Examples of discrete Hamiltonians?

Question

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.

Answer 1

This is covered completely in the Feynman Lectures on Physics III: the Hamiltonian for an electron in a magnetic field is, up to a constant

$$ B\cdot \sigma $$

Where B is the field, and $\sigma$ is a Pauli spin matrix. Without loss of generality, take the B field to point in the x-z plane, and then the Hamiltonian is a real matrix

$$ \begin{pmatrix}B_z & B_x\\ B_x &- B_z\end{pmatrix}$$

The interpretation of the on-diagonal matrix elements is that they are the energy of the spin states, ignoring transitions, in this case the interaction of the magnetic moment with the z-direction field. The interpretation of the off-diagonal elements is that they tell you the transition rate between up and down spin. In this case, you can just rotate the x and z directions to make B all in the z-direction.

All that happens is that the initial electronic spin wavefunction precesses, meaning that the two vector describing the initial electron spin-wavefunction is a rotation of the vector (1,0) describing an electron with spin in the z-direction, rotated using the $\sigma$ matrices to point in some other direction, and the direction in which this spin vector points precesses around the B field direction.

The general solution to the 2-component quantum system is covered well in Nielsen and Chuang. It is described by Pauli-matrices/quaternions/3-sphere variables (these are all equivalent up to notation), and it is used to build intuition for qubits.

To build intuition for two interacting electrons, consider the two spin Hamiltonian:

$$ \sigma_1 \cdot \sigma_2 $$

Were $\sigma_1$ acts on the first electron's spin, while $\sigma_2$ acts on the second electron's spin (you should assume they are attached to separated spinless nuclei, so that they are in a different spatial wavefunction, otherwise Pauli exclusion will require that the wavefunction in spin is antisymmetric). This Hamiltonian describes a dipole-dipole energy for two electrons interacting at a distance. You can use the same Hamiltonian to understand the fine-structure splitting in Hydrogen.

This is a 4 by 4 matrix whose eigenstates are the spin singlet and spin triplet, and solving this will help you understand why the theory of quantum angular momentum addition is important.

Answer 2

There are different ways of getting discrete systems. Generally, there is a Hamiltonian that defines the unperturbed state of a system (e.g., an atom trap, or a quantum storage device). If the system cannot move or break up (and is not too large), its spectrum is typically discrete, and the levels of interest (below some maximal excitation energy) can be labelled. These labels are the indices with which the components of your state vectors are labelled. They span a finite-dimensional vector space, and the operators on it are matrices. The diagonal matrix whose entries are the energy levels defines the unperturbed Hamiltonian $H_0$. If now an interaction is switched on, $H_0$ is changed by some Hermitian interaction operator $V$, which usually is a nondiagonal matrix. Depending on one's experimental skill, one can create systems where $V$ has some desired properties. If the interaction is controlled by an external control, it becomes time-dependent.

The components $V_{jk}$ of the matrix $V$ are the matrix elements $\langle j|V|k\rangle$, and their absolute squares have a physical meaning in terms of transition rates.

A simple example is a laser, which is typically represented by a 2 level or 3 level system interacting with an external field. Another example is a silver atom in the doubly degenerate ground state (because its nuclear spin is 1/2), which responds to an external magnetic field and thus gives rise to the Stern-Gerlach experiment.

Note that a moving system that hangs together, when considered in its rest frame, becomes nonmoving, and then the above applies. In experiments such as Stern--Gerlach, or most quantum optics experiemnts, the motion is dewscribed classically, and only the finitely many nonmoving degrees of freedom are described by quantum mechanics.

To get more complicated systems, one takes a system consisiting of several parts with few levels, and takes their tensor product as the Hilbert space. The corresponding matrices are now sums of Kronecker products of small matrices acting on the individual parts. This is the playing ground for entanglement and quantum computing.