# Hydrogenic Radial Functions via Supersymmetry

Hydrogenic Radial Functions via Supersymmetry

An application of supersymmetric quantum mechanics enables all the bound-state radial functions for the hydrogen atom to be evaluated using first-order differential operators, without any explicit reference to Laguerre polynomials.

The nonrelativistic hydrogen-like system with atomic number and assumed infinite nuclear mass satisfies the Schrödinger equation , in atomic units . Separation of variables in spherical polar coordinates gives (r,θ,ϕ)=(r)(θ,ϕ). Defining the reduced radial function , the radial equation can be expressed as . For the case (the 1, 2, 3, 4, … states) the radial function has the nodeless form (r)=const.

Z

--+(r,θ,ϕ)=0

1

2

2

∇

Z

r

2

Z

2

2

n

ψ

nlm

ℏ=m=e=1

ψ

nlm

R

nl

Y

lm

P(r)=rR(r)

-+-+(r)=0

2

d

d

2

r

l(l+1)

2

r

2Z

r

2

Z

2

n

P

nl

l=n-1

s

p

d

f

P

n,n-1

n

r

-Zr/n

e

As shown in the Details section, operators and are defined with the effect of lowering or raising the quantum number by 1, respectively, when applied to the radial function (r), namely, (r)=const(r) and (r)=const(r). The constants are most easily determined after the fact by the normalization conditions (r)dr=1.

A

l+1

+

A

l

l

P

nl

A

l+1

P

n,l

P

n,l+1

+

A

l

P

nl

P

n,l-1

∞

∫

0

2

P

nl

In this Demonstration, you can plot any radial function with to and show the result of applying (red curve) or (blue curve). The plots pertain to the case . You can also choose to view the results as formulas (with variable ) or on an energy-level diagram.

n=1

4

A

l

+

A

l

Z=1

Z