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

Xiao Lu  Zhan-Jiang Lian  Xue-Wei Li  Zao-Chun Gao<EMAIL_ADDRESS>Yong-Shou
Chen  China Institute of Atomic Energy, P.O. Box 275 (10), Beijing 102413, PR
China

###### Abstract

Angular momentum projection plays a key role in studying nuclei and other
quantum many-body systems with rotational invariance. In this Letter, we
report a new fundamental feature of the angular momentum projection, which
clearly tells us the frequently discussed $K$-mixing in the angular momentum
projected nuclear wave function can be safely removed. At a given spin $J$,
one can generate $2J+1$ angular momentum projected states from a deformed
Slater determinant. These projected states are labeled with different $K$
numbers. Usually, a nuclear wave function with $K$-mixing can be expressed as
an expansion in terms of all these $2J+1$ projected states. The coefficients
in this expansion can be obtained by solving the Hill-Wheeler equation.
Strikingly, we found that such nuclear wave function with $K$-mixing can
always be equivalently replaced by a single projected state generated from
another Slater determinant. Consequently, such nuclear wave function can be
significantly simplified, especially for high-spin states. This also reminds
us that the $K$-mixing in the angular momentum projected nuclear wave
functions, adopted by many present-day nuclear models, does not carry any
physical meaning, and is essentially different from that $K$-mixing caused by
the Coriolis force in the cranked shell model.

###### keywords:

Angular momentum projection, $K$-mixing, shell model

††journal: Physics Letter B

Angular momentum projection is a very powerful technique in many modern
nuclear structure calculations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].
This is because the mean-field many-body wave function $|\Phi\rangle$ in form
of Slater determinant (SD) or Hartree-Fock-Bogoliubov (HFB) vacuum is widely
used in the nuclear structure studies, but it usually breaks the rotational
invariance that the nuclear system should have. The angular momentum
projection then removes the spurious part from the mean-field wave function
and the remained state with good spin $J$, usually called as the angular
momentum projected state, or simply called as the projected state, is rather
suitable for the construction of nuclear wave functions defined in the
laboratory frame of reference.

For simplicity, the mean-field many-body wave functions used in the present
calculations are SDs, so that we can omit the particle number projection and
focus on the angular momentum projection. The parity projection also can be
involved if the SD is reflection asymmetric. The nuclear wave functions should
have a set of good quantum numbers including the particle numbers ($N,Z$), the
spin ($J$), the parity ($\pi$) and the magnetic quantum number ($M$). With
these quantum numbers given, one can generate $2J+1$ projected states labeled
by $K$ from a SD, $|\Phi\rangle$. Generally, all these $2J+1$ projected states
are expected to be taken to form a nuclear wave function. Therefore, this
involves the concept of $K$-mixing, which means the nuclear wave function
contains different $K$ components. Indeed, in almost all modern nuclear
theories associated with the angular momentum projection, $K$-mixing is
believed to be necessary, since this may get the energy of the calculated
nuclear wave function lowest when the $|\Phi\rangle$ and the adopted
Hamiltonian are fixed.

It is well known that the $K$-mixing also exists in the cranked shell model
(CSM) [14]. Such $K$-mixing is caused by the Coriolis force, which is a
natural feature of the CSM. However, the Coriolis force only exists in the
rotating body-fixed frame of reference. Such inertial force should not exist
in the static laboratory frame of reference in which the present wave
functions are constructed. Obviously, the presently discussed $K$-mixing is
simply originated from the nonzero matrix elements of the Hamiltonian and/or
nonzero overlaps between different projected states rather than from the
Coriolis force. Thus, conceptually, the $K$-mixing in the angular momentum
projected wave function is different from that in the CSM.

It is interesting that, in our recent variation after projection (VAP)
calculations [13], the converged VAP wave function with $K$-mixing can be
equivalently replaced by a single projected state, which makes the VAP wave
function rather simple. This is crucial that the simplified VAP can be applied
to arbitrarily high-spin states. In this Letter, we propose a more striking
and universal argument that any angular momentum projected wave functions with
$K$-mixing, whether it is optimized or not, can be equivalently replaced by a
single angular momentum projected state with $K$ arbitrarily chosen. That
means, the $K$-mixing in any angular momentum projected wave functions can be
safely removed.

