Talk:Expectation value (quantum mechanics)

Potential merge with expected value

Untitled

This concept seems not the least bit different from the one treated in expected value. Only the terminology and some of the notation are different, this one perhaps reflecting conventions prevailing among some physicists. Michael Hardy 23:23, 24 May 2007 (UTC)

I was the one who split it out from the main expected value section. This was for 2 reasons: firstly, where it was placed in that article originally was clearly not the right place for the content. If it were to be on that page, it should be much lower down. However, secondly, one needs to have quite a lot of physics background to understand the explanation in this section, as evidenced by the current need for a physics expert to verify the content. The expected value article, on the other hand is currently (and I believe, should be) an introduction to the basic concept and some of its properties. Now, from this article, it seems to me that the expectation value of an operator is the expected value of some random variable, but it is not clear to me what that variable is. Thus, I believe that either (a) the expectation value concept in quantum mechanics is a concept which is closely related to that of an expected value, in which case it deserves its own page, or (b) it is just an application of the expected value concept for a certain variable, in which case I would be happy for it to be added to the applications section (or an examples section) of expected value, with it clearly spelt out what r.v.'s expected value we are dealing with. --Steve Kroon 06:15, 25 May 2007 (UTC)

You are correct in that this is related to the mathmatical method of expected value but this is a ratere important subject in Q.M. and deserivers to have the page expanded into a fuller treatment of ther subject. I will attempt to do so somewhat soon but I think the best solution would be a link to the page on expected value. Blue loonie 07:24, 10 June 2007 (UTC)

Excellent. I agree with the statement from Blue loonie above, this is certainly an important subject in Quantum Mechanics. While it is somewhat related to the expected value in mathematics, it's implication is very different. I'll also attempt to help with updating this article but we'll see how much time I get to do this. If anyone is watching this and can give an outline of what improvements/details should be included, please list them here. --JT 01:37, 16 August 2007 (UTC)

• In the rewrite of the article, I have tried to make clear (in the first section) that the quantum mechanical expectation value is, mathematically, not the same as the expected value in usual statistics. This is due to the "noncommutative" nature of quantum theory, to drop the usual buzzword. --B. Wolterding 11:44, 20 August 2007 (UTC)

Details

The article currently says this:

The expectation value, $\langle x\rangle \,$ of an operator, ${\hat {X}}\,$, operating on a wavefunction $\psi$ is given by

$x={\frac {\int _{-\infty }^{\infty }~\psi ^{*}{\hat {X}}\psi ~dV}{\int _{-\infty }^{\infty }~\psi ^{*}\psi ~dV}}.$

Shouldn't it say something like the following?

The expectation value, $x=\langle {\hat {X}}\rangle \,$ of an operator, ${\hat {X}}\,$, operating on a wavefunction $\psi$ is given by

$x={\frac {\int _{-\infty }^{\infty }~\psi ^{*}{\hat {X}}\psi ~dV}{\int _{-\infty }^{\infty }~\psi ^{*}\psi ~dV}}.$

As it is, it says the expectation value is called $\langle x\rangle \,$, but the operator is not called $x\,$, but rather is called ${\hat {X}}$, and then the "displayed" identity does not say "$\langle x\rangle =\cdots \,$", but instead says "$x=\cdots \,$. Michael Hardy 20:37, 25 May 2007 (UTC)

Example

Hello Optics guy07,

putting an example into the article is an excellent idea! As you noted in the edit summary, the notation was not quite in line with the rest of the article, and I tried to clean it up, and also added some context to the example.

I was not completely sure whether you intended to set up the example for a 3-dimensional or for a 1-dimensional particle. I opted for the 1-dimensional particle, in order to make the example as elementary as possible. We can change that, of course.

I also moved the note about non-normalized vectors further up in the article, since I thought it would fit better there. --B. Wolterding 16:05, 21 August 2007 (UTC)

Thanks. Yeah I didn't have time to come back and reread the article. I'm also thinking about putting something in about transforming to momentum space. -JT 20:43, 21 August 2007 (UTC)

Couple of Questions

What happens if the eigenvalues ARE degenerate? Also, is the Expectation Value an eigenvalue? —Preceding unsigned comment added by 205.250.67.77 (talk) 08:40, 6 February 2009 (UTC)

(a) Everything in the article applies also in the case of degenerate eigenvalues, except for the half-sentence after footnote [2].

(b) No, not in general. (More specifically: yes, if the state $\psi$ is an eigenstate of $A$, but otherwise usually no.) --B. Wolterding (talk) 11:56, 6 February 2009 (UTC)

Some Questions

Mathematical expectation is defined only in Lebesgue integration terms (and this is when intuitive strong law of large numbers hold, and says E(x)=<x> = long term average of x. For other integration methods strong law does not hold and does not guarantee this. How does QM define expected value? by what integration? — Preceding unsigned comment added by Itaijj (talk • contribs) 08:59, 28 June 2014 (UTC)

Blank Section: Possible Vandalism?

On 22:30, 12 May 2015, an IP address, 202.92.134.5, committed section blanking on Operational definition. This user has made only a few contributions, at least one of which was unconstructive (see the User Talk page).

The following is the relevant history diff:

?title=Expectation_value_(quantum_mechanics)&diff=662100484&oldid=659202549 (the link works, even tho it's red.) ReGuess (talk) 18:47, 23 June 2016 (UTC)