Lecture 22: Metals, Insulators, and Semiconductors

The script starts with a standard introduction to a lecture, highlighting the support for MIT OpenCourseWare and its mission to provide free educational resources. The professor then addresses the audience, mentioning a guest lecture at the end of the session and encourages questions from the audience. The first question posed by an audience member is about the physical significance of crystal momentum, which the professor agrees to explain, setting the stage for a detailed discussion on the topic.

The professor delves into the concept of crystal momentum, starting with a discussion on the physical meaning of momentum in general. He uses the example of a wave function to illustrate how momentum can be altered and relates this to the phase variation of the wave function. The lecture then transitions into the specifics of crystal momentum, explaining its role in governing the spatial phase variation within a periodic potential. The professor also touches on the conservation of crystal momentum, its time independence, and its distinction from regular momentum due to its lack of being an eigenvalue of the momentum operator.

Continuing the discussion, the professor explains the properties of crystal momentum, emphasizing its conservation modulo 2π/l, where l is the lattice spacing. This unique periodicity is contrasted with the strict conservation of regular momentum. The lecture also explores the implications of an external force on crystal momentum and how it changes according to the conservation law, highlighting the periodic nature of this change. The professor concludes this section by encouraging students to develop an intuition for crystal momentum through examples, particularly in the context of solid-state physics.

The script explores the implications of translational symmetry in quantum mechanics, particularly in the context of eigenfunctions and the impact of a periodic potential. The professor discusses how the eigenfunctions of the translation operator can be constructed and how they relate to the phase of the wave function. The lecture also addresses the subtleties of defining real functions in the presence of translational symmetry and the consequences for the current in quantum systems, leading to a deeper understanding of quantum behavior in periodic potentials.

This section delves into the effects of periodic potentials on energy and momentum in quantum systems. The professor explains how the introduction of a periodic potential can lead to the formation of energy bands and the concept of Bloch oscillations. The lecture also discusses the experimental verification of these theoretical predictions, including the use of optical lattices to simulate quantum systems and observe the expected behavior. The professor emphasizes the importance of understanding the relationship between theory and experiment in quantum mechanics.

The script examines the behavior of particles in a lattice, focusing on the quantum mechanical description of electrons in a periodic potential. The professor discusses the conditions under which current can flow in such a system, highlighting the role of superpositions and energy gaps between bands. The lecture also touches on the transparency and conductivity properties of materials, providing insights into the quantum origins of these macroscopic properties.

This part of the script discusses the quantum mechanical basis for electrical conductivity and the role of conduction and valence bands. The professor explains how the filling of these bands determines the electrical properties of materials, leading to the classification of materials as conductors, insulators, or semiconductors. The lecture also explores the concept of band gaps and their significance in the transparency and electrical behavior of materials like diamond and copper.

The script moves into a detailed discussion of band structures in three dimensions, highlighting the complexity introduced by the additional degrees of freedom. The professor explains how the interaction between different energy states can lead to overlapping bands and the potential for current flow even in the presence of a filled band. The lecture also touches on the concept of semiconductors and the role of temperature in their conductivity, providing a deeper understanding of the quantum mechanical underpinnings of material properties.

This section delves into the properties of semiconductors, focusing on the quantum mechanical explanation for their behavior. The professor discusses how the small band gaps in semiconductors allow for thermal excitation of electrons, leading to conductivity at higher temperatures. The lecture also explores the quantum mechanical description of the EPR paradox, introducing the concept of entanglement and its implications for our understanding of physical reality.

The script concludes with a discussion of the EPR paradox, a thought experiment proposed by Einstein, Podolsky, and Rosen to challenge the completeness of quantum mechanics. The professor summarizes the paradox, which involves the seemingly instantaneous correlation between the states of two entangled particles, and its implications for the principles of locality and reality in physics. The lecture also touches on the historical context and the subsequent debates surrounding the interpretation of quantum mechanics.