This first lecture is a general discussion of the nature of the laws of physics and in particular classical mechanics. The notions of configuration, reversibility, determinism, and conservation law are introduced for simple systems with a finite number of states.
This lecture introduces Lagrange's formulation of classical mechanics. That formulation is formal and elegant; it is based on the Least Action Principle. The concepts introduced here are central to all modern physics. The lecture ends with angular momentum and coordinate transforms.
This lecture focuses on the relation between continuous symmetries of the Lagrangian and conserved quantities. Generalized coordinates and canonical conjugate momentum are introduced.
This lecture starts with a thorough review of symmetries and conservation laws. Energy conservation is shown to be a consequence of time translation symmetry and the Hamiltonian is introduced.
In this lecture many mechanical practical examples are worked out. Hamilton's equations are introduced: they represent one more way to do classical mechanics. The harmonic oscillator is revisited using Hamilton's equations.
This lecture analyses the flow in phase space of multiple systems, and that flow is shown to be incompressible. Poisson brackets are introduced as yet another way to express classical mechanics formally.
Poisson brackets are another formal formulation of classical mechanics. They help make the connection between symmetries and conservation laws more explicit. The Poisson bracket of the x,y,z components of angular momentum are derived.
This lecture introduces the static electric and magnetic fields, the associated Lagrangian and the Lorentz force. The vector potential, it's gauge field and gauge invariance are also introduced.
This final lecture is a general review of all the concepts learned so far applied to a particle in electric and magnetic static fields.
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.
In the first lecture of the course Professor Susskind introduces the original principle of relativity - also known as Galilean Invariance - and discusses inertial reference frames and simultaneity. He then derives the Lorentz transformation of special relativity following the method in Einstein's original paper [check this], and introduces length contraction and time dilation, invariants, and space- and time-like intervals.
Professor Susskind starts with a brief review of the Lorentz transformation, and moves on to derive the relativistic velocity addition formula. He then discusses invariant intervals, proper-time and distance, and light cones.
Professor Susskind begins with a review of space- and time-like intervals, and explains how these intervals relate to causality and action at a distance. He then introduces space-time four-vectors and four-velocity in particular. After presenting these concepts, Professor Susskind introduces relativistic particle mechanics. He presents the action principle for a particle in free space, and derives the Lagrangian for such a particle. Building on these concepts, Professor Susskind derives the relativistic formulas for momentum and energy, and discusses relativistic mass, and how the conservation of momentum and energy are modified by relativity. He then shows the origin of Einstein's famous equation E = mc2. The lecture concludes with a discussion of massless particles under relativity.
Professor Susskind moves on from relativity to introduce classical field theory. The most commonly studied classical field is the electromagnetic field; however, we will start with a less complex field - one in which the field values only depends on time - not on any spatial dimensions. Professor Susskind reviews the action principle and the Lagrangian formulation of classical mechanics, and describes how they apply to fields. He then shows how the generalized classical Lagrangian results in a wave equation much like a multi-dimensional harmonic oscillator. Next, professor Susskind brings in relativity and demonstrates how to create a Lorentz invariant action, which implies that the Lagrangian must be a scalar. The lecture concludes with a discussion of how a particle interacts with a scalar field, and how the scalar field can give rise to a mass for an otherwise massless particle. This is the Higgs mass mechanism, and the simple time dependent field we started the lecture with is the Higgs field.
Professor Susskind begins with a discussion of how, in the case of charged particle in an electromagnetic field, the particle affects the field and vice-versa. This effect arises from cross terms in the Lagrangian. He then derives the action, Lagrangian, and equations of motion for this case, and shows that the equations of motion are wave equations with a singularity at the location of the particle. Professor Susskind then introduces the contravariant and covariant four-vector notation and Einstein's summation conventions used in the study of relativity. He then proves that scalar Lagrangians are Lorentz invariant. Finally, Professor Susskind solves the wave equation for a particle in a field and demonstrates that the solutions are sums of plane waves. The Higgs boson is the case of a charged particle with zero mass, and the resulting field derived from the equations solved here is the Higgs field. The Higgs field is the origin of the electron mass.
After a review of Einstein & Minkowski notation and an introduction to tensors, Professor Susskind derives the relativistic Lorentz force law from the Lagrangian for a particle in a vector field. At the end of the lecture, he introduces the the four fundamental principles that apply to all of modern physics: stationary action, locality, Lorentz invariance, gauge invariance.
Professor Susskind elaborates on the four fundamental principles that apply to all physical laws. He then reviews the derivation of the Lorentz force law as an example of the application of these principles. The lecture closes with an introduction to gauge invariance.
