Preliminaries

Today my particle accelerator in Geneva is not operating, and I have decided to have the day off. I take a bus ride to a nearby village, and get off at the bus-stop in the main square. The village is very pretty, and I wander around enjoying the chalets and the views. I also venture into the surrounding hillside, with its criss-crossing paths. After a couple of hours, I begin to think it is time to return to the village, to catch the bus back to Geneva. If I know that I have walked a total of 6 miles, how far am I from the bus-stop?

The answer of course is that we cannot tell – it could be anything from zero distance to 6 miles. It all depends on how straight or crooked my walk had been up till then. This is because the problem is basically one involving vectors, in which not only a magnitude is involved, but also a direction. The distance that I am from the bus-stop is given by the length of the vector formed by adding all the vectors corresponding to the various straight bits of path along which I had walked.

Vectors are thus very useful in any problem in which directions are implied.

Summary of theory

What you need to know

Definitions and properties

Operator, linear operator, functions of operators, commuting and noncommuting operators, eigenfunction, eigenvalue, degeneracy, normalised function, orthogonal functions, Hermitian operator.

Postulates of quantum mechanics

(1) The state of a system with n position variables q1, q2, … qn is specified by a state (or wave) function ψ(q1, q2, … qn). All possible information about the system can be derived from this state function. In general, n is three times the number of particles in the system. So for a single particle n = 3, and q1, q2, q3 may be the Cartesian coordinates x, y, z, or the spherical polar coordinates r, θ, φ, or some other set of coordinates.

(2) To every observable there corresponds a Hermitian operator given by the following rules:

1. (i) The operator corresponding to the Cartesian position coordinate x is x × similarly for the coordinates y and z.

2. (ii) The operator corresponding to px, the x component of linear momentum, is (ħ/i)∂/∂x – similarly for the y and z components.

3. (iii) To obtain the operator corresponding to any other observable, first write down the classical expression for the observable in terms of x, y, z, px, py, pz, and then replace each of these quantities by its corresponding operator according to rules (i) and (ii).

(3) The only possible result which can be obtained when a measurement is made of an observable whose operator is A is an eigenvalue of A.

The problems in this book are intended to cover the topics in an average second- and third-year undergraduate course in Quantum Mechanics. After a preliminary chapter on orders of magnitude, there are eight chapters on topics arranged in a fairly conventional order. The tenth and final chapter contains a selection of miscellaneous problems on the topics of the previous chapters. I have separated them from the others on the grounds of their being somewhat longer and perhaps more difficult. But you should not be deterred from trying them on that account.

The important thing for all the problems is that you do attempt them. If you attempt a problem, and think about it, but cannot solve it, and then look up the solution, you will get much more benefit than if you jump to the solution as soon as you have read the problem. If you can solve a problem, you are still advised to look at the solution, which might contain a quicker or neater method than the one you have used. (If yours is quicker or neater I shall be pleased to hear from you.) I have also included some comments at the ends of some of the solutions, which you may find useful. They relate, either to the algebraic technique, or, more commonly, to a physical interpretation or application of the result.

At the beginnings of Chapters 2 to 9, I have included sections entitled Summary of theory, and you should read the summary before trying the problems in the chapter.

This is a book about physics, consisting of two hundred problems with solutions worked out in detail. My experience in teaching physics at university level has led me to believe that many first-year undergraduates find physics interesting but hard. With perhaps one or two exceptions the concepts are straightforward enough to be grasped, and students often enjoy learning these concepts, many of which are beautiful or at least intellectually satisfying. However, students often find that translating their understanding of the ideas of physics into problem-solving can be difficult. I believe the best way to learn how to solve problems is by example, hence this book. The level of difficulty of the problems (in some cases it would be more accurate to say the level of sophistication of the solutions) is intended to be roughly that of the first year of a physics course at a British university, and the range of topics treated is correspondingly intended to reflect typical first-year syllabuses. In order to ensure that this is so, I have drawn most of these problems from recent first-year examination papers at several British universities. However, I hope that it will find some use both below and above this level. Its intention is to help students to make the transition from school physics to university physics, so it assumes a background in physics and mathematics appropriate to such a level.