It is well known that unbounded operators play a crucial role in the mathematical formulation of quantum mechanics. In some sense, unbounded operators are inevitable. Indeed, we can prove that if A and B are self-adjoint operators such that [A,B]=ih, then A and B can never both be bounded.(adsbygoogle = window.adsbygoogle || []).push({});

My question is: is there any physical implications with being unbounded? Are there any physical reasons why we should expect unboundedness instead of bounded? Or is it only the mathematical formalism that changes?

One thing I can see is the following. A bounded operator always has compact spectrum. This means that the set of eigenvalues is necessarily bounded. So if all operators in QM were bounded, then I would expect that all outcomes of experiments are bounded. So in particular, the positions and momentums of all experiments would be bounded. This might constitute a physical reason why we want to work with unbounded operators. Is this accurate? And are there more such reasons?

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# Unbounded operators in QM

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