Mathematical Details
In the Heisenberg picture of quantum mechanics the state vector, does not change with time, and an observable A satisfies
where H is the Hamiltonian and is the commutator of A and H. In some sense, the Heisenberg picture is more natural and fundamental than the Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture.
This approach has a similarity to classical physics: by replacing the commutator above by the Poisson bracket, the Heisenberg equation becomes an equation in Hamiltonian mechanics.
By the Stone-von Neumann theorem, the Heisenberg picture and the Schrödinger picture are unitarily equivalent.
Read more about this topic: Heisenberg Picture
Famous quotes containing the words mathematical and/or details:
“As we speak of poetical beauty, so ought we to speak of mathematical beauty and medical beauty. But we do not do so; and that reason is that we know well what is the object of mathematics, and that it consists in proofs, and what is the object of medicine, and that it consists in healing. But we do not know in what grace consists, which is the object of poetry.”
—Blaise Pascal (1623–1662)
“Patience is a most necessary qualification for business; many a man would rather you heard his story than granted his request. One must seem to hear the unreasonable demands of the petulant, unmoved, and the tedious details of the dull, untired. That is the least price that a man must pay for a high station.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)