Next: Complex atoms Up: Wave particle duality Previous: Two slit diffraction

## The uncertainty relations

A consequence of wave-particle duality was discovered by Heisenberg soon after quantum mechanics was developed. Suppose that we wish to measure simultaneously the position and the speed of an object. To do this we shine light at the object, but we do not measure what happens to the light after striking the object. Now, as discussed in connection with electron microscopes, light of a certain wavelength can only see objects down to the size of that wavelength; at smaller scales diffraction effects take place which wash out the image. This provides a theoretical lower limit to how accurately we can measure the object's position. On the other hand, if the light was viewed as a photon, it could give up some or all of its momentum after striking the object, as was discussed in the context of photon or Compton scattering, and this gives an uncertainty in measuring the particle's speed, since we don't know how much momentum the photon gave up. If one puts these two effects together, one finds the Heisenberg uncertainty relations:

where x is the uncertainty in measuring the particle's position, v is the uncertainty in the speed, and m is the mass.

What this means is that one cannot reduce to zero in principle the uncertainty in simultaneously measuring an object's position and speed, no matter how expensive or advanced the measuring equipment being used is. For example, to reduce the uncertainty in position one could use light of a shorter wavelength, but then the associated photon would have more energy, causing a larger uncertainty in the object's speed. Conversely, to reduce the uncertainty in speed onw could use photons with very low energy, so that they would give up very little energy when they collide with the object, but this would then mean that light of a larger wavelength is used to measure the object's position, resulting in a larger uncertainty in position.

As with the case of demonstrating the wave nature of bowling balls, the uncertainty relations have little consequence at larger scales; the theoretical limits they provide in measuring the position and speed of macroscopic objects are far smaller than today's technological limitations. However, at microscopic scales they do apply; for example, saying an electron is somewhere in an atom (about 10-10 m) means that in principle that you cannot measure its speed better than to an accuracy of about 106 m/s.

These and other aspects of wave-particle duality have troubled many people in the past, notably Einstein, who disliked intensely this apparent lack of complete knowledge we have about nature. In one exchange with Bohr, Einstein is reportedly to have said that God does not play dice with the world'', at which point Bohr apparently admonished Einstein to stop telling God what to do''. Many people have tried, and are still trying, to come up with other interpretations and/or alternatives to quantum theory, but without success. As with any scientific theory, the ultimate test of quantum theory is via experiment and, despite its strangeness, it has passed every test so far with flying colours.

Next: Complex atoms Up: Wave particle duality Previous: Two slit diffraction
modtech@theory.uwinnipeg.ca
1999-09-29