程度很差 不太了解这段文字的意义(不是翻译问题) 不知道能不能请高手解释 感谢 ◎To put the separation method of solving PDEs in perspective,let us view it as a consequence of a symmetry of the PDE.Take the stationary Schrodinger equation HΨ=EΨ as an example with a potential V(r) depending only on the radial distance r.Then this PDE is invariant under rotaions that comprise the group SO(3).Its diagonal generator is the orbital angular momentum operator 2 Lz=-i(∂/∂φ),and its quadratic(Casimir) invariant is L .Since both commute with H,we end up with three separate eigenvalue equations: 2 HΨ=EΨ, L Ψ=l(l+1)Ψ, LzΨ=mΨ. 2 2 2 2 Upon replacing Lz in L by its eigenvalue m ,the L PDE becomes Legendre's ODE ,and similarly HΨ=EΨ becomes the radial ODE of the separation method in 2 spherical polar coordinates upon substituting the eigenvalue l(l+1) for L . ◎For cylindrical coordinates the PDE is invariant under rotations about the z-axis only,which form a subgroup of SO(3),This invariance yields the generator Lz=-i(∂/∂φ) and separate azimuthal ODE LzΨ=mΨ as before.Invariance under translations along the z-axis with the generator -i(∂/∂z) gives the separate ODE in the z-variable provided the boundary conditions obey the same symmetry. The potential V=V(ρ) or V=V(z) depends on one variable,as a rule. ◎In general,there are n mutually commuting generators H with eigenvalues m i i of the (classical) Lie group G of rank n and the corresponding Casimir invariants C with eigenvalues c ,which yield the separate ODEs i i H Ψ=m Ψ C Ψ=c Ψ i i i i in addition to the radial ODE HΨ=EΨ. --

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