The discussion centers on solving Ballentine's Problem 6.3, specifically addressing the trace of the commutator of infinite-dimensional matrices ##Q## and ##P##. Participants conclude that the trace of the commutator ##[Q, P] = i\hbar## cannot be zero, as the matrices ##QP## and ##PQ## are not trace-class. The key insight is that the trace is ill-defined due to the divergence of the infinite sums involved, leading to the paradoxical conclusion that ##\hbar = 0## being invalid. The resolution lies in recognizing that the canonical commutation relations cannot be satisfied by finite-dimensional matrices.
PREREQUISITESStudents and professionals in quantum mechanics, particularly those focusing on operator theory and the mathematical foundations of quantum systems. This discussion is beneficial for anyone grappling with the subtleties of infinite-dimensional spaces and trace operations.
No, it's not. An infinite sum like ##(\sum_{n=0}^\infty 1)## is divergent.PeterDonis said:[...] taking the trace (and seeing that it gives an infinite sum of terms each of which is ##i \hbar##) is simple.
Yes, I know that. I'm just saying that that infinite sum is what formally taking the trace of the infinite identity matrix gives you. I think something like that is what Ballentine intends to illustrate with the problem under discussion.strangerep said:An infinite sum like ##(\sum_{n=0}^\infty 1)## is divergent.