In quantum mechanics, a two-state system (also known as a two-level system) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit. Two-state systems are the simplest quantum systems that are of interest, since the dynamics of a one-state system is trivial (as there are no other states the system can exist in). The mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As a result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states. A very well known example of a two-state system is the spin of a spin-1/2 particle such as an electron, whose spin can have values +ħ/2 or −ħ/2, where ħ is the reduced Planck constant. The two-state system cannot be used as a description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two-state system are oscillatory. Analytical solutions for stationary state energies and time-dependence Representation Supposing the two available basis states of the system are and , in general the state can be written as a superposition of these two states with probability amplitudes , Since the basis states are orthonormal, where and is the Kronecker delta, so . These two complex numbers may be considered coordinates in a two-dimensional complex Hilbert space. Thus the state vector corresponding to the state is and the basis states correspond to the basis vectors, and If the state is normalized, the norm of the state vector is unity, i.e. . All observable physical quantities, such as energy, are associated with hermitian operators. In the case of energy and the corresponding Hamiltonian, , this means i.e. and are real, and . Thus, these four matrix elements produce a 2×2 hermitian matrix, The time-independent Schrödinger equation states that ; substituting for in terms of the basis states from above, and multiplying both sides by or produces a system of two linear equations that can be written in matrix form, or