Answer to Question #126194 in Atomic and Nuclear Physics for Jawad

Question

A nuclear scientist claimed that he has accurately measured the energy and momentum of radiation (alpha’ beta’ gamma-ray etc.) using his most sensitive geiger muller counter. Can you believe his claim is correct? If yes give a proper justification of his claim and if no then also justify it

Answer 1

Yes, the claim is true.

Reason:

From the principle of Quantum Mechanics any two physical quantity can be measured simultaneously if their Quantum Poisson's bracket corresponding to the operator of those quantities is zero i.e it's commutes.

Now,

$$\hat T=-\frac{\hbar}{2m}\nabla^2$$

And

$$\hat p=-i\hbar\nabla$$

Thus,

$$[\hat T,\hat p]=\hat T\hat p-\hat p\hat T\\ \implies [\hat T,\hat p]=-\frac{\hbar}{2m}\nabla^2(-i\hbar\nabla)-i\hbar\nabla(\frac{\hbar}{2m}\nabla^2)=0$$

Hence, we are done.