Hamiltonian of an electron under EM radiation

In summary: This is because there are other forms of canonical momentum, including the electron's in an EM field. In the Lagrangian, one finds the canonical momentum as a derivative of the kinematic momentum. And in the Hamiltonian, the canonical momentum is added to the kinetic and potential energies.
  • #1
blue_leaf77
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I might have learned what I am going to ask during my electrodynamics class long time ago but just that do not remember it now.
I always wonder why does an electron moving in space with EM radiation have Hamiltonian of the form
## H = \left( \mathbf{p}-e\mathbf{A}/c \right)^2/2m +e\phi## where ##\mathbf{A}## and ##\phi## are vector and scalar potentials, respectively? I want to study it myself and now I'm having the EM book by Griffith, in case you know that such derivation exists in that book I would prefer that you tell me which chapter it is, otherwise I'm fine if you want to explain it here instead.
 
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  • #2
Write down the Lagrangian of the charged particle in an EM field and compute the canonical momentum. It differs from the usual momentum expression.
Then construct the Hamiltonian.
 
  • #3
Ok I guess I need to go to classical mechanics.
After some reading, I found that in the case of electron in an EM field it seems it's the canonical momentum ##p_i## that enters in the usual commutation relation with ##x_i##, not the kinematic momentum. Why is this so? The commutation relation between p and x originally follows from the definition of momentum as the translation operator. But in the derivation process, at least in the book I read, the author didn't made any reference as to whether canonical or kinematic momentum that defines the translation operator. The fact that in the case of electron's Hamiltonian in an EM field it's the canonical momentum that enters in the commutation relation with x, do we define it simply by an analogy with the classical Poisson bracket? I was just guessing though that the ##p_i## in classical Poisson bracket is the canonical one, I haven't checked myself.
 
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  • #4
Any idea?
 
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1. What is the Hamiltonian of an electron under electromagnetic (EM) radiation?

The Hamiltonian of an electron under EM radiation is a mathematical operator that describes the total energy of the electron in a system that includes both the electron's kinetic energy and its potential energy due to the presence of EM radiation. It is commonly denoted as H and is an important concept in quantum mechanics.

2. How does the Hamiltonian change when an electron is under the influence of EM radiation?

The Hamiltonian of an electron under EM radiation changes due to the additional potential energy term that is added to the Hamiltonian. This term takes into account the interaction of the electron with the electric and magnetic fields of the EM radiation, and it affects the dynamical behavior of the electron.

3. What is the significance of the Hamiltonian in the study of quantum mechanics?

The Hamiltonian is a fundamental concept in quantum mechanics as it describes the total energy of a system, including the energy of the electron under EM radiation. It is used to solve the Schrödinger equation, which is a key equation in quantum mechanics that describes the time evolution of a system's wave function.

4. How does the Hamiltonian of an electron under EM radiation relate to the energy levels of an atom?

The Hamiltonian of an electron under EM radiation is related to the energy levels of an atom through the quantization of energy. The energy levels of an atom are determined by the values of the Hamiltonian operator, which correspond to the allowed energy states of the electron in the atom.

5. Can the Hamiltonian of an electron under EM radiation be used to predict the behavior of electrons in a material?

Yes, the Hamiltonian of an electron under EM radiation can be used to predict the behavior of electrons in a material. By solving the Schrödinger equation with the Hamiltonian, we can determine the wave function of the electrons in the material, which allows us to make predictions about their behavior and properties such as conductivity and magnetism.

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