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:
“Only in a house where one has learnt to be lonely does one have this solicitude for things. One’s relation to them, the daily seeing or touching, begins to become love, and to lay one open to pain.”
—Elizabeth Bowen (1899–1973)
“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)
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“I think the worst thing this nation could do for humanity would be to leave any uncertainty as to our will, our purpose and our capacity to carry out our purpose.”
—Hubert H. Humphrey (1911–1978)
“Whether our feet are compressed in iron shoes, our faces hidden with veils and masks; whether yoked with cows to draw the plow through its furrows, or classed with idiots, lunatics and criminals in the laws and constitutions of the State, the principle is the same; for the humiliations of the spirit are as real as the visible badges of servitude.”
—Elizabeth Cady Stanton (1815–1902)