Let us start with the angular momentum projection operator, which is defined
as

$\displaystyle P^{J}_{MK}=\frac{2J+1}{8\pi^{2}}\int d\Omega
D^{J*}_{MK}(\Omega)\hat{R}(\Omega),$ (1)

where $\Omega=(\alpha,\beta,\gamma)$ stands for the Euler angles, and

$\hat{R}(\Omega)=e^{-\frac{i}{\hbar}\alpha\hat{J}_{z}}e^{-\frac{i}{\hbar}\beta\hat{J}_{y}}e^{-\frac{i}{\hbar}\gamma\hat{J}_{z}}$
(2)

is the rotation operator. $D^{J}_{MK}(\Omega)=\langle
JM|\hat{R}(\Omega)|JK\rangle$ is the Wigner $D$-function. $\hat{J}_{x}$,
$\hat{J}_{y}$ and $\hat{J}_{z}$ are the angular momentum operators defined in
the laboratory frame of reference.

Suppose that there is a reference state, $|\Phi\rangle$. One can perform the
angular momentum projection on this $|\Phi\rangle$ and the projected state
$P^{J}_{MK}|\Phi\rangle$ is an eigenstate of
$\hat{J}^{2}=\hat{J}^{2}_{x}+J^{2}_{y}+J^{2}_{z}$ and $\hat{J}_{z}$ with the
eigenvalues being $J(J+1)\hbar^{2}$ and $M\hbar$, respectively. However, the
$K$ number in the $P^{J}_{MK}|\Phi\rangle$ state is no longer a good quantum
number of any operator. This can be clearly understood from the following
general identity [1],

$\displaystyle
P^{J}_{MK}\hat{R}(\Omega_{0})|\Phi\rangle=\sum_{K^{\prime}=-J}^{J}D^{J}_{KK^{\prime}}(\Omega_{0})P^{J}_{MK^{\prime}}|\Phi\rangle,$
(3)

where $|\Phi\rangle$ can be an arbitrary nuclear many-body wave function. If
one assumes the state $P^{J}_{MK}|\Phi\rangle$ as an eigenstate of an operator
$\hat{J}_{3}$ ($\hat{J}_{3}$ is supposed to be the projection of the angular
momentum on the third axis in the body-fixed frame of reference), so that

$\displaystyle\hat{J}_{3}P^{J}_{MK}|\Phi\rangle=KP^{J}_{MK}|\Phi\rangle,$ (4)

then by applying $\hat{J}_{3}$ to Eq. (3), one can immediately find

$\displaystyle
KP^{J}_{MK}\hat{R}(\Omega_{0})|\Phi\rangle=\sum_{K^{\prime}=-J}^{J}K^{\prime}D^{J}_{KK^{\prime}}(\Omega_{0})P^{J}_{MK^{\prime}}|\Phi\rangle,$
(5)

which is clearly contradictory to Eq. (3). This proves that $K$ can not be a
good quantum number in any $P^{J}_{MK}|\Phi\rangle$ state. It is known that if
$|\Phi\rangle$ is reduced to be a SD, then $\hat{R}(\Omega_{0})|\Phi\rangle$
in Eq. (3) is also a SD.

Before rotation, we assume the body-fixed frame of reference for the deformed
$|\Phi\rangle$ state exactly coincides with the laboratory frame of reference.
This means $\hat{J}_{3}=\hat{J}_{z}$. Moreover, the rotation operator
$\hat{R}(\Omega)$ is written in the laboratory frame of reference. Therefore,
it is enough to study the angular momentum projection only in the laboratory
frame of reference, and the body-fixed frame of reference can be totally
forgotten.

To learn more natures of the projected state, let us assume
$|\Phi_{K_{0}}\rangle$ to be an axially deformed state with
$\hat{J}_{z}|\Phi_{K_{0}}\rangle={K_{0}}|\Phi_{K_{0}}\rangle$. Then from Eq.
(3), one can immediately have

$\displaystyle
P^{J}_{MK}\hat{R}(\Omega_{0})|\Phi_{K_{0}}\rangle=D^{J}_{KK_{0}}(\Omega_{0})P^{J}_{MK_{0}}|\Phi_{K_{0}}\rangle,$
(6)

for arbitrary $K$ and arbitrary $\Omega_{0}$. This means all projected states
$P^{J}_{MK}\hat{R}(\Omega_{0})|\Phi_{K_{0}}\rangle$ with different $K$ are
exactly identical, and are the same as $P^{J}_{MK_{0}}|\Phi_{K_{0}}\rangle$.
It is enough that one can arbitrarily pick up just one of these
$P^{J}_{MK}\hat{R}(\Omega_{0})|\Phi_{K_{0}}\rangle$ states or
$P^{J}_{MK_{0}}|\Phi_{K_{0}}\rangle$ to construct the same nuclear wave
function. Early examples supporting this argument can be found in Refs. [5,
6].

