UDC 621.319.7

PHYSICS

I. M. TERNOV, V. G. BAGROV

MOTION OF A NEUTRAL FERMION POSSESSING AN ANOMALOUS MAGNETIC MOMENT IN AN ELECTRIC FIELD

(Presented by Academician N. N. Bogolyubov, 18 X 1965)

Let us consider the motion of a neutral Dirac particle with anomalous magnetic moment \(\mu=-\mu_0\) \((\mu_0>0)\) in a static electric field having a component along the \(z\)-axis of the Cartesian coordinate system, \(\mathcal E(z)\). A solution of the Dirac equation (see \({}^{1}\))

\[ i\hbar \frac{\partial}{\partial t}\Psi=\hat{\mathcal H}\Psi =\{c(\hat{\boldsymbol{\alpha}}\mathbf p)+\rho_3mc^2+\mu_0\rho_2\sigma_3\mathcal E\}\Psi \tag{1} \]

will be an eigenfunction of the Hamiltonian operator \(\hat{\mathcal H}\), as well as of the operators \(\hat p_x\) and \(\hat p_y\) that commute with it:

\[ \hat{\mathcal H}\Psi=E\Psi;\qquad \hat p_x\Psi=\hbar k_1\Psi;\qquad \hat p_y\Psi=\hbar k_2\Psi . \tag{2} \]

To characterize the spin states it is expedient to introduce the polarization-tensor operator \(\hat\Lambda=\rho_3[\hat{\boldsymbol{\sigma}}\mathbf p]_3\), which also commutes with the Hamiltonian and characterizes the projection of the spin onto the direction perpendicular to the external field and to the velocity:

\[ \hat\Lambda\Psi=\hbar\lambda\zeta\Psi \tag{3} \]

(for the transformation properties of this operator, see \({}^{2}\)). Taking (1), (2), and (3) into account, the wave function may be represented in the form

\[ \Psi=e^{-icKt}\frac{e^{i(k_1x+k_2y)}}{\sqrt{L_1L_2}} \begin{pmatrix} f_1(z)\\ i\zeta e^{i\varphi}f_1(z)\\ if_2(z)\\ \zeta e^{i\varphi}f_2(z) \end{pmatrix}, \tag{4} \]

where \(c\hbar K=E\) is the particle energy; \(k_1=k\cos\varphi\); \(k_2=k\sin\varphi\); \(\lambda=k=\sqrt{k_1^2+k_2^2}\); \(\zeta=\pm1\) corresponds to the two possible spin orientations. In this case the functions \(f_1\) and \(f_2\) must be determined from the system of equations

\[ \left\{\frac{d}{dz}\pm\left(\frac{\mu_0}{e\hbar}\mathcal E(z)+\zeta K\right)\right\}f_{1,2} \mp (K-k_0)f_{2,1}=0, \tag{5} \]

in which the upper and lower signs refer to the first and second component, respectively.

The fundamental possibility of the interaction of a neutral fermion with a static electric field of Coulomb type was noted in \({}^{3}\) (see also \({}^{4}\)). We consider here the case when the electric field \(\mathcal E(z)\) has the form

\[ \frac{\mu_0}{e\hbar}\mathcal E(z)=\gamma z+\gamma_0\qquad(\gamma>0). \tag{6} \]

Then the system of equations (5) admits the exact solution

\[ f_1=\frac{1}{2}\sqrt{1+\frac{k_0}{K}}\,U_n(t),\qquad f_2=\frac{1}{2}\sqrt{1-\frac{k_0}{K}}\,U_{n+1}(t), \tag{7} \]

in which \(U_n(t)=\sqrt[4]{\gamma/\pi}\,(2^n n!)^{-1/2}e^{-t^2/2}H_n(t)\) are Hermite functions, while the variable \(t\) is equal to

\[ t=\sqrt{\gamma}\,z+(\xi k+\gamma_0)/\sqrt{\gamma}. \tag{8} \]

In this case the energy spectrum is completely discrete and depends only on a single quantum number \(n=0,1,2,\ldots\)

\[ K=\sqrt{k_0^2+2\gamma(n+1)}. \tag{9} \]

The solutions obtained physically correspond to the motion of the particle along a circle lying in a plane parallel to the \(z\)-axis and passing through the vector \(\mathbf{p}=(p_1,p_2)\). The radius of this circle is

\[ R^2=2\,\overline{(z-\bar z)^2} =\frac{2}{\gamma}\left(n+1-\frac{k_0}{2K}\right) =\frac{1}{\gamma}\left(\frac{K^2-k_0^2}{\gamma}-\frac{k_0}{K}\right), \tag{10} \]

and the position of the center of the orbit is characterized by the quantity

\[ z_0=\bar z=-(\xi k+\gamma_0)/\gamma. \tag{11} \]

For definite values of the momentum \(k=\sqrt{k_1^2+k_2^2}\), the spin projection \(\xi\), and the quantity \(\gamma_0\) \((\gamma_0=-\xi k)\), the center of the circle is at the origin of coordinates.

Moscow State University
named after M. V. Lomonosov

Received
13 X 1965

CITED LITERATURE

1. W. Pauli, Rev. Mod. Phys., 13, 203 (1941).
2. A. A. Sokolov, I. M. Ternov, DAN, 153, 1052 (1963).
3. E. L. Foldy, Phys. Rev., 83, 688 (1951).
4. I. A. Akhiezer, V. B. Berestetskii, Quantum Electrodynamics, Moscow, 1959, § 15.