Why is average in QM applied to a function different from math definition

[random_soldier1337]

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.

Edited October 6, 2018 by random_soldier1337

[Prometheus]

I'm not sure what you mean by divided by the difference of limits, i've not come across this before. Maybe if you show the maths notation it will be easier to understand what you mean.

In words, i would say the definition is the integral of a probability density function (pdf) times x over some range. In the case of QM the pdf is the square of the wave function (which is the same as the complex conjugate of the wave function times the wave function). They are the same.

 

Man, when you try putting these concepts to words you really see the elegance of mathematical notation. 

[studiot]

39 minutes ago, Prometheus said:

I'm not sure what you mean by divided by the difference of limits, i've not come across this before. Maybe if you show the maths notation it will be easier to understand what you mean.

That is perfectly normal practice.

What the OP means is that the average or mean (value) of a function over an interval (a, b) is given by

[math]Av(f(x) = \frac{{\int_a^b {f(x)dx} }}{{b - a}}[/math]

 

 

It is worth making the point that for waves symmetrical about the x axis this value is zero over a whole cycle, so we take the root mean square value instead.

This is how the product of the function appears in the integral, except that for  complex valued function squaring it will yield another complex value in general and we want a real valued answer. This is done by multiplying by the complex conjugate instead of itself.

[swansont]

10 hours ago, random_soldier1337 said:

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.

There's more than one type of average, and what you describe is not an "average" in QM. The integral of |psi|^2 gives you a probability, and integrating over all space gives an answer of one, by definition. 

[random_soldier1337]

Thank you. I think I have a bit more of an understanding.

However, in the following link to the wikipedia topic, in the equation for the expectation value of Q, why is the equation formulated as such (x being replaceable by any general function f(x)):

https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)#Example_in_configuration_space

I may be mistaken but I thought in general in mathematics, expectation value was another term for the mean or the average.

[swansont]

5 hours ago, random_soldier1337 said:

Thank you. I think I have a bit more of an understanding.

However, in the following link to the wikipedia topic, in the equation for the expectation value of Q, why is the equation formulated as such (x being replaceable by any general function f(x)):

https://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)#Example_in_configuration_space

I may be mistaken but I thought in general in mathematics, expectation value was another term for the mean or the average.

Expectation value equation is not what you described in your first post (from what I can tell) But it is what Prometheus described.