Therefore, it is shown that the $K$ number in $P^{J}_{MK}|\Phi\rangle$ has no
physical meaning. Correspondingly, the $K$-mixing in Eq. (3) simply caused by
a rotation, $\hat{R}(\Omega_{0})$, can be removed due to the existence of the
equivalent $P^{J}_{MK}\hat{R}(\Omega_{0})|\Phi\rangle$ projected state.
However, for a nuclear wave function with the following form,

$\displaystyle|\Psi_{J\pi
M\alpha}\rangle=\sum_{K^{\prime}=-J}^{J}f^{J\pi\alpha}_{K^{\prime}}P^{J\pi}_{MK^{\prime}}|\Phi\rangle,$
(7)

the $K$-mixing seems more complicated than Eq. (3). Here,
$P^{J\pi}_{MK}=P^{J}_{MK}P^{\pi}$ and $P^{\pi}=\frac{1}{2}(1+\pi\hat{P})$ is
the parity projection operator. $|\Phi\rangle$ is assumed to be a SD. $\alpha$
is used to label the states with the same $J$, $\pi$ and $M$. If
$f^{J\pi\alpha}_{K^{\prime}}$ in Eq. (7) happens to be
$D^{J}_{KK^{\prime}}(\Omega_{0})$, then $|\Psi_{J\pi M\alpha}\rangle$ can be
simply compacted into a single projected state according to Eq. (3). However,
in practical calculations, the $f^{J\pi\alpha}_{K^{\prime}}$ coefficients and
the corresponding energy $E^{J\pi}_{\alpha}$ are determined by solving the
Hill-Wheeler equation of order $2J+1$,

$\displaystyle\sum_{K^{\prime}=-J}^{J}(H^{J\pi}_{KK^{\prime}}-E^{J\pi}_{\alpha}N^{J\pi}_{KK^{\prime}})f^{J\pi\alpha}_{K^{\prime}}=0,$
(8)

where
$H^{J\pi}_{KK^{\prime}}=\langle\Phi|\hat{H}P^{J\pi}_{KK^{\prime}}|\Phi\rangle$
and $N^{J\pi}_{KK^{\prime}}=\langle\Phi|P^{J\pi}_{KK^{\prime}}|\Phi\rangle$.
For convenience, we assume $E^{J\pi}_{1}\leq E^{J\pi}_{2}\leq\cdots\leq
E^{J\pi}_{2J+1}$. The $f^{J\pi\alpha}_{K^{\prime}}$ coefficients are imposed
to satisfy the normalization condition, so that $\langle\Psi_{J\pi
M\alpha}|\Psi_{J\pi M\alpha}\rangle=1$.

Here, we are only interested in the lowest state ($\alpha=1$). It is
interesting that, in our previous VAP calculations, the wave function in Eq.
(7) with $f^{J\pi 1}_{K^{\prime}}$ coefficients satisfying Eq. (8) still can
be equivalently replaced by a single projected state when the VAP energy
reaches a minimum [13]. This clearly shows that the $K$-mixing can be safely
removed from the VAP calculations.

When the adopted Hamiltonian is changed, the original optimized wave function
is no longer an optimized one for the new Hamiltonian. This implies the
$K$-mixing can be removed not only from the optimized wave function, but also
from those without optimization.

To check this conjecture, one needs to calculate the overlap between
$|\Psi_{J\pi M1}\rangle$ and the equivalent single projected state
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$, which is defined as

$O_{K}=\frac{|\langle\Psi_{J\pi
M1}|P^{J\pi}_{MK}|\Phi^{\prime}\rangle|}{\sqrt{\langle\Phi^{\prime}|P^{J\pi}_{KK}|\Phi^{\prime}\rangle}},$
(9)

where $|\Phi^{\prime}\rangle$ is another fully symmetry-unrestricted SD that
needs to be varied so that $O_{K}$ can be maximized. Ideally, if $O_{K}=1$,
then $P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ is exactly equivalent to
$|\Psi_{J\pi M1}\rangle$. In this Letter, we try to demonstrate a very
interesting phenomenon that the maximum of $O_{K}$ are always very close to 1
for arbitrary $|\Psi_{J\pi M1}\rangle$ and $K$.