After a brief review of gauge invariance, Professor Susskind describes the introductory paragraph of Einstein's 1905 paper "On the Electrodynamics of Moving Bodies," and derives the results of the paragraph in terms of the relativistic transformation of the electromagnetic field tensor. This paragraph asks the fundamental question "what is the difference between a charge moving in a magnetic field, and a fixed charge in a changing magnetic field." The answer to this fundamental question must be "nothing" if the principle of relativity is true. This conclusion is what led Einstein to develop the special theory of relativity. Professor Susskind then moves on to present Maxwell's equations. He discusses the definition of charge and current density that appear in them, and then derives the relationship between these quantities. This relationship is the continuity equation for charge and current, and represents the principle of charge conservation. The lecture concludes with the presentation the first two Maxwell equations in relativistic notation. This single equation is the Bianchi identity, and this identity makes it clear that magnetic charge sources (monopoles) and magnetic current do not exist.
Professor Susskind begins the lecture by solving Maxwell's equations for electromagnetic plane waves. He then uses the principles of action, locality and Lorentz invariance to develop the Lagrangian for electrodynamics for the special case without charges or currents. Using the Euler-Lagrange equations with this Lagrangian, he derives Maxwell's equations for this special case. Finally, Professor Susskind adds the Lagrangian term for charges and currents by using the principle of gauge invariance, and again uses the Euler-Lagrange equations to derive Maxwell's equations in relativistic notation.
Professor Susskind begins the final lecture with a review and comparison of the three different concepts of momentum: mechanical momentum from Newtonian mechanics, canonical momentum from the Lagrangian formulation of mechanics, and momentum that is conserved by symmetry under translation invariance from Noether's theorem. He then develops the connection between Lagrangian and Hamiltonian mechanics and field theory in more detail than in previous lectures. Professor Susskind moves on to develop the concepts of energy and momentum density, and then applies these concepts to electromagnetic fields. He concludes the course with an introduction to energy and momentum flux, and the stress-energy tensor.
The principle of equivalence of gravity and acceleration, or gravitational and inertial mass is the fundamental basis of general relativity. This was Einstein's key insight. Professor Susskind begins the first lecture of the course with Einstein's derivation of this equivalence principle. He then moves on to the mathematics of general relativity, including generalized coordinate transformations and tensor analysis. This topic includes the important point that the determination as to whether a spatial geometry is flat (i.e. Euclidean) is equivalent in some respects to the determination of whether an object is in a gravitational field, or merely an accelerated reference frame.
This lecture focuses on the mathematics of tensors, which represent the core concepts of general relativity. Professor Susskind opens the lecture with a brief review the geometries of flat and curved spaces. He then develops the mathematics of covariant and contravariant vectors, their coordinate transformations, and their relationship to tensors. In the second half of the lecture, Professor Susskind defines tensor operations including addition, multiplication, and contraction, and discusses the properties of the metric tensor.
In this lecture, Professor Susskind presents the mathematics required to determine whether a spatial geometry is flat or curved. The method presented is to find a diagnostic quantity which, if zero everywhere, indicates that the space is flat. This method is simpler than evaluating all possible metric tensors to determine whether the space is flat. The diagnostic that we are looking for is the curvature tensor. The curvature tensor is computed using covariant derivatives which require the computation of the Christoffel symbols. The Christoffel symbols are computed using the equation for covariant derivative of the metric tensor for Gaussian normal coordinates. We take the second covariant derivative of a vector using two different orders for the indices, and subtract these two derivatives to get the curvature tensor. If the curvature tensor is equal to zero everywhere, the space is flat. Professor Susskind demonstrates the intuitive picture of this computation using a cone, which is a flat two-dimensional space everywhere except at the tip.
Professor Susskind begins the lecture with a review of covariant and contravariant vectors and derivatives, and the method for determining whether a space is flat. He then introduces the concept of geodesics, which are the straightest paths between two points in a given space. A geodesic is a path that is locally as straight as possible, which means that the derivative of the tangent vector is equal to zero at every point. Professor Susskind then moves on to relate the mathematics of Riemannian geometry (which we have been studying so far) to spacetime. Spacetime is represented by Minkowski space, which has a different metric from that of flat Riemannian space in that the coefficient of the time dimension is negative. Minkowski space is the geometry of special relativity. The rest of the lecture presents uniformly accelerated reference frames and how they transform under special relativity. Professor Susskind shows how uniformly accelerated reference frames produce the same equations of motion as those for a uniform gravitational field, thereby beginning to establish the basis for the equivalence principle which is at the heart of general relativity.
In this lecture, Professor Susskind derives the metric for a gravitational field, and introduces the relativistic mathematics that describe a black hole. He begins by reviewing the concept of light cones and space- and time-like intervals from special relativity. He then moves on to review the flat space-time metric and geodesics, and the connection between the mathematics of geodesics and the Lagrangian formulation of classical mechanics. This leads to the mechanics of a particle moving in a gravitational field, and then to the derivation of the metric for a gravitational field, also known as the Schwarzschild metric. These are the fundamental mathematics that show the equivalence of a gravitational field and curved space-time. The metric for a gravitational field has an undefined value at a particular radius from the center of a gravitating body. Where this radius occurs outside of the body, the body is a black hole, and the radius defines the location of the event horizon. The lecture concludes with an introduction to some of the very strange properties of a black hole, including that, to an outside observer, the velocity of light slows and light rays become stuck at the horizon.
Professor Susskind continues the discussion of black hole physics. He begins by reviewing the Schwarzschild metric, and how it results in the event horizon of a black hole. Light rays can orbit a black hole. Professor Susskind derives the equations of motion for such an orbit using classical mechanics and the conservation of energy and angular momentum. This derivation yields the photon sphere at the orbital radius of a light ray around a black hole. Professor Susskind then moves on to the physics of the event horizon of a black hole. An in-falling observer experiences nothing unusual at the event horizon, but to an outside observer, it takes an infinite amount of time for the in-falling observer to reach the horizon. The physics of the horizon are analyzed using the hyperbolic coordinates of a uniformly accelerated reference frame. One inside the horizon, in-falling objects cannot avoid the singularity at the center of a black hole because the radial dimension effectively becomes a time dimension and the singularity is a point in the future of every event.
Professor Susskind continues the in-depth discussion of the physics of black holes. He begins with the Schwarzschild metric and then applies coordinate transformations to demonstrate that spacetime is nearly flat in the vicinity of the event horizon of a large black hole. In other words, nothing special happens at the event horizon for an observer falling towards the black hole. Professor Susskind then uses spacetime diagrams with hyperbolic coordinates to describe the physics of falling through the event horizon and into the black hole. The students' have many questions about the unusual properties of black holes.
Professor Susskind begins the lecture with a review of Kruskal coordinates, and how they apply to the study of black holes. He then moves on to develop a coordinate system which allows the depiction of all of spacetime on a finite blackboard. This results in a Penrose diagram for flat spacetime. The Penrose diagram for black holes leads to an understanding of wormholes, also known as Einstein-Rosen bridges. Professor Susskind then describes the process of black hole formation through the simplest possible mechanism: an infalling sphere of radiation. This process is studied by marrying a Penrose diagram for the flat spacetime inside the sphere, with a Penrose diagram for the black hole under formation outside the sphere of radiation. The boundary between the two diagrams is the radiation sphere itself, and this approach demonstrates how the black hole horizon develops and begins to expand even before the black hole itself forms.
Professor Susskind derives the Einstein field equations of general relativity. Beginning with Newtonian gravitational fields, an analogy with the four-current, and the continuity equation, he develops the stress-energy tensor (also known as the energy momentum tensor). Putting these concepts together and generalizing the Newtonian field equation leads to the definition of the Ricci tensor, the Einstein tensor, and ultimately the Einstein field equations. These equations equate curvature of spacetime as expressed by the Einstein tensor, with the energy and momentum within that spacetime as expressed by the stress–energy tensor.
Professor Susskind demonstrates how Einsteins's equations can be linearized in the approximation of a weak gravitational field. The linearized equation is a wave equation, and the solution to these equations create the theory of gravitational radiation and gravity waves. Gravity waves represent waves in the curvature of spacetime and thus are effectively tidal forces that change over time. Gravity waves propagate at the speed of light. A rotating binary pulsar is the most likely source of detectable gravity waves. Professor Susskind closes the final lecture of the course by developing the Einstein-Hilbert action for general relativity, and discussing how minimizing this action leads to the Einstein field equations.
Professor Susskind introduces the topic of modern Cosmology, which started with the discovery of cosmic microwave background radiation in 1964. However, this lecture focuses on the classical or Newtonian view of the universe. Beginning with the assumption of an isotropic universe with gravity as the only significant cosmic-scale force, Professor Susskind derives the equation for the Hubble constant, and demonstrates that the universe cannot be static - that it must be expanding or contracting.
