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The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is = where is the electron charge, is the position of the electron relative to the nucleus, = | | is the magnitude of the relative position, the potential term is due to the Coulomb interaction, wherein is the permittivity of free space and = + is the 2-body ...
For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space. [42] The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for ...
From the equations, the power series must start with at least an order of to satisfy the real part of the equation; for a good classical limit starting with the highest power of the Planck constant possible is preferable, which leads to = = and = = (), with the following constraints on the lowest order terms, () = (()) and () =
The phase-space formulation is a formulation of quantum mechanics that places the position and momentum variables on equal footing in phase space.The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.
Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum ...
In particular, this means that a system of N interacting particles in 3 dimensions is described by one vector whose components in a basis where all the X are diagonal is a mathematical function of 3N-dimensional space describing all their possible positions, effectively a much bigger collection of values than the mere collection of N three ...