In the practical calculations, instead of directly maximizing $O_{K}$, we
equivalently minimize the reciprocal of $O^{2}_{K}$,

$R_{K}=\frac{1}{O^{2}_{K}}=\frac{\langle\Phi^{\prime}|P^{J\pi}_{KK}|\Phi^{\prime}\rangle}{\langle\Phi^{\prime}|P^{J\pi}_{KM}|\Psi_{J\pi
M1}\rangle\langle\Psi_{J\pi M1}|P^{J\pi}_{MK}|\Phi^{\prime}\rangle},$ (10)

so that the optimization algorithm in our VAP calculation can be directly
adopted. The gradient and the Hessian matrix of $R_{K}$ with respect to the
variational parameters can be easily obtained using the available matrix
elements required in our VAP algorithm. Then a new VAP calculation is
established but for the minimization of $R_{K}$.

The $|\Psi_{J\pi M1}\rangle$ wave functions are selected in attempt to cover a
wide range of nuclear states, so that the present conclusion can be as general
as possible. One can start from a randomly chosen $|\Psi_{J\pi M1}\rangle$
with $|\Phi\rangle$ fully symmetry-unrestricted, then this $|\Psi_{J\pi
M1}\rangle$ is varied in a VAP calculation, so that the corresponding energy
reaches a minimum. Along the path of the VAP iteration, one can harvest a
series of different $|\Psi_{J\pi M1}\rangle$ wave functions with the final one
optimized. Each of them can be put into Eq. (10) and the $R_{K}$ can be
minimized by varying $|\Phi^{\prime}\rangle$. Here, we perform the VAP
calculations for 26Mg in the $sd$ model space and 48Cr in the $pf$ model
space. Thus the parity projection can be omitted here. The adopted shell model
interactions are USDB [15] and GXPF1A [16], respectively. The spins of the
calculated wave functions for both nuclei are $J=8$. The calculated results
are shown in Fig. 1. For the $|\Psi_{J\pi M1}\rangle$ at each VAP iteration,
whose energy is shown in Fig. 1 (a) or (b), there are $2J+1=17$ $R_{K}$
quantities with $K=-J,-J+1,\cdots,J$, which can be minimized independently.
Then the corresponding 17 maximized $O_{K}$ quantities for each $|\Psi_{J\pi
M1}\rangle$ can be obtained and are shown in Fig. 1 (c) or (d). It is very
striking that almost all calculated $O_{K}$ values are very close to 1. This
means at each VAP iteration, the calculated $|\Psi_{J\pi M1}\rangle$ wave
function can be equivalently replaced by a single projected state
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ with $K$ randomly taken. As an extra
confirmation, one can calculate the energies of
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ states, which are also shown in Fig. 1
(a) and (b). It is seen that the energies of the
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ projected states perfectly coincide with
those of the corresponding $|\Psi_{J\pi M1}\rangle$ wave functions.

Figure 1: (Color online) Energies of the $K$-mixing $|\Psi_{J\pi M1}\rangle$
wave function (black circles) and the corresponding equivalent
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ projected states (different colored
symbols) as functions of the VAP iteration for the $J^{\pi}=8^{+}$ states in
(a) 26Mg and (b) 48Cr, respectively. The corresponding $O_{K}$ overlaps are
shown in (c) 26Mg and (d) 48Cr, respectively.

From Fig. 1, one can see that the equivalence of $|\Psi_{J\pi M1}\rangle$ and
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ is not just limited to the converged VAP
wave functions, and is expected to be always valid. Now, let’s generate the
$|\Psi_{J\pi M1}\rangle$ wave function from a deformed SD, in which the single
particle wave functions are the lowest ones calculated from a deformed single
particle potential $\hat{H}_{s.p.}$, so that we know the explicit shape of the
SD. Here, the $\hat{H}_{s.p.}$ including the quadrupole and octupole
deformation terms can be written as,

$\displaystyle H_{s.p.}$ $\displaystyle=$ $\displaystyle
h_{0}+\varepsilon_{3}\hbar\omega\rho^{2}\sqrt{\frac{4\pi}{7}}Y_{30}$ (11)
$\displaystyle-\frac{2}{3}\hbar\omega\rho^{2}\varepsilon_{2}\sqrt{\frac{4\pi}{5}}[\cos\gamma
Y_{20}$ $\displaystyle-\frac{\sin\gamma}{\sqrt{2}}(Y_{22}+Y_{2-2})],$

where, $\varepsilon_{2}$, $\gamma$ and $\varepsilon_{3}$ are the quadrupole,
triaxial and octupole deformation parameters, respectively. $h_{0}$ is simply
the one-body term of the adopted shell model Hamiltonian. For the oscillator
frequency $\omega$, we simply take $\hbar\omega=41.0A^{-1/3}$MeV and $A$ is
the mass number. $\rho=\sqrt{\frac{m\omega}{\hbar}}r$ is dimensionless.

