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5: Multi-Particle Systems

In this chapter, we shall extend the single particle, one-dimensional formulation of non-relativistic quantum mechanics, introduced in the previous chapters, in order to investigate one-dimensional systems containing multiple particles.

• 5.1: Fundamental Concepts of Multi-Particle Systems
We have already seen that the instantaneous state of a system consisting of a single non-relativistic particle, whose position coordinate is $$x$$ , is fully specified by a complex wavefunction $$\Psi(x,t)$$ . This wavefunction is interpreted as follows.
• 5.2: Non-interacting Particles
For the case of non-interacting particles, the multi-particle Hamiltonian of the system can be written as the sum of N independent single-particle Hamiltonians. Moreover,  the energy of the whole system is simply the sum of the energies of the component particles.
• 5.3: Two-Particle Systems
Consider a system consisting of two particles, mass m₁  and m₂ , interacting via a potential V(x₁−x₂) that only depends on the relative positions of the particles. .  in the center of mass frame, two particles of mass m₁ and m₂ , moving in the potential V(x₁−x₂) , are equivalent to a single particle of mass μ , moving in the potential V(x) , where x=x₁−x₂.
• 5.4: Identical Particles
Wavefunctions of systems containing many identical particles are symmetric or anti-symmetric under interchange of the labels on any two particles is determined by the nature of the particles themselves . Wavefunctions that are symmetric under label interchange are said to obey Bose-Einstein statistics , and are called bosons. For instance, photons are bosons. Wavefunctions that are anti-symmetric under label interchange are said to obey Fermi-Dirac statistics , and are called fermions.
• 5.E: Multi-Particle Systems (Exercises)