  A new non-perturbative approach is proposed to solve time-independent
Schr\"{o}dinger equations in quantum mechanics and chromodynamics (QCD). It is
based on the homotopy analysis method (HAM), which was developed by the author
for highly nonlinear equations since 1992 and has been widely applied in many
fields. Unlike perturbative methods, this HAM-based approach has nothing to do
with small/large physical parameters. Besides, convergent series solution can
be obtained even if the disturbance is far from the known status. A nonlinear
harmonic oscillator is used as an example to illustrate the validity of this
approach for disturbances that might be more than hundreds larger than the
possible superior limit of the perturbative approach. This HAM-based approach
could provide us rigorous theoretical results in quantum mechanics and
chromodynamics (QCD), which can be directly compared with experimental data.
Obviously, this is of great benefit not only for improving the accuracy of
experimental measurements but also for validating physical theories.
