Abstract
In classical mechanics an image has rotational symmetry if there is a center point around which the object is turned a certain number of degrees and the object still looks the same, i.e., it matches itself a number of times while it is being rotated. In the language of quantum mechanics, isotropy of space means that the system Hamiltonian keeps invariant by a rotation. In our case the Schrödinger equation with the spherically symmetric fields possesses this kind of property. If the Hamiltonian has rotational symmetry, we can show that the angular momentum operators L commute with the Hamiltonian, which means that the angular momentum is a conserved quantity, i.e., d L/dt=0. Thus, this constant of the motion enables us to reduce the D-dimensional Schrödinger equation to a radial differential equation. This may be explained well from the rotation group theory. In this Chapter, we shall study the rotation operator, the generalized orbital angular momentum operators in higher dimensions, the linear and radial momentum operators, the generalized spherical harmonic polynomials and the Schrödinger equation for a two-body system.
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Notes
- 1.
As an illustration, we present it in three-dimensional case. The unit vectors in spherical polar coordinates are given by
$$ \mathbf{\hat{r}}=\vec{i}\sin\theta \cos\varphi+\vec{j}\sin\theta\sin\varphi+\vec{k}\cos\theta,$$(3.44)and
$$ \everymath{\displaystyle}\begin{array}{l}\frac{\partial}{\partial x}=\sin\theta \cos\varphi \frac{\partial}{\partial r}+\cos\theta \cos\varphi\frac{\partial}{\partial\theta}-\frac{\sin\varphi}{r\sin\theta}\frac{\partial}{\partial\varphi}, \\[6pt]\frac{\partial}{\partial y}=\sin\theta \sin\varphi \frac{\partial}{\partial r}+\cos\theta\sin\varphi\frac{\partial}{\partial \theta}+\frac{\cos\varphi}{r\sin\theta}\frac{\partial}{\partial \varphi}, \\[6pt]\frac{\partial}{\partial z}=\cos\theta \frac{\partial}{\partial r}-\sin\theta \frac{1}{r}\frac{\partial}{\partial \varphi}.\end{array}$$(3.45)If we take ψ(r,θ,ϕ)=R(r)Θ(θ)Φ(ϕ), then we have
$$ \everymath{\displaystyle}\begin{array}{l}\nabla\cdot \hat{\mathbf{r}}\psi(r, \theta,\varphi)=\biggl(\frac{\partial }{\partial r}+\frac{2}{r}\biggr)\psi(r, \theta, \varphi), \\[6pt]\hat{\mathbf{r}}\cdot \nabla \psi(r, \theta,\varphi)=\frac{\partial }{\partial r}\psi(r, \theta,\varphi). \end{array}$$(3.46)
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Dong, SH. (2011). Rotational Symmetry and Schrödinger Equation in D-Dimensional Space. In: Wave Equations in Higher Dimensions. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1917-0_3
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DOI: https://doi.org/10.1007/978-94-007-1917-0_3
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