Search results
Results From The WOW.Com Content Network
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. In particular, if a function is given in position space, f(r ...
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.
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle. [1]
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 ...
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below). An operator in n-dimensional space can be written:
Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry) of space (position in space is the canonical conjugate quantity to momentum). That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position; this is a special case of Noether's theorem . [ 25 ]
It is for this reason that the momentum operator is referred to as the generator of translation. [2] A nice way to double-check that these relations are correct is to do a Taylor expansion of the translation operator acting on a position-space wavefunction.
Nevertheless, as explained in the introduction, for states that are highly localized in space, the expected position and momentum will approximately follow classical trajectories, which may be understood as an instance of the correspondence principle. Similarly, we can obtain the instantaneous change in the position expectation value.