CoulombF

CoulombF[l,η,r]

gives the regular Coulomb wavefunction .

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
CoulombF[l,η,r] is a solution of the ordinary differential equation .
CoulombF[l,η,r] is proportional to near .
CoulombF[l,η,r] tends to for large and some phase shift .
CoulombF has a branch cut discontinuity in the complex plane running from to .
For certain special arguments, CoulombF automatically evaluates to exact values.
CoulombF can be evaluated to arbitrary numerical precision.
CoulombF automatically threads over lists.

Examples

open allclose all

Basic Examples (4)

Evaluate numerically:

Evaluate to arbitrary precision:

Plot the Coulomb wavefunction for repulsive () and attractive () interactions:

Complex plot:

Series expansion at the origin:

Asymptotic behavior for large radius:

Scope (19)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values (3)

Limiting value at the origin:

For zero value of the parameter η, CoulombF reduces to a spherical Bessel function:

Find the first positive zero of CoulombF:

Visualization (3)

Plot the CoulombF function:

Plot the real part of CoulombF[2,0,x+I y]:

Plot the imaginary part of CoulombF[2,0,x+I y]:

Polar plot with :

Function Properties (7)

Function domain of CoulombF:

CoulombF is an analytic function of η:

CoulombF[2,0,x] is not injective:

CoulombF[2,0,x] is neither non-negative nor non-positive:

CoulombF[2,0,x] has both singularities and discontinuities at zero:

CoulombF is neither convex nor concave:

TraditionalForm formatting:

Series Expansions (1)

Find the Taylor expansion using Series at zero and at infinity:

Plots of the first three approximations for CoulombF around :

Function Representations (1)

Representation through other Coulomb functions:

Applications (2)

Wavefunction for the radial Schrödinger equation with Coulomb potential between two point particles with charges and separated by a distance and energy of relative motion :

Verify that the wavefunction satisfies the Schrödinger equation for specific values of the energy and separation:

Construct a WKB approximation of CoulombF:

Properties & Relations (2)

CoulombF is a linear combination of CoulombH1 and CoulombH2:

CoulombF is related to Hypergeometric1F1Regularized in some region of the complex plane:

However, the stated definition has the branch cut at , while the built-in CoulombF has the branch cut at :

Top

CoulombF

Details

Examples

Basic Examples (4)