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This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.
the representation of the position operator in the momentum basis is naturally defined by (^) = (^), for every wave function (tempered distribution) ; p {\displaystyle \mathrm {p} } represents the coordinate function on the momentum line and the wave-vector function k {\displaystyle \mathrm {k} } is defined by k = p / ℏ {\displaystyle \mathrm ...
between the position operator x and momentum operator p x in the x direction of a point particle in one dimension, where [x, p x] = x p x − p x x is the commutator of x and p x , i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and is the unit operator. In general, position and momentum are vectors of operators and their ...
Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with dimension of mass ⋅ length ⋅ time −1. Mathematically, the duality between position and momentum is an example of Pontryagin duality .
The Newton–Wigner position operators x 1, x 2, x 3, are the premier notion of position in relativistic quantum mechanics of a single particle. They enjoy the same commutation relations with the 3 space momentum operators and transform under rotations in the same way as the x, y, z in ordinary QM.
Since translation operators all commute with each other (see above), and since each component of the momentum operator is a sum of two scaled translation operators (e.g. ^ = (^ ((,,)) ^ ((,,)))), it follows that translation operators all commute with the momentum operator, i.e. ^ ^ = ^ ^ This commutation with the momentum operator holds true ...
Let ψ be the wavefunction for a quantum system, and ^ be any linear operator for some observable A (such as position, momentum, energy, angular momentum etc.). If ψ is an eigenfunction of the operator A ^ {\displaystyle {\hat {A}}} , then
Informally stated, with certain technical assumptions, every representation of the Heisenberg group H 2n + 1 is equivalent to the position operators and momentum operators on R n. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra ) on a symplectic space of dimension 2 n .
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