Lecture 5 | The Theoretical Minimum

The first paragraph introduces the topics to be covered in the lecture, including uncertainty, the Schrödinger equation, and the evolution of quantum systems over time. It emphasizes the general principles of quantum mechanics, such as the role of observables and states, and the concept of distinguishable states. The lecture also hints at exploring systems with multiple spins, highlighting the complexity of states as they increase from single to multiple entities.

The second paragraph delves into the uncertainty principle, discussing how certain pairs of observables cannot be simultaneously measured with precision. It explains the conditions under which observables can be simultaneously measured—namely, when the operators representing these observables commute. The concept of a basis of vectors, which are simultaneous eigenvectors of distinct observables, is also explored, leading to the theorem that commuting observables allow for simultaneous measurability.

The third paragraph contrasts the conditions for simultaneous measurability with the behavior of operators that do not commute, such as the Pauli spin matrices. It reinforces the idea that non-commuting operators cannot be simultaneously measured, which is a core aspect of the uncertainty principle. The paragraph also touches on the classical physics perspective, where observables like position and momentum can be known with certainty, unlike in quantum mechanics.

The fourth paragraph discusses the time evolution of quantum states, focusing on the Schrödinger equation's role in governing this process. It covers the conservation of orthogonality in states and the U operator, which projects a state from an initial time to a later time. The concept of continuity in state evolution is introduced, leading to the derivation of the Hamiltonian as a Hermitian operator, which is crucial for the deterministic evolution of quantum states.

The fifth paragraph examines the Hamiltonian in more detail, emphasizing its role in the deterministic evolution of a quantum system when no measurements are being made. It distinguishes between the deterministic evolution of the state vector and the probabilistic nature of measurement outcomes. The Hamiltonian is identified as a generator of time evolution, and its hermiticity is linked to the conservation of probability.

The sixth paragraph explores the evolution of expectation values over time. It explains how the state vector's time dependence leads to changes in expectation values and how these changes can be computed using the time-dependent Schrödinger equation. The paragraph also introduces the concept of the Heisenberg picture of quantum mechanics, where the observables evolve while the states remain constant.

The seventh paragraph focuses on commutators and their significance in quantum mechanics. It establishes the rules for commutators, drawing parallels with the rules for Poisson brackets from classical mechanics. The paragraph also introduces the Heisenberg equations of motion, which describe how the average values of observables change over time, and highlights the importance of commutators in these equations.

The eighth paragraph applies the concepts discussed to a practical scenario: a single spin in a magnetic field. It outlines the Hamiltonian for a spin in a magnetic field, derives the time evolution of the spin components, and explains how the spin precesses around the magnetic field direction. The solution to the Schrödinger equation for this system is presented, illustrating how expectation values of spin components change over time.

The ninth paragraph contemplates the brittleness of quantum mechanics as a theory and contrasts it with other physical theories that allow for more deformation. It also addresses questions about the real numbers in quantum states, the unitarity of the Hamiltonian, and the connection between quantum mechanics and classical mechanics. The paragraph ends with a discussion on the probabilistic nature of quantum mechanics and its experimental validation.

The tenth paragraph further explores the precession of a quantum spin in a magnetic field, drawing analogies with classical angular momentum. It discusses the relationship between the spin operators and their commutation relations, which are reminiscent of the Poisson brackets for angular momentum. The paragraph also touches on the cyclic nature of these commutation relations and their implications for the dynamics of spin.

The eleventh paragraph discusses the most general form of the Hamiltonian for a single spin system. It explains that the Hamiltonian can be a linear combination of the Pauli matrices and that adding a magnetic field in any direction results in the spin precession around that direction. The paragraph also addresses questions about the dimensionality of state spaces and the relationship between real space and state space vectors.

The twelfth paragraph ponders why the magnetic field is treated classically rather than as a quantum operator. It discusses the division between the quantum system and the measurement apparatus, and the justification for treating heavy objects like magnets classically due to their minimal quantum fluctuations. The paragraph also raises questions about the nature of quantum measurement and the probabilistic outcomes of quantum events.

The final paragraph addresses the possibility of a quantum spin emitting a photon when in a magnetic field. It explains that while the spin can undergo transitions between energy states, the probability of emitting a photon per unit time can be very small, leading to a spin that precesses many times before a photon is emitted. The paragraph concludes with a discussion on the probabilistic nature of quantum mechanics and the role of statistics in understanding experimental outcomes.