Quantum mechanics is everywhere in science fiction but in some ways a passing familiarity with the weird phenomena can even detract from real understanding. You have to buckle down, do the hard work and study the subject thoroughly – but even this may not be enough.
You could take an undergraduate-level course in quantum mechanics (QM), be able to do the questions and even pass the exam but still have very little idea of what quantum mechanics actually means as a coherent mathematical theory. Everyone agrees that QM is hard but the hardness is not really in the difficulty of the mathematics involved; it’s rather in trying to understand how all the unfamiliar QM concepts fit together and how they connect to what we experience in the familiar classical world.
You can’t learn QM from scratch with Gary Bowman’s book. But if you are midway through such a course, or are revising for the exam, or just trying to make sense out of it afterwards, Bowman has written this book for you. Chapter by chapter he seeks to pull out the central concepts, explain clearly what they mean and show how they fit together. Let’s see how the journey looks.
In chapter 1, ‘Three Worlds’, Gary lays out the stall. World 1 is classical Newtonian mechanics. The world is made up of particles with properties. The properties give rise to the forces which cause the particles to move on trajectories according to dynamical laws. The physicist’s job is to understand the laws and use them to make predictions. World 2 adds the concept of fields as the mechanism of force transmission: we’ve reached classical field physics with Maxwell and Einstein.
World 3 is much more weird. There is now no more talk of particle trajectories, and even the concept of particle itself seems neither clear nor central. Particle properties and configurations now generate a probability function which permits us to calculate the chances of finding a particular particle in a particular dynamical state (position, momentum, energy etc).
Welcome to today’s dominant physics paradigm, that of quantum mechanics.
As this book is about the conceptual foundations of QM, in chapter 2 the author gets straight down to defining the three quantum postulates:
Postulate 1: The Quantum State
All the information about a system (particle/set of particles in a configuration) is contained in the associated quantum state Ψ. The evolution of the quantum state is determined by the time-dependent Schrödinger equation (where H is the Hamiltonian = energy operator):
iℏdΨ/dt = HΨ.
Postulate 2: Observables, Operators and Eigenstates
The quantum state Ψ is not directly observable. Properties of the quantum system which are observable are represented by Hermitian operators. If A is a Hermitian operator and a particular quantum state Ψj satisfies:
AΨj = ajΨj
then Ψj is called an eigenstate of A and the real eigenvalue constant aj is what will be measured if the system is in the eigenstate Ψj.
Postulate 3: Quantum Superpositions
If a measurement of the observable corresponding to the operator A is made on the normalised quantum state ψ given by the superposition:
ψ = Σ cnψn
where the ψns are eigenstates of A and the cns are expansion coefficients, then aj, the eigenvalue of ψj will be obtained with probability |cj|2. The system will be left in the state ψj immediately after the measurement.
So if you know your quantum mechanics you will be nodding your head in agreement, and if you don’t you will be confirmed in your (correct) belief that quantum mechanics is hard.
In the next few chapters Bowman explore the concepts introduced above. He starts with the quantum state, reviewing distributions, probabilities, uncertainties, expected values and the connection between the quantum and classical domains including the historical ‘hidden variables’ dispute. He then moves on to review vector spaces before introducing function spaces, the Dirac bra-ket notation and the inner product <α|β>. This is a very thorough and clear presentation of the meaning and usage of Dirac notation and the concept of different representations of the same quantum state, illustrated with the infinite potential well.
In chapter 5, Bowman moves on to operators and, unusually, starts with the commutator considered as an operator. We then embark on a detailed exploration of Hermitian operators and their connection with wavefunctions, and meet the projection operator |ξ><ξ| and unitary operators (there’s an expanded treatment in a useful appendix).
In a first course on QM you are frequently told about the early disputes between Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics. As you will travel the wave mechanics route in your course, you will wonder what on earth matrix mechanics was all about. In chapter 6 the author makes good this omission and you will see just how useful matrix mechanics is, especially for changing representations (basis). When you studied spin or polarization, you will now realize that it was Heisenberg’s approach you were using.
Chapter 7 is a reality check on the meaning of the uncertainty relationships (chapter 11 extends this discussion to say some very interesting things about the role of time in quantum mechanics). Chapter 8 is a thorough and succinct overview of orbital and spin angular momentum (L2 and Lz mostly).
In chapter 9 we look in detail at the time-independent Schrödinger equation, that workhorse of bound states, and look in detail at how boundary conditions lead to energy quantisation. We then extend the discussion to barrier penetration and tunnelling. As always, Bowman is keen to tease out the implications of the mathematics and to correct oversimplifications and common misunderstandings.
Chapter 10 asks the deceptively simple question; ‘why is the quantum state complex?’ The answer is phase and the opportunities it provides for interference to occur before probabilities are computed. Here is the route to the essential weirdness of quantum mechanics.
In chapter 11 we discuss time evolution of quantum states and the time-dependent Schrödinger equation of Postulate 1. Time-varying states come from superpositions with varying phase time-evolution and the author extends the usual treatment to include the unitary time-evolution operator Ut = e-iHt/ℏ which is derived in the appendix.
The final chapter returns to the position representation of wavefunctions looking at wavepackets. It finishes with a reprise of the relationship between bra-ket notation and wavefunctions, drawing out some sophisticated issues. The appendices review the mathematics of calculus, complex numbers, the simple harmonic oscillator and unitary transforms.
In summary, this gem of a book should be read by everyone as they finish their first course on quantum mechanics. It will anchor revision and shed new light everywhere on what has been learned.
Note: this is not a book about the interpretation of quantum mechanics in the sense of how we might understand the meaning of the quantum state |ψ> in physical reality. Bowman takes a robustly practical view: quantum mechanics tells you what to expect if you do many identical experiments on the same quantum state, by giving you the probability distribution of the observables you’ll measure. He calls this the statistical interpretation and it’s how practising scientists and engineers actually use QM.