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Angular momentum projection plays a key role in studying nuclei and otherquantum many-body systems with rotational invariance. In this Letter, we reporta new fundamental feature of the angular momentum projection, which clearlytells us the frequently discussed $K$-mixing in the angular momentum projectednuclear wave function can be safely removed. At a given spin $J$, one cangenerate $2J+1$ angular momentum projected states from a deformed Slaterdeterminant. These projected states are labeled with different $K$ numbers.Usually, a nuclear wave function with $K$-mixing can be expressed as anexpansion in terms of all these $2J+1$ projected states. The coefficients inthis expansion can be obtained by solving the Hill-Wheeler equation.Strikingly, we found that such nuclear wave function with $K$-mixing can alwaysbe equivalently replaced by a single projected state generated from anotherSlater determinant. Consequently, such nuclear wave function can besignificantly simplified, especially for high-spin states. This also reminds usthat the $K$-mixing in the angular momentum projected nuclear wave functions,adopted by many present-day nuclear models, does not carry any physicalmeaning, and is essentially different from that $K$-mixing caused by theCoriolis force in the cranked shell model.

Removal of $K$-mixing in angular momentum projected nuclear wave functions