Advanced Undergraduate Quantum Mechanics pp 329-344 | Cite as

# Two-Level System in a Periodic External Field

## Abstract

I have already mentioned somewhere in the beginning of this book that while vectors representing states of realistic physical systems generally belong to an infinite-dimensional vector space, we can always (well, almost, always) justify limiting our consideration to a subspace of states with a reasonably small dimension. The smallest nontrivial subspace containing states that can be assumed to be isolated from the rest of the space is two-dimensional. One relatively clean example of such a subspace is formed by two-dimensional spinors in the situations when one can neglect interactions between spins of different particles as well as by the spin–orbital interaction. An approximately isolated two-dimensional subspace can also be found in systems described by Hamiltonians with discrete spectrum, if this spectrum is strongly non-equidistant, i.e., the energy intervals between adjacent energy levels △_{ i } = *E*_{i+1} − *E*_{ i } are different for different pairs of levels. Two-level models are very popular in various areas of physics because, on one hand, they are remarkably simple, while on the other hand, they capture essential properties of many real physical systems ranging from atoms to semiconductors.