### 5. Composite Systems

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 Schr\"odinger 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 (Section 5.5). The density operator generalizes the orthogonal projection operator onto the subspace spanned by a "pure state". 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.

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• 5.1. States of Two-Particle Systems 212
• 5.1.1. The Hamiltonian for two free particles 212
• 5.1.2. The Schrödinger equation of a two-particle system 213
• 5.1.3. Two-particle Hilbert space 214
• 5.1.4. The interpretation of two-particle wave functions 215
• 5.2. Hilbert Space of a Bipartite System 216
• 5.2.1. Construction of the tensor product 216
• 5.2.2. Orthonormal basis of the tensor product space 217
• 5.2.3. Entangled states 218
• 5.2.4. Example: Two-qubit system 219
• 5.2.5. Example: Particle with spin 220
• 5.3. Interacting Particles 221
• 5.3.1. Two-particle interactions 221
• 5.3.2. Separation of the center-of-mass motion 222
• 5.4. Observables of a Bipartite System 223
• 5.4.1. Tensor product of operators 223
• 5.4.2. Local manipulations 225
• 5.5. The Density Operator 227
• 5.5.1. What does entanglement mean for the subsystems? 227
• 5.5.2. Expectation values of subsystem observables 229
• 5.5.3. Trace-class operators 230
• 5.5.4. Density operator of a subsystem 232
• 5.6. Pure and Mixed States 233
• 5.6.1. State of a subsystem 233
• 5.6.2. Canonical form of the density operator 234
• 5.6.3. Ensemble interpretation of mixed states 236
• 5.7. Preparation of Mixed States 238
• 5.7.1. Preparing an ensemble in a mixed state 238
• 5.7.2. Ambiguity of realizing mixed states 240
• 5.7.3. Example: Mixed qubit states 241
• 5.8. More About Bipartite Systems 244
• 5.8.1. Normal form of the state vector 244
• 5.8.2. Maximally entangled states 247
• 5.8.3. Purification 248
• 5.8.4. Projection postulate for mixed states 249
• 5.9. Indistinguishable Particles 250
• 5.9.1. (Anti-)symmetric states of bipartite systems 250
• 5.9.2. Example: Bosons and fermions 254
• 5.9.3. Example: Two-qubit system 255
• 5.10. Special Topic: Multiparticle Systems with Spin 256
• 5.11. Special Topic: Addition of Angular Momenta 259
• 5.11.1. Total angular momentum 259
• 5.11.2. Eigenvalues of J3 260
• 5.11.3. The quantum numbers of the total angular momentum 261
• 5.11.4. Clebsch-Gordan coefficients 264
• 5.11.5. Angular momentum plus spin 1/2 266