In Chap. 18, we limit the Cooper pair box to a two-level system to solve for its eigenenergies and eigenstates analytically. We also derive its eigenstates under certain conditions. Particularly, we see that it has the least charge sensitivity when it is operated at the sweet spot, and we can further reduce its charge sensitivity by increasing the Josephson-to-charging energy ratio. This can be achieved by having a large shunt capacitor and it is called the transmon qubit. However, we are not sure if the conclusions from the two-level system analytical solutions are correct because a two-basis-state approximation is used. Moreover, without studying higher levels, we do not understand its anharmonicity and cannot appreciate its role as an artificial atom due to its uneven energy spacings like in a natural atom. In this chapter, we will diagonalize the Hamiltonian numerically so that we can include more levels. We will compare the results to the conclusions we drew in the previous chapter. Finally, we will perform expansion for transmon to extract its parameters in analytical form which can be used to design superconducting circuits.