Two-state quantum system

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In quantum mechanics, a two-state system is a system which has two possible states. More formally, the Hilbert space of a two state system has two degrees of freedom, so a complete basis spanning the space must consist of two independent states.

The physics of a quantum mechanical two-state system is trivial if the states are degenerate, that is, if the states have the same energy. However, if there is an energy difference between the two states, then nontrivial dynamics can occur.

1 Two-state dynamics
2 Examples of two-state quantum systems
3 See also
4 Links

[edit] Two-state dynamics

If the time-independent Hamiltonian is $H$ , and we label the two levels as $\left.|a\right\rangle$ and $\left.|b\right\rangle$ with corresponding orthonormal energy eigenvalues $E a$ and $E b$ , then the dynamics of the system can be specified as follows:

At some time $t 0$ , let the system be in an arbitrary (and completely general) state, $\left.|\psi(t_0)\right\rangle=c_a\left.|a\right\rangle+c_b\left.|b\right\rangle$ then after evolving under $H$ , at time $t$ , the state will be $\left.|\psi(t)\right\rangle = c_a e^{-i\frac{E_a(t-t_0)}{\hbar}}\left|a\right\rangle+ c_b e^{-i\frac{E_b(t-t_0)}{\hbar}}\left|b\right\rangle$

The physics of two state systems can be usefully applied to multi-state systems where the system is known to have only enough energy available to excite the lowest two states, thus effectively creating a two state system. In fact, in nature, it is difficult to identify any true two-state systems; merely systems where the energetics of the circumstances isolate two particular states.

The set of all states in a two-level system can be mapped onto a representation known as the Bloch sphere.

[edit] Examples of two-state quantum systems

Spin-1/2 particles are two-state quantum systems when only the spin degree of freedom is considered.
The "inverting" degree of freedom in an ammonia molecule; the nitrogen at the vertex of an ammonia molecule exhibits two molecular states - "up" and "down", corresponding to opposite positions with respect to the plane of the three hydrogen atoms. In an electric field, these two states are non-degenerate.
Two-level systems are important in the field of quantum computing as are used to implement qubits.

[edit] See also

[edit] Links

http://www.itp.tu-berlin.de/zweiniveau.html A Java applet that visualizes various effects of two-state systems (laser driven). English user interface, but German documentation.
http://www.itp.tu-berlin.de/6854.html extended version of the applet. Includes electron phonon interaction.