5. Resonance V and Atoms I

The script begins with an introduction to MIT OpenCourseWare, highlighting its commitment to providing educational resources freely under a Creative Commons license and inviting donations to support the cause. It then transitions into a lecture on quantum mechanics, specifically revisiting the Landau-Zener transition discussed in a previous session. The professor emphasizes the coherent process of amplitude transfer between states during the transition and introduces the concept of Rabi frequency during the effective time of population transfer. The lecture also hints at the density matrix formalism as a more comprehensive approach to describe quantum systems beyond pure states.

This paragraph delves into the density matrix formalism, a tool for describing quantum systems that may not be in a pure state. The professor explains the limitations of the Schrödinger equation in depicting processes like decoherence and loss of particles, and how the density operator can account for these phenomena. The introduction of the density matrix is formalized through the expectation value of observables, with the diagonal elements representing populations and off-diagonal elements representing coherences. The paragraph also discusses the incorporation of probabilities for systems not in a pure state, highlighting the dual averaging process inherent in quantum mechanics and the density matrix's ability to handle both.

The professor continues the discussion on the density matrix, focusing on its properties and its role in describing the time evolution of quantum systems. It is highlighted that the density matrix is normalized to unity, reflecting the total probability of finding the system in one of its states. The trace of the density matrix squared is also discussed, indicating the distinction between pure and mixed states. The paragraph further explores the use of the density matrix in non-unitary time evolution, particularly for open systems, and introduces the concept of a two-level system's Hamiltonian using Pauli matrices.

The script moves on to describe the most general Hamiltonian for a two-level system using the Bloch vector and Pauli matrices. The professor explains how the Hamiltonian can be expressed as a scalar product of the vector of Pauli matrices and a vector of coefficients. The density matrix for a two-level system is also expanded in terms of the unit matrix and Pauli matrices, with the trace of the density matrix being normalized to 1. The paragraph concludes with the insertion of the Hamiltonian and density matrix into the equation of motion, leading to the Bloch equation and its implications for the time evolution of quantum systems.

This paragraph introduces the Bloch equations, which describe the time evolution of the density matrix for a two-level system. The professor provides a high-level overview of the derivation of these equations, starting from the equation of motion for the density matrix and incorporating relaxation processes. The Bloch equations are presented as a generalization of the behavior of a classical magnetic moment in a magnetic field, with the Bloch vector undergoing precession. The paragraph also discusses the physical meaning of the Bloch vector and its components, as well as the importance of the equations in quantum mechanics and optics.

The script addresses the need to include relaxation processes in the description of quantum systems to account for phenomena like decoherence and spontaneous emission. The professor introduces damping terms in a phenomenological manner to describe how systems return to thermal equilibrium. The paragraph explains the concept of relaxation times T1 and T2, which correspond to the damping of population differences and the loss of coherences, respectively. The Bloch equations are then reformulated to include these damping terms, providing a more accurate model of real-world quantum systems.

The professor discusses the factors that determine the T1 and T2 relaxation times in quantum systems. T1 is associated with energy decay and is influenced by processes such as collisions, while T2 is related to the loss of phase coherence, which can be affected by inhomogeneous environments or fluctuating fields. The script also touches on the subtlety of the definitions of T1 and T2 and how they relate to the damping of the wave function and its square, respectively. The discussion provides insights into the different processes that can contribute to relaxation in quantum systems.

The script shifts focus from the discussion of two-level systems and resonance to the study of atomic structure, starting with the hydrogen atom. The professor sets the agenda for the next lectures, which will cover electronic structure, fine structure, hyperfine structure, and the effects of external fields on atomic energy levels. The hydrogen atom is highlighted as a fundamental system for understanding atomic structure due to its simplicity and the challenges it presents in experimental physics.

This paragraph provides an in-depth look at the hydrogen atom, discussing its energy levels described by the Rydberg formula and the solutions to the Schrödinger equation that yield the atomic eigenfunctions. The professor emphasizes the importance of understanding the scaling of length scales and energy levels in hydrogen-like atoms, introducing the Bohr radius as a natural length scale and discussing the implications of the Virial theorem on the expectation value of the inverse radius. The paragraph also touches on the radial part of the wave function and its significance in understanding the size of the hydrogen atom.

The script concludes with a discussion on how the properties of hydrogen-like atoms scale with the nuclear charge z and the principal quantum number n. The professor explains that the length scales contract by a factor of z, leading to an increase in energy levels by a factor of z squared. The probability of finding an electron at the nucleus is also examined, revealing its dependence on z and n. The insights provided in this paragraph are crucial for understanding the behavior of electrons in the presence of the nucleus and have implications for phenomena such as hyperfine structure and quantum defects.