1
$\begingroup$

In quantum mechanics, we learn that for any Hamiltonian with a symmetry, there exists a unitary operator associated with that symmetry. Consider the parity operator which is defined by its operation on the position basis, $$\Pi\left|x\right>=\left|-x\right>.$$ One can show that $\Pi$ is Hermitian with eigenvalues of $\pm1.$ If Hermitian operators are associated with measurable quantities, what is the measurable quantity associated with parity? How would you go out and measure parity in the lab?

More generally, one of the postulates of quantum mechanics states that every measurable quantity $\mathcal{A}$ is associated with a Hermitian operator $A.$ We could write $$\mathcal{A}\implies A.$$ Another way to frame this question is to ask whether the converse is also true: that is, for every Hermitian operator $A,$ can we associate a measurable quantity $\mathcal{A}$? Does $$ A \implies \mathcal{A}?$$

$\endgroup$
1
  • $\begingroup$ Parity has eigenvectors and eigenvalues, so expectation values between suitable states... Parity eigenvalues of particles can be measured that way. Expectation values of suitable operators is normally what's involved in predicting the outcome of every measurement. $\endgroup$ Sep 17 at 20:09
1
$\begingroup$

Concerning Parity: The exact way in which you experimentally observe parity is quite dependant on the specific system that you consider. For example relative alignments of spin are something which you can measure and then relate to statements about parity. To be concrete, in the famous Wu experiment (Experimental Test of Parity Conservation in Beta Decay), parity violation of the weak interaction is demonstrated by measuring the angular distribution of electrons that are emitted in beta decays from a Cobalt sample that is prepared to have a certain spin alignment.

Concerning the question about arbitrary Hermitian operators: In a (not very meaningful) way I would say that the answer is yes. Each Hermitian operator divides the Hilbert space of possible quantum states into orthogonal eigenspaces. In principle there should always be measurements that correspond to projections onto these eigenspaces i.e. measurements of a quantity corresponding to the operator. I suppose though that this is simply not a really physical question. After all, you will never be able to exactly prepare any idealised self contained quantum systems without outside interference and also never be able to exactly realise any Hermetian operator as a measurement -- the best you can get is experiment and theory being "close enough" to each other for what ever it is you are doing. Especially, this means that it does not really make sense to consider every Hermetian operator that some tormented mind may conceive.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.