section_path: Examples/Free particle
section_name: Free particle
paragraph_text: Free particle
  
  The simplest example of a quantum system with a position degree of freedom is a free particle in a single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy:
  
  The general solution of the Schrödinger equation is given by
  
  which is a superposition of all possible plane waves e i (k x − ℏ k 2 2 m t) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}}, which are eigenstates of the momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k}. The coefficients of the superposition are ψ ^ (k, 0) {\displaystyle {\hat {\psi }}(k,0)}, which is the Fourier transform of the initial quantum state ψ (x, 0) {\displaystyle \psi (x,0)}.
  
  It is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet:
  
  which has Fourier transform, and therefore momentum distribution
  
  We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making a {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle.
  
  As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that the position becomes more and more uncertain. The uncertainty in momentum, however, stays constant.