Relativistic Ground State of Hydrogen-Like Atoms

Z

1

2

3

4

5

r

a

0

0.1

0.2

0.3

0.4

0.5

r

2

ϕ

2

This Demonstration computes the probability of the upper (red) and lower (gold) components of the ground state of a relativistic hydrogen-like atom of charge

Ze

as a function of

Z

, up to

Z=1/α≈137

, in accordance with the Dirac equation. This is compared to the probability in the standard nonrelativistic theory. The Dirac equation develops a singularity when

Z>1/α

.