# Hamiltonian for Charged Particles + EM-Field

• Cryo
In summary: The Hamiltonian for the system of free particles and electromagnetic field is given by:##H=\sum_i \frac{m \dot{r}_i^2}{2} + \int d^3 r \left(\frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right)##In summary, the Hamiltonian for the system of free particles and electromagnetic field (##\mathbf{E}## - electric, ##\mathbf{B}## - magnetic) is given by equation (1) where ##\sum_i## is over all particles, ##m## is the mass of the particles, and ##\dot{\mathbf
Cryo
Gold Member
Summary:

I have found the Hamiltonian for the free particles and the electromagnetic field (##\mathbf{E}## - electric, ##\mathbf{B}## - magnetic) to be (non-relativistic !):

##H=\sum_i \frac{m \dot{r}_i^2}{2} + \int d^3 r \left(\frac{\epsilon_0}{2} E^2 + \frac{1}{2\mu_0} B^2\right)## (1)

where ##\sum_i## is over all the particles, ##m## is the mass of the particles (identical), and ##\dot{\mathbf{r}}_i## is the velocity of ##i##-th particle. Integral ##\int d^3 r## runs over volume large enough for fields to vanish outside it (all particles are containted within this volume).

Is this a correct result? Normally one has things like ##\mathbf{J}.\mathbf{A}## (for current density ##\mathbf{J}## and vector potential ##\mathbf{A}##), but here I am considering both charges and fields to be completely free, thus it seems reasonable that one either stores energy as kinetic energy of charges, or in the electro-magnetic field, which includes the field produced by the charges.

Details. Lagrangian and equations of motion

I start with the Lagrangian for the full system:

##L=\sum_i \frac{m\dot{r}_i^2}{2} + \int d^3 r \left(\frac{\epsilon_0}{2} E^2 - \frac{1}{2\mu_0} B^2 + \mathbf{A}.\mathbf{J} - \phi \rho\right)##

Where scalar and vector potentials are ##\phi## and ##\mathbf{A}##, whilst charge and current densities are ##\rho## and ##\mathbf{J}##. All of these are to be varied in order to bring action to the extremum.

Varying the scalar and vector potential we can readily obtain the Maxwell's equations for vacuum + charge and current density, assuming ##\mathbf{E}=-\boldsymbol{\nabla}\phi-\dot{\mathbf{A}}## and ##\mathbf{B}=\boldsymbol{\nabla}\times\mathbf{A}##. Substituting

##\mathbf{J}=\sum_i q\dot{\mathbf{r}} \delta\left(\mathbf{r}-\mathbf{r_i}\right)##
##\rho=\sum_i q\delta\left(\mathbf{r}-\mathbf{r_i}\right)##

With ##q## being the electrical charge of each particle, we can reduce the Lagrangian to:

##L=\sum_i \left[\frac{m\dot{r}_i^2}{2} + q\dot{\mathbf{r_i}}.\mathbf{A}_i-q\phi_i\right] + \int d^3 r \left(\frac{\epsilon_0}{2} E^2 - \frac{1}{2\mu_0} B^2 \right)## (2)

Where ##\phi_i=\phi\left(t,\,\mathbf{r}_i \right)## and same for vector potential.

This, in turn permits deriving the forces on each charge by variation with respect to ##\mathbf{r_i}^k##

##m\ddot{\mathbf{r}}_i=q\mathbf{E}_i + q \mathbf{\dot{r}}_i\times\mathbf{B}_i## (3)

Details. Hamiltonian

I have obtained Hamiltonian by finding what quantity was conserved as a result of invariance with respect to time-translations, i.e. I started with

##\frac{dL}{dt}=\dots##

and simplified it until I had ##\frac{d}{dt}\left(L-\sum_i \left(q\dot{\mathbf{r}}_i.\mathbf{A}_i - q\phi_i\right)+\int d^3 r\frac{1}{\mu_0}B^2 \right)=0##

It is also easy to verify the reverse:

##\frac{dH}{dt}=\sum_i \mathbf{\dot{r}}_i.m\mathbf{\ddot{r}}_i + \int d^3 r \left(\epsilon_0 \mathbf{E}.\mathbf{\dot{E}}+\frac{1}{\mu_0} \mathbf{B}.\mathbf{\dot{B}}\right)##

From Maxwell's equations we find (ignoring surface effects):

##\int d^3 r \left(\mathbf{E}.\epsilon_0 \mathbf{\dot{E}}\right)=\int d^3 r \left(\frac{-1}{\mu_0}\mathbf{B}.\mathbf{\dot{B}}-\mathbf{E}.\mathbf{J}\right)##

Thus, using Eq. (3):

##\frac{dH}{dt}=\sum_i \mathbf{\dot{r}}_i.\left(q\mathbf{E}_i+q\mathbf{\dot{r}}_i\times\mathbf{B}_i\right)+ \int d^3 r \left(-\mathbf{E}.\mathbf{J}\right)=0##

So, indeed this quantity is conserved. Is this the Hamiltonian?

vanhees71 and Dale
Looks good!

Cryo

## 1. What is a Hamiltonian for charged particles + EM-field?

The Hamiltonian for charged particles + EM-field is a mathematical representation of the total energy of a system consisting of charged particles (such as electrons) and an electromagnetic field. It takes into account the kinetic energy of the particles as well as the potential energy due to their interaction with the electromagnetic field.

## 2. How is the Hamiltonian for charged particles + EM-field derived?

The Hamiltonian for charged particles + EM-field is derived from the classical Hamiltonian, which is a function of the generalized coordinates and momenta of the particles. It is then modified to include the effects of the electromagnetic field, such as the electric and magnetic fields.

## 3. What is the role of the Hamiltonian in quantum mechanics?

In quantum mechanics, the Hamiltonian for charged particles + EM-field is used to describe the time evolution of a quantum system. It is a fundamental operator that determines the state of the system at any given time and is used to calculate the probabilities of different outcomes of a measurement.

## 4. How does the Hamiltonian for charged particles + EM-field relate to the Schrodinger equation?

The Schrodinger equation, which describes the time evolution of a quantum system, can be written in terms of the Hamiltonian for charged particles + EM-field. This equation is used to calculate the wave function of the system, which contains information about the probability of finding the particles in different states.

## 5. What are the applications of the Hamiltonian for charged particles + EM-field?

The Hamiltonian for charged particles + EM-field has many applications in various fields, including quantum mechanics, solid state physics, and particle physics. It is used to study the behavior of charged particles in different environments, such as in a magnetic field or in the presence of an electric potential. It is also used in the development of technologies such as semiconductors and lasers.

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