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A quantum-mechanical system consisting of a nucleus of mass  $  M $
carrying a charge of  $  + Ze $
and one electron of mass  $  m $
with a charge of  $  - e $, 
which interact in accordance with Coulomb's law, i.e. which attract each other with a force which is inversely proportional to the square of the distance between the nucleus and the electron. In the special case of  $  Z = 1 $, 
when the nucleus is a proton, the hydrogen-like atom is an ordinary hydrogen atom. Meso-atoms (a  $  \mu $-
meson in the Coulomb field of a nucleus) and positronia (a system consisting of an electron and a positron) may also be considered as hydrogen-like atoms. The hydrogen-like atom problem is an exactly solvable special case of the general [[Two-body problem|two-body problem]] in classical mechanics and in quantum mechanics, and is the quantum-mechanical analogue of the classical problem of Keppler in the theory of the motion of two masses acted upon by the universal forces of gravity. After the motion of the centre of inertia has been separated, the quantum-mechanical hydrogen-like atom problem is reduced in a non-relativistic approximation to solving the [[Schrödinger equation|Schrödinger equation]] for a particle with reduced mass  $  m _ {0} = mM/( m+ M) $
moving in a field of central forces with a Coulomb potential:

$$ \tag{* }
\left \{ 
\frac{\hbar  ^ {2} }{2m _ {0} }

\Delta _ {\mathbf r} +
V ( \mathbf r ) - E  \right \} \ 
\psi ( \mathbf r )  =  0; \ \ 
V ( \mathbf r )  =  - 
\frac{Ze  ^ {2} }{\mathbf r }
 .
$$

The limitations on the wave functions  $  \psi ( \mathbf r ) $
for a solution of (*) to exist, imposed by the physical conditions, are: 1) for  $  E _ {n} = - Z  ^ {2} e  ^ {4} m/2 \hbar  ^ {2} n  ^ {2} $
and integer  $  n \geq  1 $(
discrete spectrum of energy  $  E $); 
2) for any  $  E > 0 $(
continuous energy spectrum). Solutions belonging to discrete spectra correspond to stationary connected states of the electron in the hydrogen-like atom and are  "randomly degenerate" , i.e. states with different quantum values of the orbital moment  $  l = 0 \dots n- 1 $, 
and not only of its projection  $  m _ {l} $(
$  - l \leq  m _ {l} \leq  l $; 
$  m _ {l} $
an integer) onto some axis (ordinary degeneration), have the same energies  $  E _ {n} $.  
"Random degeneration"  is a consequence of the fact that in the special case of the Coulomb potential the Schrödinger equation (*) is invariant not only with respect to the group  $  O( 3) $
of orthogonal transformations, which is true for any potential of central forces, but also with respect to the transformations of the larger group  $  O ( 4) $. 
Solutions of a continuous spectrum correspond to the ionized states of the hydrogen-like atom, i.e. unconnected states of the electron, and are degenerate with infinite multiplicity — all states with integer values  $  l \geq  0 $
and integer values  $  m _ {l} $
for a given  $  l $, 
$  - l \leq  m _ {l} \leq  l $, 
are possible.

The relativistic effects in a hydrogen-like atom include the dependence of the mass on the velocity and the spin properties of the electron and of the nucleus; they can be allowed for if the Schrödinger equation (*) is replaced by the relativistic [[Dirac equation|Dirac equation]] for an electron in the field of the Coulomb potential of the nucleus.

Allowance for relativistic effects and for the electron spin yields corrections of  $  E _ {n} $, 
which depend on  $  l $
and on the complete moment  $  j $
of the electron, defined in terms of  $  l $
and the spin of the electron and, by the same token, eliminate the random degeneration of the energy levels of the hydrogen-like atom and determine the so-called fine structure of the discrete spectrum of the energy levels of the hydrogen-like atom. Allowance for the nuclear spin and for the related magnetic moment interacting with the electron revolving around the nucleus, as well as allowance for the finite dimensions of the nucleus and for the possibility of a quadrupole moment and other higher multi-pole nuclear moments, introduce additional corrections to  $  E _ {n} $, 
which define the so-called hyperfine structure of the energy levels of the hydrogen-like atom.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Sokolov,   Yu.M. Loskutov,   I.M. Ternov,   "Quantum mechanics" , Moscow  (1965)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Messiah,   "Quantum mechanics" , '''1''' , North-Holland  (1961)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Flügge,   "Practical quantum mechanics" , Springer  (1974)</TD></TR></table>