Figure 2: (Color online) Upper panel: The
$\langle\Phi|P^{J\pi}_{KK}|\Phi\rangle$ norms of the projected states with (a)
$J^{\pi}=4^{+}$ and (b) $J^{\pi}=16^{+}$, which are generated from the
deformed SD in 60Zn and are used to form the $|\Psi_{J\pi M1}\rangle$ wave
function. The single particle states in the SD are calculated from Eq. (11)
with $\varepsilon_{2}=0.3$ and $\gamma=40^{\circ}$. Middle panel: The
corresponding $O_{K}$ overlaps for (c) $J=4^{+}$ and (d) $J=16^{+}$. Lower
panel: The energies of the calculated $|\Psi_{J\pi M1}\rangle$ (black line)
and the equivalent $P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ projected states (red
circle) for (c) $J=4^{+}$ and (f) $J=16^{+}$, respectively. The GXPF1A
interaction is adopted.

In Fig. 2, we take the $|\Psi_{J\pi M1}\rangle$ wave functions with
$J^{\pi}=4^{+}$ and $16^{+}$ in 60Zn as examples. The $pf$ model space are
adopted. For simplicity, we use the same $|\Phi\rangle$ SD for both
$J^{\pi}=4^{+}$ and $16^{+}$ . The single particle states in this
$|\Phi\rangle$ are calculated from Eq. (11) with $\varepsilon_{2}=0.3$ and
$\gamma=40^{\circ}$, so that we know $|\Phi\rangle$ has a clear triaxial
shape. This is in accordance with the distribution of the
$\langle\Phi|P^{J\pi}_{KK}|\Phi\rangle$ norms, which are calculated and shown
in Fig. 2 (a) and (b). Then, $|\Psi_{J\pi M1}\rangle$ is obtained from Eq. (8)
with $\hat{H}$ taken to be the GXPF1A interaction. Similar to the examples in
Fig. 1, one can minimize the $R_{K}$ value for each $K$ by varying the
$|\Phi^{\prime}\rangle$ SD in Eq. (10) but with $|\Psi_{J\pi M1}\rangle$
fixed. Then the corresponding maxima of $O_{K}$ can be obtained. It is seen
from Fig. 2 (c) and (d) that all the calculated $O_{K}$ maxima are also very
close to 1, indicating the $|\Psi_{J\pi M1}\rangle$ with explicit $K$-mixing
can still be equivalently replaced by a single
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ projected state with $K$ arbitrarily
chosen. The energies of $P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ are also
calculated and are compared with that of the $|\Psi_{J\pi M1}\rangle$ in Fig.
2 (e) and (f). It is found that all the energy differences between
$P^{J\pi}_{MK}|\Phi^{\prime}\rangle$ and $|\Psi_{J\pi M1}\rangle$ are within
100keV. The same calculations are also performed for the odd-A 61Zn and odd-
odd 58Cu. The results are shown in Fig. 3, which are very similar to those in
Fig. 2.

Figure 3: (Color online) Same as Fig. 2, but for odd-$A$ nucleus 61Zn and odd-
odd nucleus 58Cu.

As a final example, we study the wave functions in 68Se, which are constructed
with the JUN45 interaction [17] in the $jj44$ model space. The $jj44$ model
space constitutes of $f_{5/2}$, $p_{3/2}$, $p_{1/2}$ and $g_{9/2}$ orbits.
This raises the possibility that the $|\Phi\rangle$ SD may be both non-axial
and reflection asymmetric. Thus the parity projection is required here. To
generate such a SD, we simply take $\varepsilon_{2}=0.45$, $\gamma=40^{\circ}$
and $\varepsilon_{3}=0.3$. Then this SD is projected onto spin $J=16$ with
both parities. The $\langle\Phi|P^{J\pi}_{KK}|\Phi\rangle$ norms are shown in
Fig. 4 (a) and (b). Without exception, all the calculated $O_{K}$ values are
still very close to 1. This means even for a more complicated $|\Psi_{J\pi
M1}\rangle$ wave function with a reflection asymmetric SD, one can still find
an equivalent single projected SD to take place of the $K$-mixing $|\Psi_{J\pi
M1}\rangle$ wave function.

