# Confusion of measuring two quantities on a quantum system

Let's say there are two observables corresponding to two operators A and B, and let's say my system is in a state Phi where with probability 1 if I measure A I get 3 (let's say 3 Joules), If I measure B I get 4 (let's say 4 m/s). If I measure A and then B I would get 3 Joules for that energy measurement and 4 m/s for the speed measurement, however, mathematically, I would write:

BA Phi=12 Phi

So the measurements kind of mixed up, I don't understand this.

This question arised from problem 3.5 of Zetilli's book

• What exactly is the question? If $|\phi\rangle$ is an eigenstate of both $A$ and $B$ with eigenvalues 3 and 4, it's also an eigenstate of $AB$ (and also $BA$) with eigenvalue 12. What is it that bothers you about this? – Javier Feb 4 '19 at 1:32
• The mathematical representation of measuring A and B isn't $AB \Phi = 12 Phi$. $AB \Phi$ means you are applying the operator $AB$ to the $\Phi$ state. A measurement has no actual mathematical representation. (since you can't know what you are getting, only the probabilities which are represented in $\Phi$). – IvanMartinez Feb 4 '19 at 5:13

It seems the problem is in distinguishing measuring $$A$$, measuring $$B$$ and measuring $$BA$$. If you have an apparatus that measures $$BA$$, then you’d get $$12 J m/s$$, and there is no way you can “separate out” the $$A$$ and $$B$$ part: presumably the apparatus to measure $$BA$$ would yield a single pulse of some height (or whatever other signal) from which you’d deduce the outcome is $$12 J m/s$$, and you’d have no way of knowing if this were $$3\times 4$$ or $$2\times 6$$.
You could separate this out if you measured $$A$$, recorded the outcome, then measured $$B$$ and recorded the outcome. That’s not quite the same as measuring $$BA$$, which is technically a different operator and thus would require a different setup than measuring $$A$$ or measuring $$B$$ alone. Measuring $$A$$ then $$B$$ is two measurements, whereas measuring $$BA$$ is a single measurement.
• of course you need to keep track of the units, v.g. the angular momentum operator $\hat L_z$ really have eigenvalues $m\hbar$, not just $m$. your factor here would be $12 Jm/s$, not $12$. Maybe edit your question to clarify... – ZeroTheHero Feb 4 '19 at 1:49