As an approximation, the conjugated backbone can be considered as a real-world example of the "electron-in-a-box" solution to the Schrödinger equation; however, the development of refined models to accurately predict absorption and fluorescence spectra of well-defined oligo(thiophene) systems is ongoing.
This selection rule arises from a first-order perturbation theory approximation of the time-dependent Schrödinger equation.
This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.
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Or two charged particles, each with a well-defined angular momentum, may interact by Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle Schrödinger equation.
Erwin Schrödinger was vacationing in Arosa at Christmas 1926 when he made his breakthrough discovery of wave mechanics.
This allows an easy form to express the solution of the Schrödinger equation for the time dependent Hamiltonian.
The pseudopotential is an attempt to replace the complicated effects of the motion of the core (i.e. non-valence) electrons of an atom and its nucleus with an effective potential, or pseudopotential, so that the Schrödinger equation contains a modified effective potential term instead of the Coulombic potential term for core electrons normally found in the Schrödinger equation.
The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus).