  We examine the validity of the spherical approximation
$\gamma_{s}=(2\gamma_{2}+3\gamma_{3})/5$ in the Luttinger-Kohn Hamiltonian in
calculating the subband dispersions of the hole gas. We calculate the realistic
hole subband dispersions (without the spherical approximation) in a cylindrical
Ge nanowire by using quasi-degenerate perturbation theory. The realistic
low-energy hole subband dispersions have a double-well anticrossing structure,
that consists with the spherical approximation prediction. However, the
realistic subband dispersions are also nanowire growth direction dependent.
When the nanowire growth direction is restricted in the (100) crystal plane,
the detailed growth direction dependences of the subband parameters are given.
We find the spherical approximation is good approximation, it can nicely
reproduce the real result in some special growth directions.
