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The Schrödinger equation for the electron in a hydrogen atom (or a hydrogen-like atom) is = where is the electron charge, is the position of the electron relative to the nucleus, = | | is the magnitude of the relative position, the potential term is due to the Coulomb interaction, wherein is the permittivity of free space and = + is the 2-body ...
Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. (Image not to scale) A hydrogen atom is an atom of the chemical element hydrogen.The electrically neutral hydrogen atom contains a single positively charged proton in the nucleus, and a single negatively charged electron bound to the nucleus by the Coulomb force.
which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found.
In 1926, Schrödinger published the famous wave equation now named after him, the Schrödinger equation. This equation was based on classical conservation of energy using quantum operators and the de Broglie relations and the solutions of the equation are the wave functions for the quantum system. [16]
The Coulomb wave equation for a single charged particle of mass is the Schrödinger equation with Coulomb potential [1] (+) = (),where = is the product of the charges of the particle and of the field source (in units of the elementary charge, = for the hydrogen atom), is the fine-structure constant, and / is the energy of the particle.
An additive term 1 / R , which is constant for fixed internuclear distance R, has been omitted from the potential V, since it merely shifts the eigenvalue. The distances between the electron and the nuclei are denoted r a and r b. In atomic units (ħ = m = e = 4 π ε 0 = 1) the wave equation is
The principal quantum number arose in the solution of the radial part of the wave equation as shown below. The Schrödinger wave equation describes energy eigenstates with corresponding real numbers E n and a definite total energy, the value of E n. The bound state energies of the electron in the hydrogen atom are given by: = =, =,,, …
The benzene molecule consists of 12 nuclei and 42 electrons. The Schrödinger equation, which must be solved to obtain the energy levels and wavefunction of this molecule, is a partial differential eigenvalue equation in the three-dimensional coordinates of the nuclei and electrons, giving 3 × 12 = 36 nuclear plus 3 × 42 = 126 electronic, totalling 162 variables for the wave function.