The positive-energy continuum states of a hydrogen-like system are described by the eigenfunctions

with corresponding eigenvalues

, (

).

are the same spherical harmonics that occur for the bound states. In atomic units

, the radial equation can be written

. The solutions with the appropriate analytic and boundary conditions have the form

. These functions are deltafunction-normalized, such that

. They have the same functional forms (apart from normalization constants) as the discrete eigenfunctions under the substitution

.

You can plot the continuum function for various choices of

, ℓ, and

. The asymptotic form for large

approaches a spherical wave of the form

. The terms in the argument of cos, in addition to

, represent the phase shift with respect to the free particle. Coulomb scattering generally involves a significant number of partial waves.

A checkbox lets you compare the lowest-energy bound state with the same ℓ, that is, the wavefunctions for

,

,

, ⋯ (magnified by a factor of

for better visualization). The

solution, obtained as the limit of the discrete function as

or of the continuum function as

, has the form of a Bessel function:

.