Cthugha said:
Happiness said:
Therefore, there exists an allowed value of k such that a solution to [5.48] is not a solution to [5.49].
That is not correct. In the band gap region, you arrive at a pair of values of \lambda that is real, but not 1, which corresponds to exponentially growing or decaying modes. Out of these, the decaying mode is usually the physically relevant one.
What you said seems to contradict Griffiths's statement below:
Griffiths said states with energies in the gap region are physically impossible, but you said these states are physically relevant.
A definition of "growing mode" and "decaying mode" would be helpful, and also a concise explanation how they are "exponential".
But in any case, my following sentence
Happiness said:
Therefore, there exists an allowed value of k such that a solution to [5.48] is not a solution to [5.49].
already set the discussion: we only consider an allowed value of k. But your reply is about disallowed values of k being physically relevant. But I wasn't making any claim about these disallowed values of k!
You seem to be very proselytising in sharing and showing your knowledge in numerous things, like decaying modes, optical materials, modular arithmetic, etc., which may be good in a way, but it may sometimes distract the attention away from the main issue in the discussion.
I am sure you didn't mean to contradict Griffiths. But to new learners unfamiliar with "decaying modes", it certainly seems so. Therefore, more mindfulness would be beneficial, more mindfulness about how much your readers could understand your statements, and how effective your statements are in communicating the ideas across to your readers. Also, would they cause more confusion to your readers? Are they really helpful and relevant to the core of the issue? Are they the simplest way to resolve the issue?
Simplicity and conciseness are very expensive gifts. (in tribute to Warren Buffett's quote on honesty)