# For any two unitarily equivalent observables, can both be measured by the same experimental apparatus?

If we have an observable $$A$$, and a unitary operator $$\hat U$$, one can easily show that both $$\hat A$$ and $$\hat U \hat A \hat U^{\dagger}$$ have the same spectrum - in fact, they are called unitarily equivalent.

For example, in Stern-Gerlach experiment, one can easily show that $$S_x$$ and $$S_z$$ operators are unitarily equivalent by the rotation operator.

Motivated by the above example, as in that case, $$S_x$$ and $$S_z$$ are physically the same operators in the sense that any proper of one should be a proper of the other as well, since one can free to choose a new $$z'$$ axis in the $$x$$ axis, i.e the choice of our axis was arbitrary, so in that sense, they are the same operators, by when we use them at the same time, they just correspond to a different dimensions of the system.

As @JohnForkosh interpreted my intuition in a more clear & correct sense, this equivalent might be captured by the property that $$S_x$$ and $$S_z$$ can be measured by the same experimental apparatus, see his comment below.
• 1. It is not clear to me here what you mean by "some arbitrary choice", nor what you mean by $S_x$ and $S_z$ being "physically the same operators". They are not the same operators - a state that is an eigenstate of one is not in general an eigenstate of the other! 2. Consider the pre-eminent example of unitarily equivalent operators: Position and momentum, with the Fourier transform being the unitary transformation between them. You'll be hard-pressed to find anyone willing to say that position and momentum are "physically the same". – ACuriousMind Jan 22 at 17:46
• @ACuriousMind I didn't know that Fourier transform was a unitary operator, I just say the case $S_x$ and $S_z$, and the from the way how the professor presented, that was the natural question to ask. – onurcanbektas Jan 22 at 18:24