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.
Read more about this topic: Zero-point Energy
Famous quotes containing the words relation to the, relation to, relation, uncertainty and/or principle:
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
—Terri Apter (20th century)
“It was your severed image that grew sweeter,
That floated, wing-stiff, focused in the sun
Along uncertainty and gales of shame
Blown out before I slept. Now you are one
I dare not think alive: only a name
That chimes occasionally, as a belief
Long since embedded in the static past.”
—Philip Larkin (1922–1986)
“... it is not the color of the skin that makes the man or the woman, but the principle formed in the soul. Brilliant wit will shine, come from whence it will; and genius and talent will not hide the brightness of its lustre.”
—Maria Stewart (1803–1879)