Electron in a field: Canonical momentum versus kinematical momentum

In summary, the canonical momentum is the derivative of the Lagrangian with respect to velocity and in systems without a magnetic field, it is equal to the kinematic momentum. However, in systems with a magnetic field, the vector potential also contributes to the canonical momentum. For more information, see the link provided.
  • #1
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Can anyone explain to me the definition of canonical and kinetic momentum?

The Schrödinger equation for an electron in a laser field can be written in the length gauge or the velocity gauge.
In the litterature it is often said that in the length gauge the kinematical momentum is equal to the canonical momentum k=p, while in the velocity gauge the kinematical momentum is the canonical momentum added by the vector potential k=p+A.

Can anyone elaborate on this or has any ref I can look at?
 
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  • #2
The canonical momentum is defined as the derivative of the Lagrangian with respect to velocity. In a system where the potential doesn't have a velocity dependence (ie. one that doesn't have a magnetic field), this will just be the kinematic momentum, p = mv.

For a system with a charged particle in a magnetic field, the potential will have a term with the dot product of the velocity with the vector potential. So when you take a derivative with respect to velocity, in addition to the mv that you get from the kinetic energy, you also get a contribution from the vector potential.

See here for example:
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/ParticleMagneticField.htm
 

FAQ: Electron in a field: Canonical momentum versus kinematical momentum

What is the difference between canonical momentum and kinematical momentum for an electron in a field?

Canonical momentum is the momentum of a particle that takes into account its position and the field it is in, while kinematical momentum is the momentum of a particle based solely on its velocity. In an electron in a field, canonical momentum takes into account the electron's position and the electric or magnetic field it is in, while kinematical momentum only considers the electron's velocity.

How does the electric field affect the canonical momentum of an electron?

The electric field affects the canonical momentum of an electron by exerting a force on the electron, causing it to accelerate. This acceleration changes the electron's position, and therefore, its canonical momentum. The strength and direction of the electric field will determine the magnitude and direction of the change in canonical momentum.

Can the canonical momentum of an electron ever be equal to its kinematical momentum?

No, the canonical momentum and kinematical momentum of an electron can never be equal. This is because the canonical momentum takes into account the electron's position, which is always changing due to the presence of the electric or magnetic field. Kinematical momentum, on the other hand, only considers the electron's velocity, which can remain constant even as the electron's position changes.

How does the magnetic field affect the canonical momentum of an electron?

The magnetic field affects the canonical momentum of an electron by exerting a force on the electron perpendicular to both the magnetic field and the electron's velocity. This force causes the electron to follow a circular path, which changes its position and therefore, its canonical momentum. The strength and direction of the magnetic field will determine the magnitude and direction of the change in canonical momentum.

How is the canonical momentum of an electron related to its energy?

The canonical momentum of an electron is related to its energy through the Hamiltonian, which is a mathematical function that describes the energy of a system. The Hamiltonian includes terms for both the kinetic energy (related to the kinematical momentum) and the potential energy (related to the canonical momentum), showing the direct relationship between these two types of momentum and the electron's overall energy.

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