Two-state quantum system

As a result, the dynamics of a two-state system can be solved analytically without any approximation.

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.

Now, if we follow the same derivation, but before acting with the Hamiltonian on the individual states, we multiply both sides by

The above matrix equation should thus be interpreted as a restrictive condition on a general state vector to yield an eigenvector of

The reason is that, in some more complex problems, the state vectors may not be eigenstates of the Hamiltonian used in the matrix.

One place where this occurs is in degenerate perturbation theory, where the off-diagonal elements are nonzero until the problem is solved by diagonalization.

The eigenvectors represent the stationary states, i.e., those for whom the absolute magnitude of the squares of the probability amplitudes do not change with time.

The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system is given by

The allowed energy levels of the system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way.

, and we are interested in the probability of occupation of each of the basis states as a function of time when

It can be seen that such a time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of Larmor precession)[3] The above method can be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic field in the previous case) if the interaction is given by an appropriate coupling term that is analogous to the magnetic moment.

Nuclear magnetic resonance (NMR) is an important example in the dynamics of two-state systems because it involves the exact solution to a time dependent Hamiltonian.

and a transverse rf field B1 rotating in the xy-plane in a right-handed fashion around B0:

is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field.

[citation needed] This is the fundamental basis for NMR, and in practice is accomplished by scanning

To remove the time dependence from the problem, the wave function is transformed according to

and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields

Adding this equation to its own conjugate transpose yields a left hand side of the form

As previously mentioned, the expectation value of each Pauli matrix is a component of the Bloch vector,

All that is left to obtain the final form of the optical Bloch equations is the inclusion of the phenomenological relaxation terms.

As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator in the Heisenberg picture.

Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues.

This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of charge qubits.

In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than the ones of interest.

Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems.

It can be used to illustrate fundamental quantum mechanical phenomena such as the interference exhibited by particles of the polarization states of the photon,[5] but also more complex phenomena such as neutrino oscillation or the neutral K-meson oscillation.

Two-state formalism can be used to describe simple mixing of states, which leads to phenomena such as resonance stabilization and other level crossing related symmetries.

Phenomena with tremendous industrial applications such as the maser and laser can be explained using the two-state formalism.

Qubits, which are the building blocks of a quantum computer, are nothing but two-state systems.