# Why is the average in QM of a function different from the mathematical definition?

Math definition:

integral of function within limits divided by difference of limits.

QM definition:

integral of complex conjugate of wave equation times function times wave equation within limits of minus to plus infinity.

fresh_42
Mentor
Math definition:

integral of function within limits divided by difference of limits.

QM definition:

integral of complex conjugate of wave equation times function times wave equation within limits of minus to plus infinity.
Because both use different metrics. The QM definition is mathematical, too. It's the ##L_2## norm of a wave function, it's height if you will. There is no average going on here. What you called mathematical average, is the Euclidean height if a certain area is imagined as a rectangle. So the difference is simply the different ruler you use for a measurement. The Euclidean area isn't of much help for wave functions, the ##L_2## norm is. It's a bit as if you asked, why astronomers don't use miles or kilometers to measure intergalactic distances.

random_soldier
I think I understand somewhat. Do you have a quick reference for what the L2 norm of a wave function is? I think I would get a better understanding with that.

fresh_42
Mentor
I think I understand somewhat. Do you have a quick reference for what the L2 norm of a wave function is? I think I would get a better understanding with that.
Well, it's basically functional analysis and perhaps you want read it in a book (should be somewhere in the first 40 pages).
Maybe these insight article(s) is of help: https://www.physicsforums.com/insights/hilbert-spaces-relatives/ (2 parts).

Why especially this function norm is used is a bit excessive to explain for short. It starts with the fact, that Lebesgue integrals rather than Riemann integrals are used, that it should work for real and complex numbers, that the function space is infinite dimensional, that it should be a norm for the metric used, that functions which are almost everywhere identical to zero have to be factored out, and that we want to have a complete space, in which Cauchy sequences converge. So a lot of technical aspects are more important than a geometric area. In a way it is still an area, because we interpret it as a summation of probabilities.

random_soldier
PeroK
Homework Helper
Gold Member
2020 Award
Math definition:

integral of function within limits divided by difference of limits.

QM definition:

integral of complex conjugate of wave equation times function times wave equation within limits of minus to plus infinity.
First, you have to stop thinking like that. QM is built on an area of mathematics called Linear Algebra, which includes the theory of Hilbert Spaces and, in particular the ##L_2## space. See here:

http://mathworld.wolfram.com/L2-Function.html

Note that the wave-functions in QM are analogous to vectors: and, in fact, have the same properties as vectors.

Wherever you are learning QM, it should be putting the definitions in a proper mathematical context.

random_soldier
I see. Well, I don't know if I should go into that sort of detail. I am actually reading a text for a class, Introduction to Nuclear Physics by Kenneth S. Krane. It's not a for a QM class nor do I think it's a good source for that now that you mention all this.

martinbn
First, you have to stop thinking like that. QM is built on an area of mathematics called Linear Algebra, which includes the theory of Hilbert Spaces and, in particular the L2L2L_2 space.
Technically it is functional analysis.

fresh_42
vanhees71
Gold Member
I've no clue how you come to the "mathematical definition" in #1. For a given probability distribution ##W(x)## of a random variable ##X## the expectation value (or "average" by definition is given by
$$\langle X \rangle = \int_{\mathbb{R}} \mathrm{d} x x W(x).$$
This is independent of QT.

In QT, if ##X## is a position variable, the probability distribution is given via Born's rule ##W(x)=|\psi(x)|^2##.

martinbn