I'm not a physicist, but rather a control (feedback) systems engineer eager to understand more than just a cursory explanation of quantum mechanics. The StackExchange has been an excellent forum for helping me piece together facts that are not always expressed well in the textbooks or other online information, and I have great appreciation for whoever conceived and continues to maintain its existence. Thank you. But on to my questions..
I've read that one can take the mathematics of Schrodinger's wave equation and Heisenberg's matrix equations and show that they are 'equivalent'. In control systems we model physical (macro)systems by either a set of differential equations or as a state space system, and indeed can show equivalence between these models whether linear or nonlinear.
- Specifically between Schrodinger and Heisenberg mathematics, are they each modeling the same quantum wave mechanics, and is this equivalence shown in the same manner that I show equivalence in the physical systems modeling I've described above (differential equations and state space systems)? If not then what is it that is shown to be equivalent between these two systems of mathematics?
Since I am a control systems engineer I'm familiar with linear systems theory, state space systems, the concept of observability, and the construction of the observability matrix from a linear state space model. In linear systems theory the observability matrix is derived from the system matrix (holds information on the eigenvalues) and the output coupling matrix ( a matrix that couples the system states to measureable outputs). The observability matrix basically determines what states within the system can be measured (observed).
- Does the Heisenberg description of quantum mechanics also lead to such an observability matrix (or some equivalent) explaining why only probability amplitude can be observed (and not the complex phases)?
In controls we can also take the system models and express them in terms of a Hamiltonian, and change the basis of a system description using matrix rotations. But we typically don't use 'Bra' and 'ket' notation - < and >.
- So are control systems engineers and quantum physicists basically using the same mathematical tools to describe system models, but perhaps using slightly different terminology?