  The Wigner's theorem, which is one of the cornerstones of the mathematical
formulation of quantum mechanics, asserts that every symmetry of quantum system
is unitary or anti-unitary. This classical result was first given by Wigner in
1931. Thereafter it has been proved and generalized in various ways by many
authors. Recently, G. P. Geh\'{e}r extended Wigner's and Moln\'{a}r's theorems
and characterized the transformations on the Grassmann space of all rank-$n$
projections which preserve the transition probability. The aim of this paper is
to provide a new approach to describe the general form of the transition
probability preserving (not necessarily bijective) maps between Grassmann
spaces. As a byproduct, we are able to generalize the results of Moln\'{a}r and
G. P. Geh\'{e}r.