After reviewing the basic equation for an expanding universe, Professor Susskind solves the equation explicitly for a zero energy universe, and then extends the derivation to universes with non-zero energy. These universes can take two forms: matter-dominated universes, and radiation-dominated universes. The radiation-dominated form characterizes our early universe, and the matter-dominated form characterizes our universe today - or so cosmologists believed until observations led to theories of dark energy in the 1990s.
Professor Susskind presents three possible geometries of homogeneous space: flat (infinite), spherical (positively curved and finite), hyperbolic (negatively curved and infinite). He develops the metric for these three spatial geometries in spherical coordinates, and describes methodologies for determining which geometry represents our universe. To date, we have not been able to detect curvature in our universe. Professor Susskind concludes the lecture by introducing a time-dependent scale factor into the metric for each of these geometries.
The time-time component of Einstein's field equations for general relativity relate energy density to the geometry of space. Using this energy density, Professor Susskind presents the thermodynamic equation of state which relates the energy density to pressure. He uses this equation to derive how the density of the universe relates to the scale factor for matter- and radiation-dominated universes, and gives a brief preview of how dark energy affects this analysis.
After a review of the equations of state presented in the last lecture, Professor Susskind derives the density parameter for an energy dominated universe. He then introduces the concept of vacuum energy, which is represented by the cosmological constant, and is one possible form of dark energy, and describes the characteristics of the universe under various values of the cosmological constant and curvature.
Professor Susskind develops the energy density allocation equation, and describes the historical progress of the effort to find the correct values for the terms in this equation. This analysis combined with observations of the motion of stars and galaxies led cosmologists to postulate the existence of dark matter. Observations of luminosity and red-shift have led to the correct solution for today's universe - which is dominated by dark energy.
Professor Susskind examines the temperature history of our universe. The universe switched from radiation-dominated to matter-dominated when it was about one million times hotter than it is today. At that time, matter and energy decoupled from each other and the universe became transparent to radiation when it was about one thousand times hotter that it is today.
Professor Susskind opens the lecture with one of the fundamental questions in cosmology: why are there more protons than anti-protons in the universe today? The answer lies in theory of baryogenesis in the very early universe. This theory leads to the Sakharov conditions, and the fact that the only general symmetry requirement under all conditions is complete charge-parity-time (CPT) symmetry. Professor Susskind then introduces the concept of inflation in the early universe. He derives the basic equations of motion for inflation by starting with the equations for a falling particle in a gravitational field and a viscous fluid.
Professor Susskind introduces the theory of cosmological inflation under which the early universe underwent an exponential expansion during which it doubled in size every 10-32 seconds and expanded by at least a factor of e90. This all occurred prior to the Big Bang and during this period the universe cooled very rapidly and was essentially empty. In the early phase of this rapid expansion, the universe was hot enough that magnetic monopoles were formed. However these were the only particles formed. The rest of the energy of the universe was contained in the potential energy of the inflaton field, and it wasn't until the Big Bang that this potential energy was converted into the particles that form the matter that makes up the galaxies in the universe today. This theory of rapid explnsion explains the lack of observed magnetic monopoles and the uniformity of the cosmic microwave background radiation. The rapid expansion dispersed the monopoles, smoothed out the distribution of photons, and also flattened space itself.
Professor Susskind describes the theory whereby inhomogeneities in the universe were caused by quantum fluctuations in the early universe. He begins with the mathematical equations of a damped harmonic oscillator, and relates that simple system to a dissipative inflaton field with spatial dependencies. The quantum fluctuations in the zero point energy states of this field lead to inhomogeneities in the universe. These inhomogeneities explain both the small variations in the cosmic microwave background, as well as the formation of galaxies from regions of high energy density.
Professor Susskind introduces statistical mechanics as one of the most universal subjects in modern physics in terms of it's ability to explain and predict natural phenomena. He begins with a brief introduction to probability theory and then moves on to draw the connection between the concept of laws of motion as rules for updating states of a system, and the probability of being in a given state. Proper laws of physics are reversible and therefore preserve the distinctions between states - i.e. information. In this sense, the conservation of information is more fundamental that other physical quantities such as temperature or energy. Professor Susskind then moves on to continuous systems and phase space, and Liouville's theorem. The lecture concludes with the presentation of formulas for computing entropy, and some examples.
Professor Susskind presents the physics of temperature. Temperature is not a fundamental quantity, but is derived as the amount of energy required to add an incremental amount of entropy to a system. As the energy of a system increases, the number of possible states of a system increases, which means that the entropy increases. This is the concept behind the second law of thermodynamics, and implies that temperature is always positive. Note: this lecture is only 1 hour long.
