Both Eigenvalues and Operators are "Observables"?

Question

I am having a bit of difficulty wading through the what seems to be multiple usages for Observables in Quantum Mechanics.

" Mathematically observables are postulated to be Hermitian operators.. " However it seems as if the observables are actually the eigenvalues of the operator: Operator Postulate as well as, according to wikipedia "a system observable is a measurable operator, or gauge, where the property of the system state can be determined by some sequence of physical operations" Wikipedia Observable. But here is a prime example of where they are calling the operator $O$ an observable, but as it "operates" on some vector and there are eigenvectors and eigenvalues (the observables), associated, I don't understand why they are using observables for the operator as well Operator as Observable. I can see that observables are associated to operators, but I don't see why they are called observables.

Any help is appreciated.

possible duplicate of Does every hermitian operator represent a measurable quantity? — ACuriousMind, Aug 13 '15 at 00:53
"However it seems as if the observables are actually the eigenvalues of the operator" - that's not the case and, further, not implied by the language at the link provided. (http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm2.html) — Alfred Centauri, Aug 13 '15 at 03:21

Ultima · Answer 1 · 2015-08-13T16:07:03.087

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Observables are measurable quantities represented by certain mathematical structures dependent on the theory.

In classical mechanics, measurable quantities are represented by functions on a phase space. In quantum mechanics, they are represented by operators on a Hilbert space. In classical mechanics, a measurement is equivalent to evaluating the function $O(q, p)$ at $(q,p)$. In quantum mechanics, a measurement projects the state $|\psi \rangle$ onto an eigenspace of $\hat{O}(\hat{q}, \hat{p})$ with some eigenvalue $O \in \mathbb{R}$.

edited Aug 13 '15 at 16:07

answered Aug 13 '15 at 02:10

Ultima

1,281

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To be slightly more precise, I would probably say that the measurement projects the state $\psi$ onto an eigenstate of $\hat{O}$. – ragnar Aug 13 '15 at 03:50
Good call, it was a mistake on my part. – Ultima Aug 13 '15 at 06:54
@Ultima I'd say it projects into an eigenspace of the operator – Timaeus Aug 13 '15 at 08:12

score 1 · Answer 2 · answered Aug 13 '15 at 10:29

Observables (I refer here to Hermitian operators) are confusingly named since they are not observable. What is observable, as you noted, is the eigenvalues that represent the outcomes. So what are observables for?

The clearest way to understand the issue is to look at Heisenberg picture observables. These observables change over time but the state does not change. This is unlike the Schrodinger picture in which the state changes and the observables don't. In classical physics you can characterise a system by a set of numbers that are all observable, and have a single value, and change over time. So one set of relevant numbers might be the positions and momenta of the system. In quantum mechanics, the situation is different. Different versions of a system can interfere with one another, so a full description of the system can't be given in terms of a single number representing its value. It turns out that the appropriate description is a Hermitian operator that changes over time.

The eigenvalue just represents the outcome that one version of you sees when you do the measurement.

For more explanation see the first lecture here

http://www.quiprocone.org/Protected/DD_lectures.htm.

score 0 · Answer 3 · answered Aug 13 '15 at 01:57

In non relativistic QM observables are represented by hermitian operators $\hat{O}$ acting on the states $\psi$, they represent measurements of some physical quantity ($\hat{O}$ represents say angular momentum, or spin). The result of the measurement will be an eigenvalue $o$ of the operator $\hat{O}$. The eigenvalues represent the actual measured value.

Both Eigenvalues and Operators are "Observables"?

3 Answers3