For the case of one particle in one spatial dimension, the definition is:
where ħ is the reduced Planck constant, i the imaginary unit, x is the spatial coordinate, and a partial derivative (denoted by
The "application" of the operator on a differentiable wave function is as follows:
Note that the definition above is the canonical momentum, which is not gauge invariant and not a measurable physical quantity for charged particles in an electromagnetic field.
At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including Niels Bohr, Arnold Sommerfeld, Erwin Schrödinger, and Eugene Wigner.
Its existence and form is sometimes taken as one of the foundational postulates of quantum mechanics.
[1] Starting in one dimension, using the plane wave solution to Schrödinger's equation of a single free particle,
where p is interpreted as momentum in the x-direction and E is the particle energy.
The first order partial derivative with respect to space is
[2] Since the partial derivative is a linear operator, the momentum operator is also linear, and because any wave function can be expressed as a superposition of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component.
These new components then superimpose to form the new state, in general not a multiple of the old wave function.
In three dimensions, the plane wave solution to Schrödinger's equation is:
where ex, ey, and ez are the unit vectors for the three spatial dimensions, hence
This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables.
For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:[3]
where ∇ is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit.
For a charged particle q in an electromagnetic field, during a gauge transformation, the position space wave function undergoes a local U(1) group transformation,[5] and
Therefore, the canonical momentum is not gauge invariant, and hence not a measurable physical quantity.
The kinetic momentum, a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the scalar potential φ and vector potential A:[6]
The momentum operator can be described as a symmetric (i.e. Hermitian), unbounded operator acting on a dense subspace of the quantum state space.
In physics the term Hermitian often refers to both symmetric and self-adjoint operators.
[7][8] (In certain artificial situations, such as the quantum states on the semi-infinite interval [0, ∞), there is no way to make the momentum operator Hermitian.
By applying the commutator to an arbitrary state in either the position or momentum basis, one can easily show that:
[10] The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once.
In quantum mechanics, position and momentum are conjugate variables.
that is, the momentum acting in coordinate space corresponds to spatial frequency,
An analogous result applies for the position operator in the momentum basis,
where δ stands for Dirac's delta function.
to be analytic (i.e. differentiable in some domain of the complex plane), one may expand in a Taylor series about x:
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.