Does the Heisenberg uncertainty principle apply to the free particle?

Question

Does the Heisenberg uncertainty principle apply to the free particle? If it does or doesn't how would one go about proving this mathematically?

Answer 1

The Heisenberg uncertainty principle follows from the axioms/postulates of quantum mechanics. This means that it is a property of the theory as a whole, regardless of the situation. Therefore, (quite trivially) the Heisenberg uncertainty principle applies to the free particle...in the quantum mechanical picture of the world.

The derivation of the Heisenberg uncertainty principle is pretty standard and can be found in most undergrad QM textbooks. (See Griffiths, Shankar, or Sakurai.)