## 4 Quantum Stability and Ordered Structures for Many-Body Systems of Trapped Ions  
Furthermore, we apply the Hessian matrix approach and method previously introduced to investigate semiclassical stability and ordered structures for strongly coupled Coulomb systems (SCCS) confined in a 3D QIT. In addition we suggest an analytical a method to determine the associated critical points. We consider a system consisting of \(N\) identical ions of mass \(m_{\alpha}\) and electric charge \(Q_{\alpha}\) (\(\alpha=1,2,\ldots,N\)), confined in a 3D RF (Paul) trap. The coordinate vector of the particle labeled as \(\alpha\) is denoted by \(\vec{r}_{\alpha}=(x_{\alpha},y_{\alpha},z_{\alpha})\). A number of \(3N\) generalized quantum coordinates \(q_{ai},i=1,2,3\), are associated to the \(3N\) degrees of liberty. We also denote \(q_{\alpha 1}=x_{\alpha}\), \(q_{\alpha 2}=y_{\alpha}\), and \(q_{\alpha 3}=z_{\alpha}\). Hence, the kinetic energy for a number of \(\alpha\) particles confined in the trap can be expressed as  
\[T=\sum_{\alpha=1}^{N}\sum_{i=1}^{3}\frac{1}{2m_{\alpha}}\,q_{\alpha i}^{2}\, \tag{41}\]  
while the potential energy is  
\[U=\frac{1}{2}\sum_{\alpha=1}^{N}\sum_{i=1}^{3}k_{i}q_{\alpha i}^{2}+\sum_{1 \leq\alpha<\beta\leq N}\frac{Q_{\alpha}Q_{\beta}}{4\pi\epsilon_{0}|\vec{r}_{ \alpha}-\vec{r}_{\beta}|}\, \tag{42}\]  
where \(\epsilon_{0}\) stands for the vacuum permittivity. For a spherical 3D trap: \(k_{1}=k_{2}=k_{3}\). In case of Paul or Penning 3D QIT \(k_{1}=k_{2}\), while the linear 2D Paul trap (LIT) corresponds to  
Figure 17: The bifurcation set for a system of two ions confined in a Paul trap. \(k_{3}=0,\ k_{1}=-k_{2}\). The \(k_{i}\) constants in case of a Paul trap result from the pseudopotential approximation. The critical points of the system result as:  
\[\sum_{\alpha=1}^{N}\Delta_{\alpha}U=0\,\ \Delta_{\alpha}=\frac{\partial^{2}}{ \partial x_{\alpha}^{2}}+\frac{\partial^{2}}{\partial y_{\alpha}^{2}}+\frac{ \partial^{2}}{\partial z_{\alpha}^{2}}\, \tag{43}\]  
\[\Delta U=0\ \text{or}\ \frac{\partial U}{\partial q_{\gamma j}}=0,\ \ \gamma=1,\ldots,N;\ j=1,2,3. \tag{44}\]  
We denote  
\[\frac{\partial q_{ai}}{\partial q_{\gamma j}}=\delta_{\alpha\gamma}\delta_{ij}\, \tag{45}\]  
where \(\delta\) stands for the Kronecker delta function. After doing the math we can write Equation (44) as  
\[\frac{\partial U}{\partial q_{\gamma j}}=\frac{1}{2}\sum_{\alpha=1}^{N}\sum_{ i=1}^{3}2k_{i}q_{\alpha i}\delta_{\alpha\gamma}\delta_{ij}-\sum_{1\leq\alpha< \beta\leq N}\frac{1}{4\pi\epsilon_{  
We search for a fixed solution \(q^{0}_{\gamma j}\) of the system of Equation (49). Then, the elements of the Hessian matrix of the potential function \(U\) in a critical point of coordinates \(q^{0}_{\gamma j}\) are:  
\[\frac{\partial^{2}U}{\partial q^{0}_{\gamma j}\partial q^{0}_{ \gamma^{\prime}j^{\prime}}}=k_{j}\delta_{\gamma\gamma^{\prime}}\delta_{j\bar{ \gamma}}+\varepsilon_{\gamma\gamma^{\prime}}\delta_{j\bar{\gamma}^{\prime}}- \left(\sum_{\alpha=1}^{N}\varepsilon_{\alpha\gamma}\right)\delta_{\gamma\gamma ^{\prime}}\delta_{j\bar{\gamma}^{\prime}}+\\ \eta_{\gamma\gamma^{\prime}}\left(q^{0}_{\gamma j}q^{0}_{\gamma^ {\prime}j^{\prime}}-q^{0}_{\gamma j}q^{0}_{\gamma^{\prime}j}+q^{0}_{\gamma^{ \prime}j}q^{0}_{\gamma^{\prime}j^{\prime}}\right)+q^{0}_{\gamma j}q^{0}_{ \gamma\gamma^{\prime}}\sum_{\alpha=1}^{N}\eta_{\alpha\gamma}\delta_{\gamma \gamma^{\prime}}+\\ \delta_{\gamma\gamma^{\prime}}\sum_{\alpha=1}^{N}\eta_{\alpha \gamma}q^{0}_{\alpha j^{\prime}}-\delta_{\gamma\gamma^{\prime}}q^{0}_{\gamma \gamma}\sum_{\alpha=1}^{N}\eta_{\alpha\gamma}q^{0}_{\alpha j^{\prime}}-\delta _{\gamma\gamma^{\prime}}q^{0}_{\gamma\gamma^{\prime}}\sum_{\alpha=1}^{N}\eta_ {\alpha\gamma}q^{0}_{\alpha j}\,. \tag{53}\]  
As it can be observed from Equation (53), our method allows one to determine (identify) the critical points of the potential function for the quantum system of \(N\) identical ions, where equilibrium configurations occur.