In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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 Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equiv ...
, the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, 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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
consisting of momentum
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s 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
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
. 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
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretic ...
,
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
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, ...
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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
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 vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
s 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
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...
and no
spin, the momentum operator can be written in the position basis as:
where is the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
operator, is the
reduced Planck constant, and is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
In one spatial dimension, this becomes
This is the expression for the
canonical momentum. For a charged particle in an
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical ...
, 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
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to i ...
(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
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation In mathematics, a unitary representation of a grou ...
translation operators. See
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.)
Canonical commutation relation
One can easily show that by appropriately using the momentum basis and the position basis:
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.
Fourier transform
The following discussion uses the
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathem ...
. One may write
so the tilde represents the Fourier transform, in converting from coordinate space to momentum space. It then holds that
that is, the momentum acting in coordinate space corresponds to spatial frequency,
An analogous result applies for the position operator in the momentum basis,
leading to further useful relations,
where stands for
Dirac's delta function.
Derivation from infinitesimal translations
The
translation operator is denoted , where represents the length of the translation. It satisfies the following identity:
that becomes
Assuming the function to be
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
(i.e.
differentiable in some domain of the
complex plane), one may expand in a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
about :
so for
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
values of :
As it is known from
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, the
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
is the generator of
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
, so the relation between translation and momentum operators is:
thus
4-momentum operator
Inserting the 3d momentum operator above and the
energy operator
In quantum mechanics, energy is defined in terms of the energy operator, acting on the wave function of the system as a consequence of time translation symmetry.
Definition
It is given by:
\hat = i\hbar\frac
It acts on the wave function (the ...
into the
4-momentum (as a
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
with
metric signature):
obtains the 4-momentum operator:
where is the
4-gradient, and the becomes preceding the 3-momentum operator. 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 derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
s for
Lorentz covariance.
The
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise form ...
and
Dirac slash of the 4-momentum is given by contracting with the
gamma matrices:
If the signature was , the operator would be
instead.
See also
*
Mathematical descriptions of the electromagnetic field
*
Translation operator (quantum mechanics) In quantum mechanics, a translation operator is defined as an operator which shifts particles and fields by a certain amount in a certain direction.
More specifically, for any displacement vector \mathbf x, there is a corresponding translation ope ...
*
Relativistic wave equations
*
Pauli–Lubanski pseudovector
References
{{Physics operator
Quantum mechanics