The Heisenberg Uncertainty Principle

Recall that a particle has wavelength l = h/p, where p is mv, the momentum, and h is Planck's constant. This is a little too simplistic. Actually a particle is or has a wave packet that consists of a smear of wavelengths over a range D l , and therefore the momentum is smeared out over a range
D p. The particle can be located anywhere in the packet. If it is not smeared at all, one pure l , then the packet is infinite in length. (This does not mean that such a thing exists, it just means we know of no limit to how long the wave packet can be.) More realistic cases are shown below, with increasing variation of l as you go down.

The top one has fairly precise p and imprecise location (small D p, large D x):

The bottom one has imprecise p and precise location (large D p, small D x).

When a boat travels in water, a group of waves is generated, and unless other waves obscure what is going on you might notice that new waves start up at the trailing edge, run through faster than the group, and die out at the leading edge. There is the (fast) phase velocity and the group velocity. Particles work the same way, and the group velocity is the velocity of the particle.

With photons, the length of the packet is also known as the coherence length, and for a gas laser, this is about 1/3 the length of the laser tube. I suspect that for a solid state laser it is about the length of the laser rod. In holography this length determines how far off you can be between the object beam and the reference beam.

Heisenberg found that the ultimate precision to aim for has D pD x = h/2p , or
about 10^-34 J s, due to the wave packet relationships discussed above. In general
D pD x > h/2p if p is in the x direction.
Another formulation is in terms of energy and time:
D ED t > h/2p .
This shows that if an unstable state has a fuzzy energy it will decay more quickly than one with a sharply defined energy.

To save writing, h/2p is called h-bar: h/2p =

Another application of this is on the vacuum. The energy is approximately zero, but we need some D E, so the perfect vacuum has particles and antiparticles popping in and out of existence. This is called the vacuum polarization (particle-antiparticle pair is a dipole). There is no rule that says nature has to be reasonable.

Let this page decay while you tunnel back to the main quantum page.

Other main physics pages:

mechanics
fluids, heat, electricity and magnetism
vibrations and waves
alphabetical junque in my index

Comments, questions: fredrick.gram at tri-c.edu, but remove “at” and use the at symbol.