I have a strong interest in the mathematical structure of quantum mechanics. I'm particularly interested in discrete systems, i.e. systems whose state is in a finite-dimensional Hilbert space. Up to now I've been thinking of the Hamiltonian in such cases as just being some arbitrary Hermitian matrix that governs the system's dynamics.
However, it would be really helpful to have some idea of what these Hamiltonian matrices and their elements represent in particular (idealised) physical situations. For example: what is the Hamiltonian for the spin state of an electron in a magnetic field (if that's a meaningful question to ask) and how is it derived? The Hamiltonian for an evolving spin state is a $2\times 2$ Hermitian matrix - do its individual elements have any particular physical significance? What about systems with more than two states? For example, can one write down a Hamiltonian for the spin states of two interacting electrons in some particular situation?
It's difficult to search for such examples, because what tends to come up are systems like the quantum harmonic oscillator, whose Hamiltonians have discrete spectra, but which nevertheless live in infinite-dimensional Hilbert spaces.