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Quantum Chemistry, Expectation Values (Specifically a particle in a 1-D Box)

Anonymous Mon 28 Feb 04:24:05 2022 No.14269142 View Reply Original Report

Quoted By: >>14269867 >>14271813

What excatly does this mean? I know how to solve for the expectation value of a given operator, but I don't actually intuitivelly know what it is, just paraphrase the book and other online sources.
>Weighted average
I'm guessing that it's the mean of all the values of the operator at it's given location, but I honestly have no clue what that means if that's even correct.

Anonymous

Anonymous Mon 28 Feb 2022 15:07:09 No.14269867 Report

Quoted By: >>14270154

>>14269142
It’s the most likely value for that observable for that given that. For example, the expected value for position of an electron of the hydrogen atom.

Anonymous

Anonymous Mon 28 Feb 2022 17:55:16 No.14270154 Report

Quoted By: >>14271703

>>14269867
So I know this is not equivalent, but I’m using this as a visualization so please bear with me. I’d say the chance of finding the observable was later on a Gaussian distribution, would it be the value at the peak? Sorry if this doesn’t make sense I can clarify if need be

Anonymous

Anonymous Tue 01 Mar 2022 06:48:44 No.14271703 Report

Quoted By:

>>14270154
That would be the expected value for that distribution yes. Imagine it like the mean, if you drew positions randomly from that distribution and averaged your results then you would get around that value (the larger your sample the closer you'd get).

Anonymous

Anonymous Tue 01 Mar 2022 08:12:56 No.14271813 Report

Quoted By:

>>14269142
It is literally the expected value of the ovservable, where the underlying probability distribution is defined by the particular quantum state ("wavefunction") at hand, in the eigenbasis of the observable. The associated random variable is the eigenvalues of the observable.
Operationally, you might imagine being able to consistently prepare the state, followed by measuring it in the eigenbasis of the observable of interest. The measurement will yield some eigenvalue of the observable. If you repeat this experiment many times and average the outcomes, this average will converge to the theoretically predicted expectation value.

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