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:
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 wavesedit
The momentum and energy operators can be constructed in the following way.[1]
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.
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 dimensionsedit
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 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.
Definition (position space)edit
For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as:[2]
The expression above is called minimal coupling. For electrically neutral particles, the canonical momentum is equal to the kinetic momentum.
Propertiesedit
Hermiticityedit
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.[6]
(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.[7] This is closely related to the fact that a semi-infinite interval cannot have translational symmetry—more specifically, it does not have unitarytranslation operators. See below.)
Canonical commutation relationedit
By applying the commutator to an arbitrary state in either the position or momentum basis, one can easily show that:
As it is known from classical mechanics, the momentum is the generator of translation, so the relation between translation and momentum operators is[8]:[further explanation needed]
^Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
^Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN 0-07-145546-9
^In the position coordinate representation, that is,
^Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
^See Lecture notes 1 by Robert Littlejohn Archived 2012-06-17 at the Wayback Machine for a specific mathematical discussion and proof for the case of a single, uncharged, spin-zero particle. See Lecture notes 4 by Robert Littlejohn for the general case.
^Bonneau,G., Faraut, J., Valent, G. (2001). "Self-adjoint extensions of operators and the teaching of quantum mechanics". American Journal of Physics. 69 (3): 322–331. arXiv:quant-ph/0103153. Bibcode:2001AmJPh..69..322B. doi:10.1119/1.1328351. S2CID 16949018.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-47322-4.