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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.
If one picks the eigenfunctions of the momentum operator as a set of basis functions, the resulting wave function () is said to be the wave function in momentum space. [5] A feature of quantum mechanics is that phase spaces can come in different types: discrete-variable, rotor, and continuous-variable.
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 ...
The Hamiltonian of the particle is: ^ = ^ + ^ = ^ + ^, where m is the particle's mass, k is the force constant, = / is the angular frequency of the oscillator, ^ is the position operator (given by x in the coordinate basis), and ^ is the momentum operator (given by ^ = / in the coordinate basis).
Specifically, the position and momentum operators in quantum mechanics, usually denoted and , satisfy the canonical commutation relation: [,] = where is the identity operator. It follows that X {\displaystyle X} and P {\displaystyle P} commute with their commutator.
The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis, | , even though, in the momentum basis, this operator amounts to a mere multiplication operator (by iħp).
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 ...