Particle in a spherically symmetric potential information
Quantum mechanical model
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Hydrogen atomic orbitals of different energy levels. The more opaque areas are where one is most likely to find an electron at any given time.
In quantum mechanics, a spherically symmetric potential is a system of which the potential only depends on the radial distance from the spherical center and a location in space. A particle in a spherically symmetric potential will behave accordingly to said potential and can therefore be used as an approximation, for example, of the electron in a hydrogen atom or of the formation of chemical bonds.[1]
In the general time-independent case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:
Here, is the mass of the particle, is the momentum operator, and the potential depends only on the vector magnitude of the position vector, that is, the radial distance from the origin (hence the spherical symmetry of the problem).
To describe a particle in a spherically symmetric system, it is convenient to use spherical coordinates; denoted by , and . The time-independent Schrödinger equation for the system is then a separable, partial differential equation. This means solutions to the angular dimensions of the equation can be found independently of the radial dimension. This leaves an ordinary differential equation in terms only of the radius, , which determines the eigenstates for the particular potential, .
^Ruedenberg, Klaus; Schmidt, Michael W. (2009-03-12). "Physical Understanding through Variational Reasoning: Electron Sharing and Covalent Bonding". The Journal of Physical Chemistry A. 113 (10): 1954–1968. doi:10.1021/jp807973x. ISSN 1089-5639. PMID 19228050.
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