I worked through the problems in Arfken the hard way; I solved the problems one-by-one starting from chapter 5 until the chapter on integral equations. I still have a list of the problems I did not solve (I used the third edition,but teach using the 6th ed.), and as I teach, I revisit them, hoping that this time, I would be able to figure it out.

On the other hand, although I have not worked through some sections in Arfken (such as the chapter on matrices and group theory), my problem-solving in other areas of theoretical physics does mean that I've actually solved them elsewhere. Sakurai's

*Modern Quantum Mechanics*, chapters 1 to 3 was where I learned about Pauli matrices and the rotation group, for example. If you compare Sakurai and Arfken's group theory chapter, there will be a lot of overlap. So dedicated study elsewhere means being able to solve Arfken's problems even though, on my first pass, I skipped through the group theory.

Another example is Arfken's chapter on the calculus of variations. I've encountered the material elsewhere, in classical mechanics (Marion's and Goldstein's books), in general relativity (Hartle and Weinberg), Finkelstein's book (

*Nonrelativistic Mechanics*, a parallel presentation of classical and quantum mechanics), and Jackson's

*Classical Electrodynamics*.

I've collected stacks of folders with solutions, and I really have no plans of uploading them. When I do teach, I do not provide an answer key either; I do mark papers and when time permits, we solve the assigned problems on the blackboard. I usually get a student to solve it in front and then act as the hostile critic so as to root out misunderstanding, and bad reasoning. When the student is stuck, I suggest possible lines of attack, or solve a related problem.

Why do I not provide fully-worked solutions? Providing worked solutions encourages memorization, rather than understanding. Also, the worked solutions sometimes end up being recycled by people who take the course later on. It's hard to think up good problems at that level; it's different with introductory physics where it is easier to invent easy problems.

On the other hand, people argue that without solutions manuals, they cannot be sure of their work. I think it's bosh. The problem sets of today are usually the research questions of yesterday. Cauchy's theorem and other things that we take for granted today did not come by divine revelation; these things were discovered by people who were willing to think for themselves.

Instead, what one needs is a method of eliminating self-deception. When I write my solutions, I do it in such a way that my solutions can serve as supplementary lecture notes. (This is actually an old idea, and I owe this statement to N. David Mermin. ) This means you have to express clearly, without any obscurities, how you actually reason from start to finish. You have to justify every step you make, by quoting previous theorems (that you've also proven for yourself!), etc. If the reasoning you present is tight, with no missing steps, then you can be pretty sure that your solution is correct. It does mean, however, that you have a good grasp of prerequisite material.

Solutions manuals pervert the learning process because being able to submit correct solutions to problem sets leads to self-deception. One learns by making mistakes and by being stuck in some areas, and trying various possible dead-end solutions paths. Learning the dead-ends is also important; if you do not learn the dead ends, then you will be compelled to learn them later on, when you actually have to use the methods in Arfken to solve your particular research problem. (And since it is a research problem, by definition, no solutions manuals exist!)

(addendum: one of my friends had this to say about solutions manuals: You want a solutions manual? Make one!)