In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:
where is Planck's reduced constant, the imaginary unit, is the spatial coordinate, and a partial derivative (denoted by ) is used instead of a total derivative () since the wave function is also a function of time. The "hat" indicates an operator. The "application" of the operator on a differentiable wave function is as follows:
In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by , i.e. it is a multiplication operator, just as the position operator is a multiplication operator in the position representation. 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. In that case, the canonical momentum is not equal to the kinetic momentum.
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.
Origin from De Broglie plane waves
The momentum and energy operators can be constructed in the following way.
One dimension
Starting in one dimension, using the plane wave solution to Schrödinger's equation of a single free particle,
where is interpreted as momentum in the -direction and is the particle energy. The first order partial derivative with respect to space is
This suggests the operator equivalence
so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the eigenvalue of the above operator.
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.
Three dimensions
The derivation in three dimensions is the same, except the gradient operator del is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is:
and the gradient is
where , , and 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.
Definition (position space)
For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:
where is the gradient operator, is the reduced Planck constant, and is the imaginary unit.
In one spatial dimension, this becomes
This is the expression for the canonical momentum. For a charged particle in an electromagnetic field, during a gauge transformation, the position space wave function undergoes a local U(1) group transformation, and will change its value. 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 and vector potential :
The expression above is called minimal coupling. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.
Properties
Hermiticity
The momentum operator is always a Hermitian operator (more technically, in math terminology a "self-adjoint operator") when it acts on physical (in particular, normalizable) quantum states.
(In certain artificial situations, such as the quantum states on the semi-infinite interval , there is no way to make the momentum operator Hermitian. This is closely related to the fact that a semi-infinite interval cannot have translational symmetry—more specifically, it does not have unitary translation operators. See below.)
Canonical commutation relation
By applying the commutator to an arbitrary state in either the position or momentum basis, one can easily show that: