under the evolution generated by . This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law. Examples 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 , which are eigenstates of the momentum operator with momentum . The coefficients of the superposition are , which is the Fourier transform of the initial quantum state . 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 smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making 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. Particle in a box The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For the one-dimensional case in the direction, the time-independent Schrödinger equation may be written With the differential operator defined by the previous equation is evocative of the classic kinetic energy analogue, with state in this case having energy coincident with the kinetic energy of the particle. The general solutions of the Schrödinger equation for the particle in a box are or, from Euler's formula, The infinite potential walls of the box determine the values of and