Angular momentum projection plays a key role in studying quantum many-body systems with rotational invariance such as atomic nuclei. With given quantum numbers $JM$, one can generate $2J+1$ angular momentum projected states labeled with $-J\le K\le J$ from a deformed Slater determinant. Usually, a nuclear wave function with $K$ mixing can be expressed as a superposition of all these $2J+1$ projected states, where the coefficients can be obtained by solving the generalized eigenvalue equation. In this Letter, we report a new fundamental feature that the frequently discussed $K$ mixing in the angular momentum projected nuclear wave function can be safely removed. Strikingly, we found that such nuclear wave function with $K$ mixing can always be well approximated by a single projected state with any given $K$. This simplification can further improve methods related to angular momentum projection in chemistry, atomic, and nuclear physics, particularly offering significant advantages in dealing with high-spin states.

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• Angular momentum
• Many-body techniques
• Nuclear structure & decays
• Shell model