In the quantum mechanics, special states such as eigenstate, basis states, and ground state are defined. I may be know these definitions, but I'm very confusing; this confusion occurred when I studying the Two-state quantum system.
**My questions are shown below**.

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**【My questions】**<br>
Is it possible to realize a Two-state quantum system having basis states $|1>$ and $|2>$ under the Hamiltonian of (1-3)' below ? If so, why?

Here, $|1>$ and $|2>$ are expressed as follows.
$$|1>:=\left(\begin{matrix}1\\0\\\end{matrix}\right) \tag{1-1}$$
$$|2>:=\left(\begin{matrix}0\\1\\\end{matrix}\right) \tag{1-2}$$

According to the [Wikipedia](https://en.wikipedia.org/wiki/Two-state_quantum_system), the Hamiltonian $H$ of the two-state quantum system, **whose basis states are $|1>$ and $|2>$** should be written in the form of (1-3). Here, ${\varepsilon_1}$ and ${\varepsilon_2}$ are real number,  $\gamma$ is a complex number, and $\bar{\gamma}$ is a conjugate complex number of the $\gamma$.

$$H={\varepsilon_1}\left|1><1|\ +{\varepsilon_2}\right|2><2|\ +\ \bar{\gamma}|1><2|\ +\ \gamma|2><1| \tag{1-3}$$

Following (1-4) is the [matrix representation](https://en.wikipedia.org/wiki/Matrix_representation) of the $H$ of (1-3)
$$\left(\begin{matrix}\varepsilon_1&\bar{\gamma}\\ \gamma &\varepsilon_2\\\end{matrix}\right)\tag{1-4}$$


Therefore, the Hamiltonian of equation (1-4) 'should be appropriate as the Hamiltonian of the two-state system; this is just a special case of ${\varepsilon_1} =\gamma=1$, and ${\varepsilon_2} =-1$ in the (1-3).
$$H=\left|1><1|\ -\right|2><2|\ +\ |1><2|\ +\ |2><1| \tag{1-3'}$$

Following (1-4)' is the matrix representation of the $H$ of (1-3)'.
$$\left(\begin{matrix}1&1\\1&-1\\\end{matrix}\right)\tag{1-4'}$$

From the assumption, the possible states are $|1>$ or $|2>$. Or, the $|1>$ or $|2>$ are the most stable states.

On the other hand, the $\varphi$ is an eigenstate of the hamiltonian $H$ iff the $\varphi$ is the eigenvector.
In this sence, eigenstates are nothing but the following $\varphi_{-\sqrt{2}}$ and $\varphi_{\sqrt{2}}$.


$$\varphi_{-\sqrt2}:=\frac{1}{\sqrt{2\left(2+\sqrt2\right)}}\left(\begin{matrix}1\\-\sqrt{2\ }\ -1\\\end{matrix}\right)$$
$$\varphi_{\sqrt2}:=\frac{1}{\sqrt{2\left(2-\sqrt2\right)}}\left(\begin{matrix}1\\ \sqrt{2\ }\ -1\\\end{matrix}\right)$$

The smallest eigenvalue is ${-\sqrt2}$, so $\varphi_{-\sqrt2}$ might be the [ground state](https://en.wikipedia.org/wiki/Ground_state), but neither  $|1>$ nor $|2>$ might be  ground state; the expected value of energy of $|1>$ and $|2>$  seems higher than $-\sqrt{2}$, 
therefore, I have no idea why $|1>$ and $|2>$ are 'preferred' state; these are unlikely to be the [stable state](https://en.wikipedia.org/wiki/Stationary_state).