Relation To The Uncertainty Principle
Zero-point energy is fundamentally related to the Heisenberg uncertainty principle. Roughly speaking, the uncertainty principle states that complementary variables (such as a particle's position and momentum, or a field's value and derivative at a point in space) cannot simultaneously be defined precisely by any given quantum state. In particular, there cannot be a state in which the system 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 system (the quantum-mechanical operator giving its energy) can be approximated as
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 is requires solving for the ground state of the system. See quantum harmonic oscillator for details.
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