6: Three-Dimensional Quantum Mechanics

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In this chapter, we shall extend our previous one-dimensional formulation of non-relativistic quantum mechanics to produce a fully three-dimensional theory.

6.1: Fundamental Concepts
This page covers wavefunctions in quantum mechanics, explaining their role in defining particle states in various dimensions and introducing key concepts such as probability density, normalization, and probability conservation. It details position and momentum operators and their commutation relations, as well as the Hamiltonian and the time-dependent Schrödinger equation.
6.2: Particle in Box
This page explores the quantum mechanics of a particle in a cubic box, emphasizing the stationary wavefunction and its boundary conditions. The wavefunction is factored by spatial dimensions, yielding second-order differential equations with sinusoidal solutions. From these, quantized energy levels are derived, expressed in relation to the integers associated with the wavefunction's modes.
6.3: Degenerate Electron Gases
This page explores the behavior of non-interacting electrons in a cubic box, highlighting the Pauli exclusion principle, degeneracy in electron gases, and single-particle energy levels. It discusses Fermi energy related to electron density, differentiating between degenerate and non-degenerate electron gas behavior based on pressure and thermal energy. The conclusion emphasizes the bulk modulus of metals, illustrating their strong resistance to compression due to high electron degeneracy.
6.4: White Dwarf Stars
This page explains the lifecycle of main-sequence stars, such as the Sun, highlighting their equilibrium between gravitational forces and thermal expansion due to nuclear fusion. It describes the transition of stars into white dwarfs when fuel is depleted, supported by electron degeneracy pressure. The concept of the Chandrasekhar limit is introduced, noting its significance for stellar collapse into neutron stars or black holes.
6.5: Exercises
This page covers the analysis of a particle in a three-dimensional isotropic harmonic oscillator potential, emphasizing energy levels and separation of variables. It addresses fermions, calculating Fermi energy under masslessness and examining the density of states for three-dimensional and two-dimensional electron gases.

Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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