What is the basic difference between the Pauli Exclusion Principle and the Heisenberg Uncertainty Principle?
The Pauli Exclusion Principle simply states that no two fermions (a family of particles with half-integer spin, which includes electrons) may exist in the same energy state.
A good example of this can be seen in the organization of the electrons of an atom into different "shells", each with it's own energy level, where no two electrons have the same energy and momentum.
For a graphic example of this phenomenon, see this website.
Heisenberg's Uncertainty Principle on the other hand relates to one of the key concepts of quantum mechanics- wave particle duality. Simply put, the Uncertainty Principle states that the more accurately you know the momentum or energy of a particle, the less accurately you will be able to know it's location in space or time.
This is due to the fact that in quantum mechanics we view particles as being "wave packets" comprised of several different waves of various frequencies (and hence different energies) all superimposed one upon the other. One of the consequences of this is that in order to create a wave packet that is more "localized" (concentrated at a single point in space or time), we have to add more and more component waves of various energies, thus making it more difficult to determine the total energy or momentum of the configuration.
Click here to see a demonstration of how to create a wave packet.
Loren Chang, Physics Undergrad. Student, UC Irvine
'As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.'