Composite Systems

Chapter summary

Up to now, we have only considered single-particle systems. Real physics starts where at least two particles are involved. As a first example of a two-particle system we consider the Schrödinger equation for two free particles. This shows us how to construct two-particle states as products of one-particle states (Section 5.1). An abstract formulation of this method is given by the tensor product of Hilbert spaces. The Hilbert space of a composite system contains not only product states, but also their linear combinations (Section 5.2). In general, these states are entangled, that is, they cannot be written as simple products. Entanglement cannot be created by local measurements or manipulations of the subsystems, but usually, an interaction between the subsystems immediately leads to entanglement (Sections 5.3 and 5.4).

The theory presented here has applications not only to atomic physics but also to quantum information theory. An entangled state of a composite system encodes information about the system as a whole that cannot be measured locally (that is, by measurements on the subsystems alone). This information describes correlations between the subsystems. In general, the state of a subsystem cannot be described by a state vector in the Hilbert space of the subsystem. If the bipartite system is in an entangled state, then any subsystem is in a statistical mixture of states. We describe this new situation by a density operator ρ (Section 5.5). The density operator generalizes the orthogonal projection operator onto the subspace spanned by a “pure state” ψ. In Section 5.6, we compare pure and mixed states and discuss the ensemble interpretation of mixed states. We describe the ambiguities in the preparation of mixed states in Section 5.7 and some mathematical aspects of bipartite systems in Section 5.8 (the normal form of the state vector, maximally entangled states, purification, and the projection postulate for mixed states).

Section 5.9 describes the physics of composite systems whose parts are indistinguishable. A major new ingredient is the symmetrization postulate, which says that all states of the composite system are either symmetric or antisymmetic under exchange of the subsystems. Hence, one may conclude that there are two families of elementary particles, bosons and fermions. Fermions obey Pauli’s exclusion principle, which states that two fermions cannot occupy the same state. Section 5.10 describes multiparticle systems with spin, thus forming the theoretical foundation of the physics of atoms, molecules, and matter. Finally, in Section 5.11 we present the addition of angular momenta as a problem where two systems are combined into a larger system.