- #36

Cthugha

Science Advisor

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This wasn't the issue I had.

Well, your issue already changed several times. This is the difference in the treatments between Griffith and Bransden/Joachain. Griffith first looks for common eigenstates of the Hamiltonian and the displacement operator for displacement by a single period and then introduces periodic boundary conditions, which is equivalent to looking for all solutions for all displacement operators that correspond to displacement by an integer number of lattice periods. Bransden/Joachain immediately assume periodic boundary conditions at the beginning of their discussion. It is a very different question whether you look for superpositions of functions that are eigenfunctions of one of the displacement operators and the Hamiltonian or superpositions of eigenfunctions of different displacement operators and the Hamiltonian.

There are two values of K for each energy E.

These differ only by sign. Your concern was that these states might result in different [itex]\lambda[/itex]. For D(a) as treated by Griffith, you end up with [itex]K=\pm \frac{2\pi}{a}[/itex], which obiously yield the same [itex]\lambda[/itex]. This will change when considering a different displacement operator.

Not quite clear what you mean. States that are eigenfunctions of D? States of the same energy?

States that are eigenfunctions of D(a) and the Hamiltonian.

Ok, there may be other ways to get [5.56]. But the book's approach is much easier. It's just one line. And this wasn't my issue. The priority should be on the accurate identifying of the cause of the confusion and addressing it using the simplest tool, without too much unnecessary information. Nonetheless, thank you for your advice.

The approach given in the book is not simpler because what I noted was exactly the approach used in the book.