a position probability distribution increases, in consequence of the fuzziness of the corresponding momentum, when there is no counterbalancing attraction (like that between the nucleus and the electron in atomic hydrogen).

Because it is easy to handle, we choose a Gaussian function

 

which has a bell-shaped graph. It defines a position probability distribution

 

If we normalize this distribution so that   then   and

 

We also have that

  •  
  • the Fourier transform of   is  
  • this defines the momentum probability distribution  
  • and  

The fuzziness of the position and of the momentum of a particle associated with   is therefore the minimum allowed by the "uncertainty" relation:  

Now recall that

 

where   This has the Fourier transform

 

and this defines the position probability distribution

 

Comparison with   reveals that   Therefore,

 

The graphs below illustrate how rapidly the fuzziness of a particle the mass of an electron grows, when compared to an object the mass of a   molecule or a peanut. Here we see one reason, though by no means the only one, why for all intents and purposes "once sharp, always sharp" is true of the positions of macroscopic objects.

 

Above: an electron with   nanometer. In a second,   grows to nearly 60 km.

Below: an electron with   centimeter.   grows only 16% in a second.

 

Next, a   molecule with   nanometer. In a second,   grows to 4.4 centimeters.

 

Finally, a peanut (2.8 g) with   nanometer.   takes the present age of the universe to grow to 7.5 micrometers.