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Concept
Momentum operator
Summary
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 ħ is Planck's reduced constant, i the imaginary unit, x is the spatial coordinate, and a partial derivative (denoted by ) is used instead of a total derivative (d/dx) 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 p, 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.
The momentum and energy operators can be constructed in the following way.
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
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.
Official source
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