An integrated approach to ladder and shift operators for the Morse oscillator, radial Coulomb and radial oscillator potentials
An integrated approach to ladder and shift operators for the Morse oscillator, radial Coulomb and radial oscillator potentials
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DOI：
10.1088/0305-4470/26/7/018
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发表时间：
1993-04
期刊：
影响因子：
--
通讯作者：
I. L. Cooper
中科院分区：
文献类型：
--
作者：
I. L. Cooper
莫尔斯振荡器、径向库仑和径向谐振子问题可以使用各种代数方法精确求解。这些问题对应于so(2,1)代数的不同实现，并且代数生成器的比较可用于识别每对系统之间的映射。在库仑势和谐振子势的情况下，所得的过渡算子充当梯子或能量变化算子，而在莫尔斯势的情况下，它们充当以恒定能量起作用的移位算子。这是 so(2,1) 动力学对称性的结果，由此莫尔斯哈密顿量只能用代数的卡西米尔算子来表达。另一种代数方法，即使用超对称量子力学方法或因式分解，在每种情况下都会产生一组移位算子。可以通过适当的映射来识别各种梯和移位算子之间的关系，并且可以概括这些结果以便将一维莫尔斯振荡器与涉及任意数量的角度维度的径向库仑和径向谐振子势相关。
The Morse oscillator, radial Coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods. These problems correspond to different realizations of the so(2,1) algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems. The resultant transition operators act as ladder, or energy changing, operators in the cases of the Coulomb and harmonic oscillator potentials, whereas they act as shift operators, acting at constant energy, in the case of the Morse potential. This is a consequence of the so(2,1) dynamical symmetry, whereby the Morse Hamiltonian is expressible solely in terms of the Casimir operator of the algebra. An alternative algebraic approach, the use of the method of supersymmetric quantum mechanics, or factorization, produces in each case a set of shift operators. Relations between the various ladder and shift operators may be identified by means of the appropriate mappings, and these results can be generalized so as to relate the one dimensional Morse oscillator to the radial Coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions.