Its fairly intuitive that for a wave mechanical system in a bound potential well, eigenfunctions stem from the idea of standing waves. And that, in order to form, standing waves need reflection. But for the case of the free particle, where waves are free to disperse, and there is no opportunity for reflection, and no standing waves are possible, how can we say that generally, eigenvalue equations are a basic fundamental of quantum theory?
You can always take boundary conditions at infinity, so again you end up with the same problem. Besides this, all quantum operators are linear, thus having eigenvectors ('states') and eigenvalues ('measurable') for each concept under investigation.
For a bound potential well, it can be said that inside the well the potential is zero and outside it is infinity, thereby creating a sort of wall. Quantum mechanically there can be no 100% reflection( there is also tunnelling effect) so imposing the condition of standing waves is not justified, but it is hardly required. The only condition to be imposed is of continuity of the wavefunction at the boundaries. Also the condition that the wavefunction goes to zero at infinity also holds. The rest follows from schrodinger equation.
You are correct that there is no complete reflection even for infinite potential well
If we start from Schroedinger's equation
H * \Psi = E * \Psi
then, if not much has changed in 30 years since I learned that story, there is a matehmatical theorem named "spectral theorem" which states that this equation has many solutions but only for certain set of values of number "E". The set can be discrete, finite, infinite, continuum or diccrete and continuum at different parts. Each solution , that is function Psi, coresponds to one value of E from that set (named: spectrum). The funny thing about it is that for physically sound operators H set of all Psi's forms a complete otrhogonal set (basis in a Hilbert space), that is, you can describe any function with that set. That is why these functions, which by the way are named "eigenvectors" are so important.
Now whether hamiltonian H describes particles that move left, or move right, or oscillate or whatever: it does not matter at all. Mathematics of the spectral theorem could not care less about what we percieve and call "motion". That is why, your question, to an extent is ill posed.
Thank you Mario.
Are you, then, saying that Schrödinger and the spectral theorem are more fundamental than eigen functions? If so, for what reason? Could it not be claimed that Schrödinger and the spectral theorem are valid only because eigenfunctions are the fundamental foundation?
Quantum Physics is essentially a Linear Science: All operators that describe a system are Linear Operators. So, study of Linear Algebra is a requirement before every discussion about Quantum Physics concepts.
Steve,
You caught me there! As a physicist I believe that reality is fundamental and that theories are just more or less accurate construct that we use to classify reality.
But:
1. I do not think that you are asking about some borderline effect that has not yet been discovered and that deviates from quantum theory. So for all we care, the reality and math theory are exactly the same thing in this discussion.
2. Point of my discussion above is that it does not matter what H describes and especially, related to your question, it does not matter whether the system is a photon in a silvered box or in open space or a goat eating grass.
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