Zero-point energy is fundamentally related to the Heisenberg uncertainty principle. Roughly speaking, the uncertainty principle states that complementary variables (such as a particles position and momentum, or a fields value and derivative at a point in space) cannot simultaneously be specified precisely by any given quantum state. In particular, there cannot exist a state in which the system simply sits motionless at the bottom of its potential well, for then its position and momentum would both be completely determined to arbitrarily great precision. Therefore, the lowest-energy state (the ground state) of the system must have a distribution in position and momentum that satisfies the uncertainty principle, which implies its energy must be greater than the minimum of the potential well.
Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving its energy) can be approximated as a quantum harmonic oscillator,
where  is the minimum of the classical potential well.
The uncertainty principle tells us that
making the expectation values of the kinetic and potential terms above satisfy
The expectation value of the energy must therefore be at least
where  is the angular frequency at which the system oscillates.
A more thorough treatment, showing that the energy of the ground state actually saturates this bound and is exactly , requires solving for the ground state of the system.