After reviewing the laws of thermodynamics, Professor Susskind begins the derivation of how energy states are distributed for a complex system system with many energy states. As the number of particles in a system grows, the distribution of states clumps more tightly around a mean. This is the Maxwell-Boltzmann distribution. The derivation requires Sterling's approximation and Lagrange multipliers.
Professor Susskind completes the derivation of the Boltzman distribution of states of a system. This distribution describes a system in equilibrium and with maximum entropy. He derives the formulas for energy, entropy, temperature, and the partition function for this distribution. He then applies these general formulas to the example of an ideal gas.
Professor Susskind derives the formula for the pressure of an ideal gas. He begins by introducing the Helmholtz free energy, and the concept of adiabatic processes. These concepts lead to the definition of pressure as the change of energy with volume at a fixed entropy, and then to the famous equation of state for an ideal gas: pV = NkT. Professor Susskind then moves on to define the concept of fluctuations., and in the process demonstrates once again that the power in statistical mechanics lies in the partition function and is derivatives. The fluctuation of energy in a system leads to the definition of the heat capacity of the system and the specific heat of a given material.
Professor Susskind derives the equations for the energy and pressure of a gas of weakly interacting particles, and develops the concepts of heat and work which lead to the first law of thermodynamics.
Professor Susskind begins the lecture with 2 examples: (1) deriving the speed of sound in an ideal gas; and (2) a single harmonic oscillator in a heat bath. The harmonic oscillator example leads to a discrepancy with empirical observation that can only be resolved through quantum mechanics. At low temperatures relative to the first excited state of the oscillator, quantum mechanics suppresses the energy of the harmonic oscillator. Through this mechanism, certain modes of oscillation are "frozen out" until the system reaches higher temperatures. Einstein proposed this quantized effect in 1907, which is one of the theories that led to the development of quantum mechanics. Professor Susskind then discusses the apparent contradiction between the second law of thermodynamics, and the reversibility of classical mechanics. If entropy always increases, reversibility is violated. The resolution of this conflict lies in the (lack of) precision of our observations. Undetectable differences in initial conditions lead to large changes in results. This is the foundation of chaos theory.
Professor Susskind develops the equation for the probability that all molecules of a gas will converge in one half of a room, and concludes that this event is possible, but that the time scale for it to occur is incredibly long. This line of reasoning leads to the resolution of the paradox between the reversibility of classical mechanics and the apparent lack of time reversibility of the second law of thermodynamics by demonstrating that statistical mechanics processes are in fact time reversible if the system is known precisely enough and the observer waits long enough. He then moves on to magnetism and begins to introduce the concepts of ferromagnetic phase transitions and spontaneous symmetry breaking. Spontaneous symmetry breaking occurs when magnets in a lattice begin to cool. With no external magnetic field, they may end up in one of two symmetrical states - e.g. all up or all down. But a very small magnetic field affecting just one of the magnets will break this symmetry and bias the system toward one of the ground states.
After reviewing the discussion of a single magnetic particle (or spin) in a heat bath, Professor Susskind continues with the development of the one-dimensional Ising model. This model does not exhibit phase transitions. He then moves on to the multi-dimensional Ising model. For any number of dimensions above one, the Ising model exhibits phase transitions at specific temperatures. This is due to the fact that, in multiple dimensions, each point in the lattice is affected by more than two neighboring lattice elements.
Professor Susskind continues the discussion of phase transitions beginning with a review of the Ising model and the mean field approximation, and then presents the temperature and magnetic field parameters of the phase transition of a magnetic material. He then moves to the physics of the liquid-gas phase transition and develops the mathematical analogy between this case that that for a magnetic material. Professor Susskind finishes the Theoretical Minimum series of courses by taking questions from the class, which leads to a discussion of the anthropic principle and fine tuning.
Leonard Susskind gives the first lecture of a three-quarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he explores light, particles and quantum field theory.
In this lecture Professor Susskind explores quantum field theory.
In this lecture Professor Susskind talks about what a quantum field is and how it is related to particles.
In this lecture Professor Susskind continues on the subject of quantum field theory.
In this lecture Professor Susskind continues on the subject of quantum field theory, more specifically, energy conservation, waves and fermions.
In this lecture Professor Susskind continues on the subject of quantum field theory, including, the Dirac equation and Higgs particles.
Leonard Susskind discusses the theory and mathematics of angular momentum.
Leonard Susskind discusses the theory and mathematics of particle spin and half spin, the Dirac equation, and isotopic spin.
Leonard Susskind discusses the equations of motion of fields containing particles and quantum field theory, and shows how basic processes are coded by a Lagrangian.
