Hamiltonian for a free electron in electromagnetic field

In summary, the conversation discusses how to derive the Hamiltonian for a free electron in an electromagnetic field mathematically. It also touches on the issue of deriving the Lagrangian for this system and how it is typically approached in literature. The conversation concludes with the suggestion to plug the final form of the Lagrangian into the Lagrange-Euler equation to obtain the expression for the Lorentz force.
  • #1
athosanian
67
8
hello, how to derive the hamiltonian for a free electron in electromagnetic field mathematically ?
for a first step what is the lagrangian for a free electron in the EM field in classical mechanics ?
the physics textbook always like to give the results directly.
 
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  • #3
thanks, I understand how to derive the Hamiltonian from Lagrangian.,
however, the file gives the Lagrangian directly without any reason.
 
  • #4
Up to now, all literatures I have read always deduce the Lagrangian from the equation of motion (Lorentz force law) by, apparently, inspection - kind of trying to seek for the right form of L and check it by plugging it into the equation of motion of L. I have never seen a reference doing the reverse way, that is deriving the Lagrangian from the energies like we usually do in systems involving gravitational potential only. Perhaps there are such references but I just never come across them. Anyway, just for a check you can plug in the final form of the Lagrangian you found there into the Lagrange-Euler equation (the equation of motion), I think you should end up with the expression of Lorenty force.
 
  • #5
yes, I put the L into the E-L equation, could derive the Newton's equaiton of motion. But I think it is very smart to work out the lagrangian.
 

FAQ: Hamiltonian for a free electron in electromagnetic field

1. What is a Hamiltonian for a free electron in an electromagnetic field?

A Hamiltonian for a free electron in an electromagnetic field is a mathematical operator that describes the total energy of a free electron in the presence of an external electromagnetic field. It takes into account the kinetic energy of the electron as well as its interaction with the electric and magnetic fields.

2. How is the Hamiltonian for a free electron in an electromagnetic field derived?

The Hamiltonian for a free electron in an electromagnetic field is derived from the Schrödinger equation, which is the fundamental equation of quantum mechanics. By including the electromagnetic potential in the equation, we can obtain the Hamiltonian operator for the system.

3. What is the significance of the Hamiltonian for a free electron in an electromagnetic field?

The Hamiltonian for a free electron in an electromagnetic field allows us to predict the behavior of an electron in the presence of an external electromagnetic field. It is a crucial tool for understanding and studying the quantum mechanical properties of particles in electromagnetic fields.

4. What are the key components of the Hamiltonian for a free electron in an electromagnetic field?

The Hamiltonian for a free electron in an electromagnetic field includes the kinetic energy operator, the potential energy operator, and the spin operator. These components take into account the motion, interaction, and intrinsic spin of the electron in the electromagnetic field.

5. Can the Hamiltonian for a free electron in an electromagnetic field be applied to other particles?

Yes, the Hamiltonian for a free electron in an electromagnetic field can be extended to other particles, such as protons or neutrons, by incorporating their unique properties into the equation. It also serves as a foundation for more complex Hamiltonians that describe the behavior of multiple particles in electromagnetic fields.

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