Lecture 9 | Quantum Entanglements, Part 1 (Stanford)

The paragraph begins with an introduction to the topic of quantum systems and their evolution over time. It emphasizes the importance of revisiting previous material to ensure a solid understanding. The lecturer discusses the state of a quantum system, described by a ket vector, and the concept of time dependence. The central role of linear operators in the time evolution of quantum states is introduced, with an explanation of the postulate that state vectors change linearly. The paragraph also covers the complex conjugate rule for bra vectors and the assumption of probability conservation, leading to the introduction of unitary operators.

This section delves into the inner product of vectors and its time invariance, which is a consequence of the assumption that the length of every vector remains unchanged. The lecturer explains how the inner product between any pair of vectors remains constant over time. The concept of the total probability and its conservation is also discussed, leading to the introduction of the general statement involving the product of state vectors and the inner product.

The paragraph introduces the concept of unitary operators, which are crucial for maintaining the probability conservation in quantum mechanics. It explains that the product of a unitary operator and its conjugate transpose results in the unit operator. The lecturer then explores the idea of allowing time to evolve for a very short amount and the concept of an infinitesimal change. The Hamiltonian is introduced as a hermitian operator representing the total energy of the system and is shown to be related to the observables that can be measured in a laboratory.

The focus of this paragraph is on the Hamiltonian and its role in quantum mechanics. It discusses the Hamiltonian as an operator that corresponds to energy and how it is derived from the condition that the expectation value of the Hamiltonian does not change with time. The conservation of energy is highlighted as a fundamental principle, and the Hamiltonian's role in describing the energy of a system is emphasized.

This section introduces the Schrodinger equation, which describes how the state vector of a quantum system changes with time. The lecturer explains the process of solving the Schrodinger equation for eigenvectors of the Hamiltonian and discusses the time dependence of these eigenvectors. The concept of energy eigenstates and their evolution with time is also covered, providing insight into the general solution of the Schrodinger equation.

The paragraph discusses the time evolution of quantum states, particularly focusing on eigenvectors of the Hamiltonian. It explains that each eigenvector evolves with its own frequency, leading to a general solution for the time evolution of any quantum state. The lecturer also touches on the Fourier series and the concept of superposition of energy eigenstates, illustrating how the Hamiltonian allows for the expansion of any state of a system in terms of its eigenvectors.

This section explores the time derivative of expectation values of observables in quantum mechanics. The lecturer derives the equation for the time derivative of the expectation value of an observable and relates it to the commutator of the Hamiltonian with the observable. The importance of the Hamiltonian in determining the rate of change of observables is emphasized, and the concept of conserved quantities is introduced.

The paragraph presents a detailed example of a quantum mechanical problem involving a single spin in a magnetic field. It discusses the Hamiltonian for a spin in a magnetic field and calculates the time dependence of the averages of the three components of the spin. The lecturer shows that the spin component along the magnetic field direction remains constant, while the components in the plane perpendicular to the field undergo precession, illustrating the motion of the spin vector.

This section deals with the problem of two electrons providing a magnetic field for each other. The Hamiltonian for this scenario is introduced, and the energy eigenvalues and eigenvectors are discussed. The concept of degenerate energy levels is explained, and the states are categorized into singlet and triplet states. The time evolution of the system is explored, showing how the probabilities of the up-down and down-up states change over time.

The final paragraph focuses on the oscillation between up-down and down-up states and how it is related to the difference in energies of the singlet and triplet states. The lecturer demonstrates how to calculate the frequency of this oscillation and emphasizes that the probabilities depend only on the difference in energies, not the overall sum. The general approach to solving quantum mechanical problems is summarized as a ritual that involves expanding in terms of energy eigenvectors, evolving with their respective phases, and then recombining in the original basis.