arXiv 1002.45072010-02-24quant-ph

2+1维Aharonov-Bohm型场中费米子的零能态

Zero-energy states of fermions in the field of Aharonov--Bohm type in 2+1 dimensions

V. R. Khalilov

备注：: 8 pages

摘要：

在考虑费米子自旋的情况下，解决了2+1维Aharonov-Bohm场中Dirac方程构造自伴哈密顿量的量子力学问题.对于上述具有粒子自旋的狄拉克哈密顿量，我们得到了自伴扩张的单参数族。通过可接受的边界条件选择自伴Hamilton扩张的正确域可以包含具有测度$rdr$的半直线上的正则和奇异（在点${\bf r}=0$）平方可积函数。我们认为奇异解存在的物理原因是费米子的自旋磁矩与Aharonov-Bohm磁场相互作用产生的附加吸引势。在一定的参数范围内存在束缚态。结果表明，零能量的费米子（粒子和反粒子）态是量子系统不稳定性和静态外场产生费米子-反费米子对的可能性的重要信号。

英文摘要：

The quantum-mechanical problem of constructing a self-adjoint Hamiltonian for the Dirac equation in an Aharonov--Bohm field in 2+1 dimensions is solved with taking into account the fermion spin. The one-parameter family of self-adjoint extensions is found for the above Dirac Hamiltonian with particle spin. The correct domain of the self-adjoint Hamiltonian extension selecting by means of acceptable boundary conditions can contain regular and singular (at the point ${\bf r}=0$) square-integrable functions on the half-line with measure $rdr$. We argue that the physical reason of the existence of singular solutions is the additional attractive potential, which appear due to the interaction between the spin magnetic moment of fermion and Aharonov--Bohm magnetic field. For some range of parameters there are bound fermionic states. It is shown that fermion (particle and antiparticle) states with zero energy are intersected what signals on the instability of quantum system and the possibility of a fermion-antifermion pair creation by the static external field.

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