Eigenfunctions are those functions that satisfy eigenvalue equations. That is, eigenfunctions are the solutions of eigenvalue equations. In quantum physics we say that because an eigenvalue equation is linear, then all linear combinations of its solutions are also solutions. What is meant by this? Is it true? Is it usefulful to teach this as a principle? Is this misleading?
The solutions of eigenvalue equations are yielded by an operator acting on a function in a unique way such that the result is a (scalar) multiple of the original function. Recalling that linear combinations are scalar multiples of sets of vectors, and the importance of Hermitian operators with respect to orthogonality and bases for functional spaces, I'd say that while it isn't necessarily important to teach it as presented, the underlying concepts are pretty important and should be taught. In my experience, physicists approach notions such as linear combinations (and linear algebra in general) somewhat differently from mathematicians (which is why I found Quantum Theory for Mathematicians helpful and why I have a personal grudge against Dirac for a notation that in an exaggerated and melodramatic sense forced me to learn linear algebra twice). Physicists tend to have started with vectors that only exist in 3D and learn that god-awful I, j, k notation for unit vectors that could be replaced by a single symbol for all unit vectors as in linear algebra, the portions of linear algebra beyond vector analysis are introduced here and there, then quantum mechanics comes along with its infinite-dimensional/function space and an emphasis on complex conjugates and a specific type of matrix and so on, all of which (in my view) tends to present linear algebra, functional analysis, and other math topics in a rather chaotic way. As a result, students tend to vary when it comes to what aspects of multivariable mathematics, analysis, etc., they are comfortable with. So what and how notions from linear algebra should be taught in an introductory QM course tend to depend upon the students' backgrounds as dictated (in part) by the required courses and courses that students tend to have taken.
But some version of the above is important to teach.
Thank you for your answer Andrew. Am I wrong? I don't see how a superposition of eigenfunctions is a solution of the eigenvalue equation. Yet every student of quantum mechanics is told that it is. I'm wondering what it is I am missing here.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Quantum Physics
- Quantum Mechanics