Hello everyone,
I would be grateful if you can help me understand what is the physical meaning of the inner product of two functions (L2-inner product)
I would like to thank you in advance for your help
Regards
Alex
Well, in quantum mechanics, it's used to compute the expectations of observables. Let's say you have a Hermitian operator $\displaystyle \hat{A}$ corresponding to some observable. You have already solved the Schroedinger equation for $\displaystyle \psi.$ What do you do? You form the $\displaystyle L^{2}$ inner product
$\displaystyle \langle\psi|\hat{A}\psi\rangle.$
This gives you the expected value from measuring the observable $\displaystyle \hat{A}.$
Dear all,
I would like also to check if I understood this correctly:
1. So If I got it right the inner product gives u a good feeling of what you expect (or what might be the result/outcome of a system that you study).
2. When I have the inner product of a function space the outcome is another function which
shows what to expect as outcome (regarding the specific domain range).
Do you agree?
Best Regards Alex
1. In the particular context of quantum mechanics, the inner product gives you expectation values, or the most probable value for measuring any particular observable. In other contexts, the interpretation of the inner product is different. For example, in normal 3D space, the inner product is the dot product (at least, it is in its most common definition), and is something of a measure of how much two vectors are pointed in the same direction.
2. The outcome of an inner product is a scalar, just like with its special case, the dot product. It is not another function or vector.
Just to add my curmudeonly point: You should understand that mathematics is NOT physics and mathematical concepts do not have specific "physical meanings" associated with them. They can, of course, have applications to physics, as in the quantum mechanical application that Ackbeet mentioned, and, in those applications, have a specific "physical meaning".
One of the interesting things about mathematics is that, once you have "invented" something to solve one problem, you find that it can be used to solve problems in many different fields. Newton and Leibniz invented Calculus to study problems in mechanics or, even more specifically, the movements of the planets. But now it is used in Biology, Economics, and even more distant fields. But if you are primarily interested in Physics, you are welcome to think about the "physical meaning" (perhap better- "physical application") of a mathematical concept.