论文标题
整数和半整数角动量的Hurwitz-Hopf地图和谐波波函数
The Hurwitz-Hopf Map and Harmonic Wave Functions for Integer and Half-Integer Angular Momentum
论文作者
论文摘要
整数的谐波波函数和半整数角动量由Euler Angles $（θ，ϕ，ψ）$表示，该$定义了$ SO（3）$的旋转，而Euclidean Norm在$ {\ Mathbb r}^3 $中。在Schwinger的经典工作之后，使用$ 2 $维谐波振荡器来生产升高和降低操作员，以将波浪函数的总角动量特征值以半单位为单位。代表空间的性质$ \ MATHCAL H $从双重覆盖组同构$ su（2）\ to So（3）$接近，并且使用Hurwitz-Hopf Map $ h：{\ Mathbb r}^4 \ to {\ Mathbb r}^3 $。它显示了如何重新考虑$ h $作为2-1组映射，$ g_0 = {\ Mathbb r}^+\ times su（2）\ to {\ Mathbb r}^+times so（3）$，将其转换为sigsionment $（z_1，z_1，z_1，z_2） $（z_1，z_2）$的复杂变量。它显示了$ g_0 $的谎言代数如何与两个Heisenberg Lie代数为$ 2 $二维（Schwinger's）的谐波振荡器，由操作员$ \ \ {z_1，z_1，z_2，z_1，z_2，\ bar {z}}整个运营商将代数关闭，要么将$ 13 $维的Lie代数或$（4 | 8）$ - 尺寸 - 尺寸lie superalgebra关闭。 $ \ Mathcal H $中的波函数可以用复杂坐标$（Z_1，Z_2）$中的多项式及其复杂的共轭物编写，并且该表示形式是通过$ G_0 $ $ G_0 $的各种最高权重（或最低权重）矢量表示明确构建的。考虑到电子旋转的新的非相关量子（Schrödinger样）方程，以$（r，θ，ϕ，ψ）$和时间$ t $进行引入和表达。特此可以根据引入的谐波波函数准确求解该方程。
Harmonic wave functions for integer and half-integer angular momentum are given in terms of the Euler angles $(θ,ϕ,ψ)$ that define a rotation in $SO(3)$, and the Euclidean norm in ${\mathbb R}^3$. Following a classical work by Schwinger, $2$-dimensional harmonic oscillators are used to produce raising and lowering operators that change the total angular momentum eigenvalue of the wave functions in half units. The nature of the representation space $\mathcal H$ is approached from the double covering group homomorphism $SU(2)\to SO(3)$ and the topology involved is taken care of by using the Hurwitz-Hopf map $H:{\mathbb R}^4\to{\mathbb R}^3$. It is shown how to reconsider $H$ as a 2-to-1 group map, $G_0={\mathbb R}^+\times SU(2)\to {\mathbb R}^+\times SO(3)$, translating it into an assignment $(z_1,z_2)\mapsto (r,θ,ϕ,ψ)$ whose domain consists of pairs $(z_1,z_2)$ of complex variables. It is shown how the Lie algebra of $G_0$ is coupled with two Heisenberg Lie algebras of $2$-dimensional (Schwinger's) harmonic oscillators generated by the operators $\{z_1,z_2,\bar{z}_1,\bar{z}_2\}$ and their adjoints. The whole set of operators gets algebraically closed either into a $13$-dimensional Lie algebra or into a $(4|8)$-dimensional Lie superalgebra. The wave functions in $\mathcal H$ can be written in terms of polynomials in the complex coordinates $(z_1,z_2)$ and their complex conjugates, and the representations are explicitly constructed via the various highest weight (or lowest weight) vector representations of $G_0$. A new non-relativistic quantum (Schrödinger-like) equation for the hydrogen atom that takes into account the electron spin is introduced and expressed in terms of $(r,θ,ϕ,ψ)$ and the time $t$. The equation may be solved exactly in terms of the harmonic wave functions hereby introduced.
