Charged Particle Motion in Electromagnetic Fields

The classical Hamiltonian for a particle of mass $m$ and charge $q$ moving under the influence of electromagnetic fields is [55]

$$H= \frac{1}{2\,m}\,({\bf p} - q\,{\bf A})\cdot({\bf p} - q\,{\bf A})\ + q\,\phi,$$ (3.71)

where ${\bf A}={\bf A}({\bf x},t)$ and $\phi=\phi({\bf x},t)$ are the vector and scalar potentials, respectively [49]. These potentials are related to the familiar electric and magnetic field-strengths, ${\bf E}({\bf x},t)$ and ${\bf B}({\bf x},t)$ , respectively, via [49]

$${\bf E} = -\nabla\phi-\frac{\partial {\bf A}}{\partial t},$$ (3.72)

$${\bf B} = \nabla\times {\bf A}.$$ (3.73)

Let us assume that expression (3.71) is also the correct quantum mechanical Hamiltonian for a charged particle moving in electromagnetic fields. Obviously, in quantum mechanics, we must treat ${\bf p}$ , ${\bf A}$ , and $\phi$ as operators that do not necessarily commute.

The Heisenberg equations of motion for the components of ${\bf x}$ are

$$\frac{dx_i}{dt} =\frac{[x_i, H]}{{\rm i}\,\hbar}.$$ (3.74)

However,

$$[x_i, H] = {\rm i}\,\hbar\,\frac{\partial H}{\partial p_i} = \frac{{\rm i}\,\hbar}{m}\,(p_i-q\,A_i),$$ (3.75)

where use has been made of Equations (3.33) and (3.71). It follows that

$$m\,\frac{d{\bf x}}{dt} = \mbox{\boldmath$\Pi$} ,$$ (3.76)

where

$$\mbox{\boldmath$\Pi$} = {\bf p} - q\,{\bf A}.$$ (3.77)

Here, $\Pi$ is referred to as the mechanical momentum, whereas ${\bf p}$ is termed the canonical momentum.

It is easily seen that

$$[{\mit\Pi}_i, {\mit\Pi}_j] = q\,[p_j, A_i] -q\,[p_i, A_j].$$ (3.78)

However,

$$[p_j, A_i] = -{\rm i}\,\hbar\,\frac{\partial A_i}{\partial x_j},$$ (3.79)

where we have employed Equation (3.34). Thus, we obtain

$$[{\mit\Pi}_i, {\mit\Pi}_j] = {\rm i}\,\hbar\,q\left(\frac{\partia... ...rac{\partial A_i}{\partial x_j}\right)= {\rm i}\,\hbar\,q\,\epsilon_{ijk}\,B_k,$$ (3.80)

because [from Equation (3.73)]

$$B_i = \epsilon_{ijk}\,\frac{\partial A_k}{\partial x_j}.$$ (3.81)

Here, $\epsilon_{ijk}$ is the totally antisymmetric tensor (that is, $\epsilon_{ijk}=1$ if $i$ , $j$ , $k$ is a cyclic permutation of $1$ , $2$ , $3$ ; $\epsilon_{ijk}=-1$ if $i$ , $j$ , $k$ is an anti-cyclic permutation of $1$ , $2$ , $3$ ; and $\epsilon_{ijk}=0$ otherwise), and we have used the standard result $\epsilon_{ijk}\,\epsilon_{iab}=\delta_{ja}\,\delta_{kb}-\delta_{jb}\,\delta_{ka}$ , as well as the Einstein summation convention (that repeated indices are implicitly summed from $1$ to $3$ ) [92].

We can write the Hamiltonian (3.71) in the form

$$H = \frac{{\mit\Pi}^{\,2}}{2\,m} +q\,\phi.$$ (3.82)

The Heisenberg equation of motions for the components of $\Pi$ are

$$\frac{d{\mit\Pi}_i}{dt}=\frac{[{\mit\Pi}_i, H]}{{\rm i}\,\hbar} + \frac{\partial {\mit\Pi}_i}{\partial t}.$$ (3.83)

Here, we have taken into account the fact that ${\mit\Pi}_i$ depends explicitly on time through its dependence on $A_i({\bf x}, t)$ . However,

$$[{\mit\Pi}_i, {\mit\Pi}^{\,2}] = {\mit\Pi}_j\,[{\mit\Pi}_i, {\mit... ...\left(\epsilon_{ijk}\,{\mit\Pi}_j\,B_k-\epsilon_{ijk}\,B_j\,{\mit\Pi}_k\right),$$ (3.84)

where use has been made of Equation (3.80). Moreover,

$$\frac{\partial {\mit\Pi}_i}{\partial t} = -q\,\frac{\partial A_i}{\partial t},$$ (3.85)

$$[{\mit\Pi}_i, \phi] = [p_i, \phi] = -{\rm i}\,\hbar\,\frac{\partial\phi}{\partial x_i},$$ (3.86)

where we have employed Equation (3.33). The previous five equations yield

$$\frac{d\mbox{\boldmath$\Pi$}}{dt} = q\,{\bf E} + \frac{q}{2\,m}\l... ...box{\boldmath$\Pi$}\times {\bf B} - {\bf B}\times \mbox{\boldmath$\Pi$}\right),$$ (3.87)

which can be combined with Equation (3.76) to give

$$m\,\frac{d^{\,2} {\bf x}}{dt^{\,2}} = q\,{\bf E}+ \frac{q}{2}\left(\frac{d{\bf x}}{dt}\times {\bf B} - {\bf B}\times \frac{d{\bf x}}{dt}\right).$$ (3.88)

This equation of motion is a generalization of the Ehrenfest theorem that takes electromagnetic fields into account. The fact that Equation (3.88) is analogous in form to the corresponding classical equation of motion (given that $d{\bf x}/dt$ and ${\bf B}$ commute in classical mechanics) justifies our earlier assumption that Equation (3.71) is the correct quantum mechanical Hamiltonian for a charged particle moving in electromagnetic fields.