In this lecture, Professor Susskind elaborates further on using field Lagrangians, the action principle, and path integrals in studying particle physics.
In the first lecture of the series Professor Susskind introduces theoretical concepts underlying the standard model. He also gives a zoological overview of the observed particles.
Isospin is introduced by analogy to spin. Color is introduced as a solution for the existence of delta particles (uuu), which would otherwise have all quarks in the same state - illegal for fermions. Gluons properties are describe by analogy to quark antiquark pairs.
This is the first of two lectures focusing on group theory. Basic concepts of group theory are introduced and connected to particle properties like spin and color.
Professor Susskind continues developing group theory, making connections from group generators and subgroups to particles. Gluons properties are explored with this framework, including confinement of quarks.
Professor Susskind describes the concept of a gauge field and it's associated symmetries. Symmetries lead to conserved charges like the electric charge and color. These concepts are used to describe the weak interaction.
In this lecture Professor Susskind continues with the weak interaction, answering the question "Why is the weak force weak?" He connects this to the form of the propagator in Feynman diagrams. The lecture ends with the introduction of explicit and spontaneous symmetry breaking.
In this lecture Professor Susskind shows how spontaneous symmetry breaking in a field theory can lead to the creation of Goldstone bosons like the photon and gluons.
The Goldstone boson gets eaten, giving mass to the gauge boson via the Higgs field.
Professor Susskind continues his description of the Higgs mechanism focusing on giving mass to fermions.
The final lecture returns to renormalization, the mass of fermions, the W boson and the expected mass scale of the Higgs. Professor Susskind finishes with the running of the coupling constants and the interesting fact that they appear to unify at an energy scale of 10^16 GeV.
In the first lecture of the series Professor Susskind introduces the concept of renormalization, which allows elimination of as yet unknown physics at very tiny scales or high energies from our calculations of physics at accessible scales. He also connects dimensional analysis to the set of possible Lagrangians for our theories.
Professor Susskind starts with the topic of rotations, showing that particles under rotation by 2π either return to their initial state (wavefunction) or to their initial wavefunction multiplied by a phase of -1. The first case corresponds to bosons and the second to fermions. He then uses this difference to show that bosons contribute positively to vacuum energy and fermions contribute negatively.
This lecture reviews the propagator and connects its form to dimensional analysis. Then loop propagators are used to introduce mass renormalization.
Professor Susskind reviews the mathematical concepts of symmetry in preparation for the development of supersymmetry. Then he introduces the mathematical concept of Grassmann numbers, which are used in the description of fermionic fields.
The lecture finishes the development of Grassmann numbers with the introduction of integration and differentiation. With these tools in place Prof Susskind describes a simplified supersymmetric model.
Professor Susskind introduces superfields and integration with Grassmann variables.
This lecture develops the notion of a Supercharge, Q+ representing the transformation from a fermion to a boson and Q the transformation from a boson to a fermion.
In this lecture, Professor Susskind generalizes supersymmetry to 3 space and 1 time dimension.
In the first half of the lecture, Professor Susskind makes an analogy between breaking supersymmetry and breaking the symmetry of a ferromagnet. In the second half of the lecture, Professor Susskind introduces GUTs as corresponding to the group SU(5) which has the subgroup SU(3)xSU(2)xSU(1) corresponding to the Standard Model.
The final lecture focuses on grand unified theories, and how their group structure connects to fermions (neutrinos, leptons and quarks) and the gauge bosons.
In the first lecture of the series Professor Susskind explains the historical origins of string theory. Hadrons are observed to come in angular momentum sequences where a plot of angular momentum against mass squared is a straight line. These plots are known as Regge plots. A mechanical model that matches the observed straight line is a stretchable string connecting two masses. The lecture ends with a model of the relativistic motion of such a string.
Professor Susskind establishes the mathematical foundation for solving the equations of motion for strings. Breaking a string into discrete mass points, he substitutes a Fourier series for the position, generating a set of independent solutions. Dirichlet and Neumann boundary conditions are explained and connected to open and closed strings.
This lecture develops an algebraic approach to the energy spectrum of strings. Raising and lowering operators are associated with the modes of the strings. The lecture finishes with the basics of string interactions.
This lecture starts with a review of Noether’s theorem, which links continuous symmetries with conserved quantities (charges). The model of a closed string is developed following the earlier procedure for open strings. The energy spectrum is similar but with a new constraint called the level matching rule linking the energy of waves in each direction around the string that is connected to Noether’s theorem.
