Particle in a spherically symmetric potential
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.
[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:
This leaves an ordinary differential equation in terms only of the radius,
If solved by separation of variables, the eigenstates of the system will have the form:
The kinetic energy operator in spherical polar coordinates is:
which is precisely the one-dimensional Schrödinger equation with an effective potential given by
The correction to the potential V(r) is called the centrifugal barrier term.
Since angular momentum operators are generators of rotation, applying the Baker-Campbell-Hausdorff Lemma we get:
also commutes with the Hamiltonian, the energy eigenvalues in such cases are always independent of
where regular solutions for positive energies are given by so-called Bessel functions of the first kind
The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in
Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range.
The resolution essentially follows that of the vacuum case above with normalization of the total wavefunction added, solving two Schrödinger equations — inside and outside the sphere — of the previous kind, i.e., with constant potential.
must be defined everywhere selected Bessel function of the first kind over the other possibilities in the vacuum case.
Bound states bring the novelty as compared to the vacuum case now that
Allowed energies are those for which the radial wavefunction vanishes at the boundary.
Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions.
, typically by using a table or calculator, as these zeros are not solvable for the general case.
, we will show that the radial Schrödinger equation has the normalized solution,
This normalization is with the usual volume element r2 dr. First we scale the radial coordinate
Consideration of the limiting behavior of v(y) at the origin and at infinity suggests the following substitution for v(y),
is a non-negative integral number, the solutions of this equations are generalized (associated) Laguerre polynomials
By making use of the orthogonality properties of the generalized Laguerre polynomials, this equation simplifies to:
Other forms of the normalization constant can be derived by using properties of the gamma function, while noting that
A hydrogenic (hydrogen-like) atom is a two-particle system consisting of a nucleus and an electron.
where In order to simplify the Schrödinger equation, we introduce the following constants that define the atomic unit of energy and length:
is physically allowed (continuous spectrum), the corresponding eigenfunctions are non-square integrable.
the inverse powers of x are negligible and the normalizable (and therefore, physical) solution for large
the inverse square power dominates and the physical solution for small
is a non-negative integer, this equation has polynomial solutions written as
