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Concept
Particle in a spherically symmetric potential
Summary
In quantum mechanics, a particle in a spherically symmetric potential is a system with a potential that depends only on the distance between the particle and a center. This is how isolated atoms are described, and plays a central role as a first approximation to the formation of chemical bonds.
In the general case, the dynamics of a particle in a spherically symmetric potential are governed by a Hamiltonian of the following form:
where is the mass of the particle, is the momentum operator, and the potential depends only on , the modulus of the radius vector. The possible quantum states of the particle can be found by using the above Hamiltonian to solve the Schrödinger equation for its eigenvalues and their corresponding eigenstates, which are wave functions.
To describe these systems, it is convenient to use spherical coordinates, , and - the time-independent Schrödinger equation for the system is then separable. 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, .
The eigenstates of the system have the form:
in which the spherical angles and represent the polar and azimuthal angle, respectively. The last two factors of are often grouped together as spherical harmonics, so that the eigenfunctions take the form:
The differential equation which characterizes the function is called the radial equation.
The kinetic energy operator in spherical polar coordinates is:The spherical harmonics satisfy
Substituting this into the Schrödinger equation we get a one-dimensional eigenvalue equation,This equation can be reduced to an equivalent 1-D Schrödinger equation by substituting , where satisfieswhich is precisely the one-dimensional Schrödinger equation with an effective potential given bywhere . The correction to the potential V(r) is called the centrifugal barrier term.
If , then near the origin, .
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