Professor Susskind examines the ground state of bosonic strings more closely, returning to the tachyon problem from lecture 3. The resolution of this problem ends up requiring the introduction of additional dimensions in which the string can vibrate or stretch. Curiously, the number of extra dimensions needed is 22.
Spin is introduced in the mass points that strings are built from, creating strings that behave like bosons and fermions. Then Professor Susskind examines the scattering characteristics of strings and shows that the Veneziano amplitude from meson scattering has string like characteristics.
Fermionic strings with Fermionic mass points are introduced. Now there are both spacelike oscillations as well as spin oscillations. The notion of a world sheet, traced out by a string over time, is introduced. Path integrals are defined with respect to the world sheet and lead to the Laplace equation.
Professor Susskind develops the concept of conformal mapping, which can be used to dramatically simplify the solution of the 2 dimensional Laplace equation encountered in Lecture 7. In this case the 2 dimensions correspond to the world sheet traced out by strings when scattering.
This lecture explores the consequences of string excitations in compact dimensions. Compact dimensions lead to quantized momentum and also to quantized energy levels due to stretching around the compact dimensions. The duality of the momentum and winding states is explored.
Professor Susskind explains the behavior of open and closed strings in periodic dimensions. The endpoints of open strings are seen to be stuck on surfaces known as D-branes. The behavior of these open strings can model more traditional field theories with the end points taking the role of particles with charges (electric or color).
Professor Susskind starts with the philosophy of reductionism, where complex objects are broken down in to larger numbers of smaller objects with more fundamental rules of behavior. While this philosophy has served physics well, Professor Susskind argues that modern theories spell the end of reductionism. What is fundamental depends on the situation or energy scale, or whatever is most “useful” for predicting behavior. This change extends to string theory, which contains dualities between strings and D1-branes. (Note: this final lecture of String Theory was delivered in Winter, 2011 as the first lecture of Cosmology and Black Holes. The lecture only appears here in this course.)
Professor Susskind describes the special theory of relativity and focuses on showing how it connects to string theory. He considers concepts such as space-time and some of Einstein's original concepts.
Professor Susskind present the mathematics of a black hole.
Professor Susskind describes the geometry of a black hole near the event horizon. He describes how standard concepts from quantum physics can explain the physics that occur at this point.
Professor Susskind again discusses black holes and how light behaves around a black hole. He uses his own theories to mathematically explain the behavior of a black hole and the area around it.
Professor Susskind describes how string theory gives a resolution to the question regarding the entropy of a black hole.
Professor Susskind continues the presentation of the theory behind calculating the entropy of a black hole.
Professor Susskind discusses the concept of cosmic horizons as well as the relationship between ultraviolet and infrared light.
Professor Susskind completes the course with a broad discussion of black holes and horizons.
The course begins with a brief review of quantum mechanics and the material presented in the core Theoretical Minimum course on the subject. The concepts covered include vector spaces and states of a system, operators and observables, eigenfunctions and eigenvalues, position and momentum operators, time evolution of a quantum system, unitary operators, the Hamiltonian, and the time-dependent and independent Schrodinger equations. After the review, Professor Susskind introduces the concept of symmetry. Symmetry transformation operators commute with the Hamiltonian. Continuous symmetry transformations are composed from the identity operator and a generator function. These generator functions are Hermitian operators that represent conserved quantities. The lecture closes with the example of translational symmetry. The generator function for translational symmetry is the momentum operator divided by ħ. Topics: Vector space Observables Hermitian operators Eigenvectors and eigenvalues Position and momentum operators Time evolution Unitarity and unitary operators The Hamiltonian Time-dependent and independent Schrödinger equations Symmetry Conserved quantities Generator functions
Professor Susskind presents an example of rotational symmetry and derives the angular momentum operator as the generator of this symmetry. He then presents the concept of degenerate states, and shows that any two symmetries that do not commute imply degeneracy. Symmetries that do not commute can form a symmetry group, and the generators of these symmetries form a Lie algebra. The angular momentum generators in three dimensions are an example of a symmetry group. Professor Susskind then derives the raising and lowering operators from the angular momentum generators, and shows how they are used to raise or lower the magnetic quantum number of a system between degenerate energy states. Due to reflection symmetry, these states must have whole- or half-integer values for the magnetic quantum number. Topics: Rotational symmetry Angular momentum Commutator Degenercy Symmetry generators Symmetry groups Lie algebra Raising and lowering operators