Figure 4: (Color online) Same as Fig. 2, but for $J=16$ in 68Se with both
parities. The single particle states in the adopted $|\Phi\rangle$ SD are
calculated from Eq. (11) with $\varepsilon_{2}=0.45$, $\gamma=40^{\circ}$ and
$\varepsilon_{3}=0.3$. The JUN45 interaction is adopted.

All the present calculations apparently show that the $K$-mixing can be
removed in the construction of nuclear wave functions in the laboratory frame
of reference. In fact, one can also understand this point from the nuclear
shell model. In the $M$-scheme shell model, the nuclear wave function is
expanded in term of a complete set of $|\Phi_{M,i}\rangle$ SDs with good $M$,

$|\Psi_{\mu JM}\rangle=\sum_{i=1}^{D_{M}}C^{\mu JM}_{i}|\Phi_{M,i}\rangle,$
(12)

where, $\mu$ is used to label the wave functions with the same $J$ and $M$.
$D_{M}$ is the dimension of the $M$-scheme configuration space. The $C^{\mu
JM}_{i}$ coefficients are obtained by the shell model dialgonalization.

One can perform the angular momentum projection on both sides of Eq. (12).
Notice that the $M$ number in Eq. (12) should be regarded as the $K$ number
before projection. Then we have

$P^{J}_{MK}|\Psi_{\mu JK}\rangle=\sum_{i=1}^{D_{K}}C^{\mu
JK}_{i}P^{J}_{MK}|\Phi_{K,i}\rangle.$ (13)

The left side in Eq. (13) can be written as,

$P^{J}_{MK}|\Psi_{\mu
JK}\rangle=\sum_{\mu^{\prime}}|\Psi_{\mu^{\prime}JM}\rangle\langle\Psi_{\mu^{\prime}JK}|\Psi_{\mu
JK}\rangle=|\Psi_{\mu JM}\rangle.$ (14)

This means Eq. (13) and Eq. (12) are identical wave functions. Thus the shell
model wave function can be written as,

$|\Psi_{\mu JM}\rangle=\sum_{i=1}^{D_{K}}C^{\mu
JK}_{i}P^{J}_{MK}|\Phi_{K,i}\rangle.$ (15)

From Eq. (15), it is clearly shown that even without $K$-mixing, one can still
construct the exact shell model wave function with sufficient number of
projected SDs. Moreover, the $K$ number in Eq. (15) can be arbitrarily chosen
to form the same $|\Psi_{\mu JM}\rangle$ wave function. This tells us the
$K$-mixing is indeed unnecessary in the shell model calculation.

Therefore, one can safely construct an approximated shell model wave function
with a small number of $|\Phi_{i}\rangle$ SDs,

$|\Psi_{JM}\rangle=\sum_{i=1}^{n}f^{JK}_{i}P^{J}_{MK}|\Phi_{i}\rangle,$ (16)

which form is exactly the same as Eq. (15). Such $|\Psi_{JM}\rangle$ wave
function can be varied so that it can be close to the corresponding exact
shell model wave function. This is exactly our VAP method used in Refs. [13,
18]

In summary, we have studied the so called $K$-mixing in the nuclear wave
functions which are defined in the laboratory frame of reference. For a given
spin $J$, the nuclear wave function is usually expressed in terms of all the
$2J+1$ projected states with different $K$ numbers. The coefficients in front
of these $2J+1$ projected basis states are usually determined by solving the
corresponding Hill-Wheeler equation. So far, such form of nuclear wave
function have been adopted by almost all nowadays nuclear structure models
involving the angular momentum projection. Previously, the $K$-mixing in such
wave functions was believed to be useful in studying the structure of nuclear
states. Here, we have demonstrated in the SD case that such nuclear wave
function with $K$-mixing can always be equivalently replaced by a single
projected SD, which means the $K$-mixing can be safely removed and the nuclear
wave function can be written in a more compact form. This interesting feature
seriously reminds us that such $K$-mixing, different from that caused by the
Coriolis force in the cranked shell model, might be a misleading in
understanding the physics of nuclei.

Acknowledgements This work is supported by the National Natural Science
Foundation of China under Grant No. $11975314$, by the Key Laboratory of
Nuclear Data foundation (JCKY2022201C158) and by the Continuous Basic
Scientific Research Project Nos. WDJC-$2019$-$13$, BJ$20002501$.

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