# Quantum Mechanics

*The theoretical minimum* is an online lecture series given by Leonard Susskind, professor of theoretical physics at Stanford University. The first volume in the book series with the same name treated *Classical Mechanics*, a course given in the fall 2011, and the current volume on Quantum Mechanics resulted from the winter 2012 lectures. The course consists of a sequence of 10 lessons, which correspond to the 10 chapters of the book. The co-author Art Friedman is computer scientist who attended the lessons. The books can be considered as the elaborated lecture notes for Susskind's courses.

The name of the series and subtitle *what you need to know to start doing physics* explains the concept. This is intended for the amateur scientist who really wants to learn and understand the subject. It is Susskind's conviction that this can only be achieved if one masters the mathematics that describe the physics. In this second volume it is assumed that the reader is familiar with elements about classical mechanics explained in volume 1 of the series. Furthermore, the reader should not be afraid of the abstraction of mathematics. The ideal situation would be that the reader is familiar with the basic mathematics at the level of a bachelor university degree. On the other hand, introducing the necessary mathematics is exactly what this course is about, so in principle no mathematics of this level is assumed. But if mathematics scares you off, then this is not for you, which implies, if we believe Susskind, that you will never be able to properly understand quantum physics.

Susskind obviously is addressing the reader who has some (classical) physics background. He starts from the intuitive physical description, and then illustrates that this can be perfectly described by introducing the necessary mathematical concepts. For example: what is spin? Not just an ordinary 3-vector. If you measure it there are only two possible outcomes, say +1 and -1. But before it is measured, it can be in different (coherent) states. The outcome of the measurement will depend on this state and on the "orientation" of the measuring apparatus. So the reader is taken along on an exploration tour and it takes a while (in fact 3 chapters) before a proper mathematical concept sublimates. In this case, the Pauli matrices. Given an orientation for the measuring apparatus, it will be possible to combine the Pauli matrices to a matrix and its eigenvalue decomposition will allow you to compute the probabilities of either outcome of the experiment. So the mathematics do not come gratuitously, but come to the help of the physicist who just needs it at some point.

On the other hand, if you are a mathematician, then the content may somewhat disappoint you because the mathematics is relatively elementary knowledge. Since I assume that a mathematician feels comfortable with abstraction, he or she might have preferred a more axiomatic approach. For example, the above state vector is a 3D unit vector and hence has two degrees of freedom. The Pauli matrices are 2x2 unitary matrices with eigenvalues +1 and -1 and the eigenvectors are orthonormal basis vectors. Of course, the line of thought that Susskind uses, has the advantage of explaining why a certain mathematical concept is needed to catch a particular physical phenomenon. So, if the mathematical abstraction is the more challenging part for the physicist, for a mathematician it also requires an effort to detach from the oh so familiar laws of mechanics in our everyday life and to accept the sometimes paradoxical physical interpretation imposed by the mathematics of the quantum mechanical game.

To summarize, this is more about mathematics for physicists than about physics for mathematicians. Whatever the approach or background, quantum mechanics remains a difficult subject because it is often counter intuitive. The reason is that humans normally observe the world they live in at a scale that is hugely different from the scale needed to describe the physics at a quantum level. As Susskind claims at some point: conceptually quantum mechanics should be the first approach to describe mechanical phenomena because that corresponds to reality and classical mechanics is a simplification that is a good approximation only at a much larger scale.

However, as is usually the case, that top-down approach is not the best way to learn things. It is much better to start with a simple special case and when that is properly understood, one may step up to generalizations. However, supposing you can assimilate all the material as it is intended by the authors, you will not be at a level where you can directly involve in current research on quantum mechanics. You will be at an elementary level still, and it will take much more mathematics to reach the level of current research. There are many more advanced courses available on the courses website, so one may expect several more volumes in the *Theoretical Minimum* book series to come.

Let's go quickly through the contents of the book. It takes three lessons/chapters to explain the notion of a state of a system. For this, Susskind uses the easiest example of a system of just one simple observable: the spin or a qubit. While explaining this concept, Susskind introduces on the mathematical side the complex numbers, vector spaces, orthogonal basis vectors and Gram-Schmidt orthogonalization, (Hermitian) operators and their eigenvalues, and of course the bra-ket notation that was introduced by P. Dirac.

The next two chapters deal with time dependency. Here the notion of Lie bracket (the authors prefer to use commutator as an alias), Hamiltonian, Schrödinger equation, Cauchy-Schwartz, unitary operators, and the general uncertainty principle is introduced.

The two following chapters are on observation and state of a combination of systems. Again, the simplest case is observing two spins. Susskind shows that the tensor product of the bases for the states of the separate systems does not form a complete basis for the states of the combined system. Hence some extra basis vectors (singlet and triplets) are needed to describe the whole state space. These vectors are responsible for entanglement. The mathematical concepts here are the outer product, density, and correlation.

The remaining chapters are about particles and waves. This duality is probably the best known aspect of quantum mechanics: a photon behaves as a wave and at the same time it is like a particle. I believe the reader who has been hanging on till this point of the course will have to fasten seat belts and shift to a higher gear now. The first thing is to move from eigenvectors of a matrix to eigenfunctions of an operator, and from finite sums to integrals. Position, velocity, momentum, Hamiltonian all become operators and can best be studied in the Fourier domain, the latter resulting for example in the classical Heisenberg uncertainty principle. To know how particles move, one has to reconsider time dependency in this continuous setting. The equation of motion with kinetic and potential energy and the classical form of the Schrödinger equation, waves, wave packages and the harmonic oscillator are the results with which this course comes to an end. Several exercises are inserted where the reader is asked to prove some of the properties or to work out some formulas. Usually they are not very hard and they help to assimilate the material. In an appendix some of the basic formulas are summarized so that they can be easily looked up, for example when going through the formulas to solve the exercises. Also the subject index is handy when studying the material.

**Submitted by Adhemar Bultheel |

**6 / May / 2014