  Projection is noninvertible. This means two different vectors may have the
same projected components. In nuclear case, one may take the intrinsic state as
a vector, and take the nuclear wave function as the projected component
obtained by projecting the former onto good quantum numbers. This immediately
comes to the conclusion that, for a given nuclear state in the laboratory frame
of reference, the corresponding intrinsic state in the intrinsic frame of
reference can not be uniquely determined. In this letter, I will show this
interesting phenomenon explicitly based on the improved variation after
projection(VAP) method. First of all, it is found that, the form of the trial
VAP wavefunction with spin $J$ can be greatly simplified by adopting just one
projected state rather than previously adopting all $(2J+1)$ spin-projected
states for each selected Slater determinant. This is crucial in the
calculations of high-spin states with arbitrary intrinsic Slater determinants.
Based on this simplified VAP, the present calculations show that orthogonal
intrinsic states (differed by $K$) may have almost the same projected
wavefunctions, indicating the uncertainty of the nuclear intrinsic states. This
is quite different from the traditional concept of intrinsic state which is
expected to be unique.