Professor Susskind uses the quantum mechanics of angular momentum derived in the last lecture to develop the Hamiltonian for the central force coulomb potential which describes an atom. The solution of the Schrödinger equation for this system leads to the energy levels for atomic orbits. He then derives the equations for a quantum harmonic oscillator, and demonstrates that the ground state of a harmonic oscillator cannot be at zero energy due to the Heisenberg uncertainty principle. Topics: Angular momentum multiplets Coulomb potential Central force problem Atomic orbit Harmonic oscillator Heisenberg uncertainty principle
Professor Susskind builds on the discussion of quantum harmonic oscillators from the last lecture to derive the higher order energy states and wave functions. He then moves on to discuss spin states of particles, and introduces the Pauli matrices, which account for the interaction of a particle's spin with an external magnetic field. By examining the energy levels of electrons in an atom, Pauli and others realized that only two electrons can be in any given state. This led both to the the exclusion principle, as well as the need for another state variable - spin - which allows two electrons in each energy level. Topics: Spin Pauli matrices Pauli exclusion principle
Professor Susskind presents the quantum mechanics of multi-particle systems, and demonstrates that fermions and bosons are distinguished by the two possible solutions to the wave function equation when two particles are swapped. When two particles are swapped, the boson wave function equation has a phase factor of +1 whereas the fermion equations has a phase factor of -1. For fermions, this results in a wave function with zero probability for two particles to be in the same state, thus demonstrating the exclusion principle. On the other hand, bosons prefer to be in the same state. This is what makes a photon (boson) laser possible, but an electron (fermion) laser impossible. The spin variable is required to allow two electrons to occupy the same state in an atom. Electrons are fermions which have half-integer spins. This implies that a rotation of the angular momentum by 2π will result in a phase change by -1. This implies that the identity operation for fermions is not a rotation by 2π, but rather a rotation by 4π, and that a rotation by 2π can be offset or canceled by a swap of two particles. This is the tale of 2 minus signs. Topics: Bosons Fermions Spin statistics Permutation groups Solitons
Professor Susskind introduces quantum field theory. Excepting gravity, quantum field theory is our most complete description of the universe. Each quantum field corresponds to a specific particle type, and is represented by a state vector consisting of the number of particles in each possible energy state. These numbers are called occupation numbers. This representation uses the same quantum mathematics as a state vector for multiple harmonic oscillators with the basis vectors being the energy state of each oscillator. Therefore the quantum field mathematics follow those introduced for harmonic oscillators in previous lectures. However, in the case of quantum field theory, the raising and lowering operators become operators which create and destroy particles in a given energy state. Topics: Quantum field theory Occupation numbers Creation and anhilation operators
Professor Susskind continues with the presentation of quantum field theory. He reviews the derivation of the creation and annihilation operators, and then develops the formulas for the energy of a multi-particle system. This derivation demonstrates the correspondence between classical and quantum field theory for many particle systems. Topics: Classical field theory
Professor Susskind answers a question about neutrino mixing and relates the oscillating quantum states of a neutrino to a precessing electron spin in a magnetic field. He then discusses a recent article about whether an electron is a sphere. After these topics, Professor Susskind continues the previous discussion about second quantization, and demonstrates that the position and momentum creation and annihilation operators are Fourier conjugates of each other. Topics: Neutrino mixing Second quantization Fourier conjugates
Professor Susskind presents the Hamiltonian for a quantum field, and demonstrates how these Hamiltonians describe particle interactions such as decay and scattering. He then introduces the field theory for fermions by deriving the Dirac equation. The theory behind the Dirac equations was the first theory to account fully for special relativity in the context of quantum mechanics. This relativistic Schrödinger equation implies the existence of antimatter. Topics: Hamiltonian Dirac equation Klein-Gordon equation Antimatter
Professor Susskind closes the course with the presentation of the quantum field theory for spin-1/2 fermions. This theory is based on the Dirac equation, which, when Dirac developed it in 1928, was the first thory to account fully for special relativity in the context of quantum mechanics. This theory explains spin as a consequaence of of the union of quantum mechanics and relativity, and also led to the theory of antimatter and ultimate discovery of the first antimatter particle - the positron. Professor Susskind begins the presentation by reviewing the Dirac equation for an electron in one dimension, and then generalizes this to derive the therory for three dimensions. This led Dirac to develop his 4x4 gamma matrices. In Dirac's theory, the mass of fermions originates from the cross term between the two chiralities in the Dirac equation. Topics: Fermion Dirac equation Pauli matrices Chirality Dirac sea Zitterbewegung Positron
Professor Susskind presents an explanation of what the Higgs mechanism is, and what it means to "give mass to particles." He also explains what's at stake for the future of physics and cosmology.