## 7 Adiabatic Evolution of the \"Invariant States\" of the Magnetic Moment Operator  
The stationary Fock states (3) are determined by integral parameters \(n_{r}=0,1,2,\ldots\) and \(m=0,\pm 1,\pm 2,\ldots\) In these normalized states, the energy and canonical angular momentum have definite values. In addition, the _mean value_ of the magnetic moment operator (13) does not depend on time in these states. Recently, a wider family of _isotropic_ states, possessing time-independent mean values of the magnetic moment operator for the time-independent Hamiltonian (2) and named as \"magnetic moment invariant states\", was found in paper [37]. They are determined by two _continuous_ positive parameters, \(G_{+}\geq 1\) and \(G_{-}\geq 1\), with the following nonzero mean values (provided \(\omega>0\)):  
\[\langle\tau_{x}^{2}\rangle=\langle\tau_{y}^{2}\rangle=\langle\hat{H}\rangle= \hbar\omega G_{+}, \tag{57}\]  
\[\langle\hat{x}\hat{p}_{y}\rangle=-\langle\hat{y}\hat{p}_{x}\rangle=\frac{ \hbar}{4}(G_{-}-G_{+}),\qquad\langle y\pi_{x}\rangle=-\langle x\pi_{y}\rangle =\frac{\hbar}{2}G_{+}, \tag{58}\]  
\[\langle p_{x}^{2}\rangle=\langle p_{y}^{2}\rangle=\omega^{2}\langle x^{2} \rangle=\omega^{2}\langle y^{2}\rangle=\frac{1}{4}\hbar\omega(G_{+}+G_{-}). \tag{59}\]  
The solution (3) results in Formulas (57)-(59) with the _integral_ positive coefficients  
\[G_{\pm}=1+2n_{r}+|m|\mp m. \tag{60}\]It is known that there exist many quite different quantum states with the same mean values of canonical operators and their powers (products). In particular, one can construct a two-parameter family of _Gaussian_ states possessing the second-order moments (57)-(59) with arbitrary coefficients \(G_{\pm}\geq 1\). Their Wigner functions have the following form [37] [here \(\mathbf{r}=(x,y)\) and \(\mathbf{p}=(p_{x},p_{y})\)]:  
\[W_{G}(\mathbf{r},\mathbf{p})=\frac{4}{G_{+}G_{-}}\exp\Biggl{\{}-\frac{(G_{+}+G _{-})\bigl{[}\omega^{2}\mathbf{r}^{2}+\mathbf{p}^{2}\bigr{]}+2(G_{+}-G_{-}) \omega(xp_{y}-yp_{x})}{2\hbar\omega G_{+}G_{-}}\Biggr{\}}. \tag{61}\]  
The quantum purity of the state (61) equals  
\[\mathcal{P}=\int W^{2}(\mathbf{r},\mathbf{p})\frac{d\mathbf{r}d\mathbf{p}}{(2 \pi\hbar)^{2}}=(G_{+}G_{-})^{-1}. \tag{62}\]  
Consequently, all Gaussian \"magnetic moment invariant states\" are _mixed_ if \(G_{\pm}>1\), in contradistinction to the _pure_ quantum states (3) with integral values of parameters \(G_{\pm}\), given by Equation (60). The meaning of parameters \(G_{+}\) and \(G_{-}\) becomes more clear, if one goes from the canonical operators in the phase space to the relative \((x_{r},y_{r})\) and guiding center \((x_{c},y_{c})\) coordinates,  
\[x_{r}=-\pi_{y}/(2m\omega),\;\;y_{r}=\pi_{x}/(2m\omega),\;\;\;x_{c}=x+\pi_{y}/( 2m\omega),\;\;y_{c}=y-\pi_{x}/(2m\omega). \tag{63}\]  
The importance of these integrals of motion was emphasized by many authors in recent decades [38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55]. Equivalent integrals of motion, obtained by the multiplication of \(x_{c}\) and \(y_{c}\) by \(m\omega\), were considered under the name \"pseudomomentum\" in papers [43; 56; 57]. The corresponding second-order moments are as follows,  
\[\langle x_{r}^{2}\rangle=\langle y_{r}^{2}\rangle=\frac{\hbar}{4\omega}G_{+}, \qquad\langle x_{c}^{2}\rangle=\langle y_{c}^{2}\rangle=\frac{\hbar}{4\omega} G_{-}, \tag{64}\]  
with zero mean values of all cross-products. Now, the inequalities \(G_{\pm}\geq 1\) follow from the commutators \([\hat{x}_{r},\hat{y}_{r}]=[\hat{y}_{c},\hat{x}_{c}]=i\hbar/(2\omega)\) and the Heisenberg-Weyl (actually, [PERSON]'s [58]) uncertainty relation  
\[\langle A^{2}\rangle\langle B^{2}\rangle\geq|\langle[\hat{A},\hat{B}]\rangle| ^{2}/4. \tag{65}\]  
The evolution of the second-order moments can be calculated in the same way as that shown in Section 6. It is sufficient to replace \(\gamma\) with \((G_{+}+G_{-})/2\) and \(m\) with \((G_{-}-G_{+})/2\) in Equations (50) and (51).