Momentum

In physics, momentum is a physical quantity related to the velocity and mass of an object.

Momentum is the Noether charge of translational invariance. As such, even fields as well as other things can have momentum, not just particles. However, in curved spacetime which is not asymptotically Minkowski, momentum isn't defined at all.

Momentum in classical mechanics

In classical mechanics, momentum (traditionally written as p) is defined as the product of mass and velocity. It is thus a vector quantity.

<math>\mathbf{p} = m \mathbf{v}<math>

Impulse

The change in momentum, called the impulse, is equal to force times the change in time.

<math>\Delta \mathbf{p} = \mathbf{F} \cdot \Delta t<math>

<math> \mathbf{I} = \mathbf{F} \cdot t<math>

The SI unit of momentum can be expressed as kg m/s, not to be confused with newton seconds as a newton is defined as mass by acceleration.

An impulse changes the momentum of an object. An impulse is calculated as the integral of force with respect to duration.

<math> \mathbf{I} = \int \mathbf{F}\,dt <math>

using the definition of force yields:

<math> \mathbf{I} = \int\frac{d\mathbf{p}}{dt}\,dt <math>

<math> \mathbf{I} = \int d\mathbf{p} <math>

<math> \mathbf{I} = \Delta \mathbf{p} <math>

See also angular momentum.

Momentum in relativistic mechanics

It is commonly believed that the physical laws should be invariant under translations. Thus, the definition of momentum was changed when Einstein formulated Special relativity so that its magnitude would remain invariant under relativistic transformations. See physical conservation law. We now define a vector, called the 4-momentum thus:

[E/c p]

where E is the total energy of the system, and p is called the "relativistic momentum" defined thus:

<math> E = \gamma mc^2 \;<math>

<math> \mathbf{p} = \gamma m\mathbf{v} <math>

where

<math> \gamma = \frac{1}{\sqrt{1 – v^2/c^2}}<math>.

Setting velocity to zero, one derives that the rest mass and the energy of an object are related by E=mc².

The "length" of the vector that remains constant is defined thus:

<math> \mathbf{p} \cdot \mathbf{p} – E^2 <math>

Massless objects such as photons also carry momentum; the formula is p=E/c, where E is the energy the photon carries and c is the speed of light.

Momentum in quantum mechanics

In quantum mechanics momentum is defined as an operator on the wave function. 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 interchangeable.

For a single particle with no electric charge and no spin, the momentum operator can be written in the position basis as

<math>\mathbf{p}={\hbar\over i}\nabla=-i\hbar\nabla<math>

where is the gradient operator. This is a commonly encountered form of the momentum operator, though not the most general one.

Origin of momentum

Momentum arises from the condition that an experiment must give the same results regardless of the position or velocity of the observer. More formally it is the requirement of invariance under translation. Classical momentum is the result of the invariance of translation in three dimensions. Relativistic momentum as proposed by Albert Einstein arises from the invariance of four-vectors under lorentzian translation. These four-vectors appear spontaneously in the Green's function from quantum field theory.

Figurative use

A process may be said to gain momentum. The terminology implies that it requires effort to start such a process, but that it is relatively easy to keep it going.

See also

• Conservation law
• Slippery slope