The discussion centers on the necessity of Hermitian operators in quantum mechanics, particularly regarding observables and their eigenstates. It is established that only Hermitian operators guarantee real eigenvalues, which align with the requirement that measured values must be real numbers. The origin of this convention is traced back to the postulates introduced in Dirac's "The Principles of Quantum Mechanics," with the assertion that the mathematical framework of quantum mechanics is built upon these foundational assumptions. The conversation also highlights the role of Positive Operator Valued Measures (POVMs) as a more general framework for measurements beyond traditional Hermitian operators.
PREREQUISITESQuantum physicists, students of quantum mechanics, and researchers interested in the foundations of quantum theory and measurement theory.
A. Neumaier said:In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part.
Nice post-editing :)A. Neumaier said:The real observables are the points on the oscillating curve; the observable complex frequency is extracted from these and produces the physical way of summarizing the behavior of the oscillator.
It is always the summary that carries the physics. Without summarizing what happens in Nature we cannot form a single concept. Every observable is an abstraction of the real thing, and as an abstraction it may be a real number, a complex number, or an even more complicated object such as a vector or a tensor.
Take a wire and bend it to a sine wave :)Derek Potter said:I am surprised that nobody else seems to be interested in whether the idea of complex observables makes sense physically.