arXiv quant-ph/96110521996-11-27quant-phhep-th

量子正则变换与Schreodinger方程的精确解

Quantum Canonical Transformations and Exact Solution of the Schreodinger Equation

Ali Mostafazadeh

备注：: 7 pages
发表信息：: J.Math.Phys. 38 (1997) 3489-3496
DOI：: 10.1063/1.531864

摘要：

利用含时么正变换研究了$H(t)=\vec R(t).\vec J$形式的显含时哈密顿量的Schreodinger方程，其中$\vec R$是时间的任意实向量值函数，$\vec J$是角动量算符.这种形式的最一般的哈密顿量的Schreodinger方程的解决方案被证明是等价的特殊情况$\vec R=(1,0,ν(t))$。这对应于$\vec J$自旋半表象的驱动二能级原子问题。它还表明，通过要求的幅度的$\vec R$依赖于它的方向，以一种特定的方式，可以解决Schreodinger方程准确。特别是，它表明，对于每一个形式的$H(t)=\vec R(t)\cdot \vec J$的哈密顿有另一个哈密顿具有相同的本征态的Schreodinger方程是完全解决。本文还指出了这些结果在平行输运方程的精确解和SU（2）主丛的精确完整性计算（Yang-Mills规范理论）中的应用。

英文摘要：

Time-dependent unitary transformations are used to study the Schreodinger equation for explicitly time-dependent Hamiltonians of the form $H(t)=\vec R(t).\vec J$, where $\vec R$ is an arbitrary real vector-valued function of time and $\vec J$ is the angular momentum operator. The solution of the Schreodinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case $\vec R=(1,0,ν(t))$. This corresponds to the problem of driven two-level atom for the spin half representation of $\vec J$. It is also shown that by requiring the magnitude of $\vec R$ to depend on its direction in a particular way, one can solve the Schreodinger equation exactly. In particular, it is shown that for every Hamiltonian of the form $H(t)=\vec R(t)\cdot \vec J$ there is another Hamiltonian with the same eigenstates for which the Schreodinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holonomy calculation for SU(2) principal bundles (Yang-Mills gauge theory) is also pointed out.

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