变化磁场中带电粒子的对称性分析
Symmetry analysis for a charged particle in a certain varying magnetic field
摘要：
我们分析了经典的运动方程的粒子运动中存在的静态磁场施加在$ z $方向，其变化为$ {1\over{x^2}} $。我们通过李群分析的方法找到对称性。在相应的量子力学情形下，利用谱生成$su(1,1)$代数的方法，在不显式求解薛定谔方程的情况下，求出了薛定谔方程的能级.列举了李点对称性。我们还发现，对于特定的特征值的向量场包含$ {1\over{x}} {{\p}\over{\p x}}$和$ {1\over {x^2}} {{\p}\over{\p {x}}}$类型的条款和有限的李积的生成元不关闭。
英文摘要：
We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the $ z $ direction, which varies as $ {1\over{x^2}} $. We find the symmetries through Lie's method of group analysis. In the corresponding quantum mechanical case, the method of spectrum generating $su(1,1)$ algebra is used to find the energy levels for the Schroedinger equation without explicitly solving the equation. The Lie point symmetries are enumerated. We also find that for specific eigenvalues the vector field contains $ {1\over{x}} {{\p}\over{\p x}}$ and $ {1\over {x^2}} {{\p}\over{\p {x}}}$ type of terms and a finite Lie product of the generators do not close.
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