37.6 - Heisenberg uncertainty principle
In prior sections, we discussed the interpretation of a matter wavefunction as relating to
the likelihood that a particle will be found in a particular location.
Now we will turn to what happens when you try to measure the location of a particle.
We will keep things simpler by assuming that the particle moves along a line, namely
the x axis.
In general, if you want to know where a particle is, you make a measurement. All
measurements are imperfect and have some uncertainty. ǻx represents the uncertainty
in position. If you were making a measurement of the momentum of the particle in the x-
direction, there would also be some uncertainty of momentum ǻpx.
A classical physicist would agree there is some uncertainty in all measurements, just as
you may have learned while doing lab assignments. Better procedures and instruments
can reduce the uncertainty. For example, if you were trying to measure an electron’s
position and momentum 5.00 seconds after it had been propelled by a given electric
field, you might set up 1000 identical trials, use the best instruments available, and
average the results for momentum and for position.
According to classical physics, these uncertainties can in principle be reduced to zero.
There are no limits to knowledge about a classical particle. A classical physicist
considering the theoretical uncertainty of position and momentum could write an
equation like:(classical physics)The German physicist Werner Heisenberg, in contrast, made the bold statement that knowledge is limited. He related the uncertainties in the
particle’s position and momentum along the same axis:This inequality is a statement of the Heisenberg uncertainty principle. It states that there is a tradeoff between reducing the uncertainty in
position and trying to do the same for momentum. One can find situations where ǻx is relatively small, but this low uncertainty in position
comes at the price of a higher uncertainty in the particle’s momentum. Or if the momentum can be well determined, then the particle’s location
will be less precisely known.
Heisenberg’s principle does not describe an “equipment problem”. The problem cannot be solved with a better microscope, or more expensive
lab equipment, or any other technique. Physicists hold that it is a fundamental property of reality: A particle’s position and momentum must
reflect a certain minimum amount of uncertainty because the particle simply cannot have both a definite position and a definite momentum. The
wave/particle is spread out in space.
One important implication of this principle is that a particle can never have zero kinetic energy. If it did, it would be stationary (and have zero
momentum), and then both its position and momentum could be determined. Consistency with the uncertainty relation requires that the particle
must have at least some amount of kinetic energy, even at zero temperature.
Again, as with much quantum physics, the departure from classical physics becomes apparent only when the mass of the particle is very small.Heisenberg uncertainty principle
ǻx = uncertainty in position
ǻpx = uncertainty in momentum
= Planck’s constant/2ʌ
(^694) Copyright 2000-2007 Kinetic Books Co. Chapter 37