Editing Zero-point energy (section)

== Atomic physics ==
{{Main|Ground state}}
[[File:QHO-groundstate-animation-color.gif|thumb|The zero-point energy {{math|''E'' {{=}} {{sfrac|''ħω''|2}}}} causes the ground-state of a harmonic oscillator to advance its phase (color). This has measurable effects when several eigenstates are superimposed.]]

The idea of a quantum harmonic oscillator and its associated energy can apply to either an atom or a subatomic particle. In ordinary atomic physics, the zero-point energy is the energy associated with the [[ground state]] of the system. The professional physics literature tends to measure frequency, as denoted by {{mvar|ν}} above, using [[angular frequency]], denoted with {{mvar|ω}} and defined by {{math|''ω'' {{=}} 2''πν''}}. This leads to a convention of writing the Planck constant {{mvar|h}} with a bar through its top ({{mvar|ħ}}) to denote the quantity {{math|{{sfrac|''h''|2π}}}}. In these terms, an example of zero-point energy is the above {{math|''E'' {{=}} {{sfrac|''ħω''|2}}}} associated with the ground state of the quantum harmonic oscillator. In quantum mechanical terms, the zero-point energy is the [[expectation value]] of the Hamiltonian of the system in the ground state.

If more than one ground state exists, they are said to be [[degenerate energy level|degenerate]]. Many systems have degenerate ground states. Degeneracy occurs whenever there exists a [[unitary operator]] which acts non-trivially on a ground state and [[commutator|commutes]] with the Hamiltonian of the system.

According to the [[third law of thermodynamics]], a system at absolute zero temperature exists in its ground state; thus, its [[entropy]] is determined by the degeneracy of the ground state. Many systems, such as a perfect [[crystal lattice]], have a unique ground state and therefore have zero entropy at absolute zero. It is also possible for the highest excited state to have absolute zero temperature for systems that exhibit [[negative temperature]].

The [[wave function]] of the ground state of a [[particle in a box|particle in a one-dimensional well]] is a half-period [[sine wave]] which goes to zero at the two edges of the well. The energy of the particle is given by:
<math display="block">\frac{h^2 n^2}{8 m L^2}</math>
where {{mvar|h}} is the [[Planck constant]], {{mvar|m}} is the mass of the particle, {{mvar|n}} is the energy state ({{math|''n'' {{=}} 1}} corresponds to the ground-state energy), and {{mvar|L}} is the width of the well.