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.
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