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5: Multi-Particle Systems

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    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
      This page covers the quantum mechanical description of non-relativistic particle systems through wavefunctions, \(\psi(x,t)\), which determine particle location probabilities and require normalization. For multiple particles, the wavefunction expands to \(\psi(x_1, x_2, \ldots, x_N,t)\) and must also be normalized.
    • 5.2: Non-interacting Particles
      This page describes the Hamiltonian for multi-particle systems, highlighting its components of kinetic and potential energy. It explains that for non-interacting particles, the Hamiltonian can be simplified to a sum of independent single-particle Hamiltonians. This leads to the multi-particle wavefunction being expressed as a product of single-particle wavefunctions, allowing the Schrödinger equation to be factored into independent equations.
    • 5.3: Two-Particle Systems
      This page explores a two-particle system with masses \(m_1\) and \(m_2\) that interact through a position-dependent potential. It explains how to transform the Hamiltonian using center of mass and relative coordinates, allowing for wavefunction factorization. Key equations show conservation of the total momentum and time-dependent Schrödinger equations.
    • 5.4: Identical Particles
      This page explores the wavefunctions of systems with identical particles, distinguishing between bosons and fermions based on their symmetrical properties. It delineates the construction of wavefunctions for two identical non-interacting particles and underscores the Pauli exclusion principle that prevents identical fermions from occupying the same state.
    • 5.E: Multi-Particle Systems (Exercises)
      This page explores the behavior and theoretical constructs of two non-interacting particles in various quantum states, focusing on distinguishable particles, indistinguishable bosons, and fermions. It details state counting differences and calculates expectation values like \(\langle (x_1-x_2)^{\,2}\rangle\), as well as energy levels and degeneracies in a 1D box, emphasizing the impact of quantum statistics arising from particle indistinguishability.

    This page titled 5: Multi-Particle Systems is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard Fitzpatrick via source content that was edited to the style and standards of the LibreTexts platform.