The time evolution of a quantum state is described by the Schrödinger equation:
and this provides the lower bound on the product of standard deviations:
The simplest example of 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 previous equation is evocative of the classic kinetic energy analogue,
The general solutions of the Schrödinger equation for the particle in a box are
As in the classical case, the potential for the quantum harmonic oscillator is given by
This problem can either be treated by directly solving the Schrödinger equation, which is not trivial, or by using the more elegant "ladder method" first proposed by Paul Dirac. The eigenstates are given by
This is another example illustrating the discretization of energy for bound states.
and the probabilities that it will be detected at the right or at the top are given respectively by
One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.
The following titles, all by working physicists, attempt to communicate quantum theory to lay people, using a minimum of technical apparatus.