  The tunneling effect is the most popular phenomenon of quantum physics and is
present in modern physical theories. Still, the most important features of this
effect are already present in toy models - low dimensional quantum mechanics
with high potential barriers. Even these simple systems cannot be solved
analytically and approximations have to be made. Tunneling is closely related
to splitting of the ground state energy which is degenerate in the classical
limit. In this thesis we use two methods to estimate the energy splitting in
one dimensional quantum mechanics. In semi-classical approximation the
splitting is calculated in the weak coupling limit. The other approach,
Tamm-Dancoff method gives very precise numerical results for finite coupling.
Comparing these two approaches we are able to confirm results of the
semi-classical approximation and determine range of its applicability. Several
potentials are considered: anharmonic double well, cosine potential and cosine
potential in periodic space with finite number of minima. We also address a
non-generic problem of anharmonic triple well potential.
