Professor Susskind opens the course by describing the non-intuitive nature of quantum mechanics.  With the discovery of quantum mechanics, the fundamental laws of physics moved into a realm that defies human intuition or visualization.  Quantum mechanics can only be understood deeply by studying the abstract mathematics that describe it.

Professor Susskind then moves on to describe how the space of states for quantum mechanics, and the rules for updating those states, are fundamentally different from those of classical mechanics.  For quantum mechanics, the space of states is a vector space versus a set of states for classical mechanics.  He then then describes the basic mathematics of vector spaces.

Professor Susskind introduces the simplest possible quantum mechanical system: a single particle with spin.  He presents the fundamental logic of quantum mechanics in terms of preparing and measuring the direction of the spin.  This fundamental logic differs from classical systems in that it is entirely about probabilities, and therefore is very different from classical boolean logic.

Professor Susskind then reviews the concept of vector spaces and describes the vector space for a single spin system.  He concludes the lecture by relating the concept of orthogonality in vector spaces to overlaps in configuration or phase space.  More precisely orthogonal vector space states correspond to a lack of overlap in configuration space.

Professor Susskind elaborates on the abstract mathematics of vector spaces by introducing the concepts of basis vectors, linear combinations of vector states, and matrix algebra as it applies to vector spaces.  He then introduces linear operators and bra-ket notation, and presents Hermitian operators as a special class of operators that represent observables.  Eigenvectors of Hermitian operators represent orthogonal vector states, and their eigenvalues are the values of the observable.

Professor Susskind then applies these concepts to the single spin system that we studied in the last lecture, and introduces the Pauli matrices as the Hermitian operators representing the three spin axis directions.

Professor Susskind opens the lecture by presenting the four fundamental principles of quantum mechanics that he touched on briefly in the last lecture.  He then discusses the evolution in time of a quantum system, and describes how the classical concept of reversibility relates to the quantum mechanical principle of conservation of information, which is actually the conservation of distinctions or distinguishability of states.  The evolution in time of a quantum system is represented by unitary operators which preserve distinctions and overlap.

Professor Susskind then derives the time-dependent Schrödinger equation, and describes how to calculate the expected value of an observable, and how it changes with time.  This discussion introduces the commutator operator.  Professor Susskind closes the lecture by showing the connection between the quantum mechanical commutator and the Poisson bracket formulation of classical physics, thus showing how the time evolution of the expected value of an observable is closely related to classical equations of motion.

Professor Susskind begins the lecture by introducing the Heisenberg uncertainty principle and explains how it relates to commutators. He proves that two simultaneously measurable operators must commute. If they don't then the observables corresponding to the two operators cannot be measured simultaneously.

He then reviews the time evolution of a system and the Schrödinger equation. Unitary operators represent the time evolution of a system, and the quantum mechanical Hamiltonian generates the time evolution. Professor Susskind reviews the derivation of the time-dependent Schrodinger equation, the computation of expectation values of observables, and the parallels between the quantum mechanical commutator and the classical Poisson bracket.

Professor Susskind then demonstrates how to solve the Schrödinger equation for a general quantum mechanical system. This solution is the origin of the connection between the energy of a system and oscillations of the wave function. This is the Heisenberg matrix formulation of quantum mechanics.

The lecture concludes by solving a practical example of a single spin in a constant magnetic field.

Professor Susskind begins the lecture with a review of the problem of a single spin in a magnetic field. He re-emphasizes that observables corresponding to the Pauli sigma matrices do not commute, which implies that they obey the uncertainty relationship, and reviews the principles by which the spin in a magnetic field will radiate a photon and transition to the lowest possible energy state.

Professor Susskind then moves on to discuss the effect of measurement on a quantum system and the concept of wave function collapse. In general, the measuring apparatus becomes part of the quantum system and the space of states for the combines system is the tensor product of the states of the individual system components. This is the concept of entanglement.

Professor Susskind demonstrates the simplest example of entanglement of a two spin system. He distinguishes the unentangled product states from the more general entangled states, and gives examples or operators and expectation values for each. The singlet and triplet states are introduced.

Professor Susskind concludes the lecture by summarizing the essence of entanglement in the principle that, although a single spin quantum mechanical system can be simulated with a classical computer, a two spin system cannot be simulated by two classical computers unless they are connected together.

This lecture takes a deeper look at entanglement.  Professor Susskind begins by discussing the wave function, which is the inner product of the system's state vector with the set of basis vectors, and how it contains probability amplitudes for the various states.  He relates these probability amplitudes to the expectation values of observables discussed in previous lectures.

He then examines more deeply the difference between product and entangled states.  For product states, the wave function factorises which allows the two (or more) sub-systems to be treated as independent systems.  He also describes the properties of a maximally entangled two spin system, and introduces the concept of density matrices, which express everything we can know about one part of an entangled system.

Professor Susskind then moves on to discuss measurement versus entanglement.  There are two views of measurement: one in which the measuring apparatus becomes entangled with the system under measurement, and the other in which the wave function of the system under measurement collapses when measured.

He then discusses locality beginning with Einstein's famously skeptical phrase "spooky actions at a distance."  He distinguishes between actual instantaneous action at a distance - which is impossible - and simple correlation.  What is strange about quantum mechanics is not correlation in entangled states, but rather that we can know everything about this system as a whole, without knowing anything about the individual states of the entangled elements.  

Professor Susskind concludes the lecture by revisiting the example of the computer simulation from the last lecture, which is an example of Bell's theorem that local hidden variables are not sufficient to explain quantum mechanics.

Professor Susskind opens the lecture by examining entanglement and density matrices in more detail.  He shows that no action on one part of an entangled system can affect the statistics of the other part.  This is the principle of locality and is directly connected to the requirement that systems evolve over time only through unitary operators.  Violating locality implies non-local hidden variables which are equivalent to wires that transmit information instantaneously.  These would allow true "spooky action at a distance," but they don't exist.

Professor Susskind then discusses the simplest possible continuous system of a particle moving in one dimension.  He presents the wave function for such a system, and discusses its Hermitian operators and observables including the operators corresponding to position, momentum, and energy.  The energy operator is the Hamiltonian, and generates the time evolution of a system.  Finally, he presents the difference between the Hamiltonian for a relativistic particle moving with a constant velocity in any reference frame (e.g. a photon or neutrino), and a non-relativistic particle (i.e. one with mass).

Professor Susskind opens the lecture with a review of the entangled singlet and triplet states and how they decay.  He then shows how Fourier analysis can be used to decompose a typical quantum mechanical wave function.

He then continues the discussion of a continuous system - a single particle moving in one dimension - and shows that the solutions to the eigenvector equations for position and momentum lead to the uncertainty principle.  In other words, the wave function solution for a specific value of momentum has probabilities for the position everywhere (in the single dimension).  This derivation shows that the position and momentum wave functions are Fourier transforms of each other.  Thus mathematically the uncertainty principle is simply a a statement about Fourier transforms.

Professor Susskind begins the final lecture of the course by deriving the uncertainty principle from the triangle inequity.  He then shows the correspondence between the motion of wave packets and the classical equations of motion.  The expectation value of position for the center of a wave packet follows the classical equations.  Heavy particles have wave packets which do not spread out over time.