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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.

N. Virgo
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  • I might be off or to broad here for you, but by the time evolution of the basisvectors given by the Schrödinger equation "$\frac{\text d }{\text d t}\psi=\frac{1}{T}\psi$", aren't the elements just some eigenfrequencies, describing (relative) rates of change of the physical configurations wandering through state space? – Nikolaj-K Jul 30 '12 at 11:43
  • @NickKidman if I understand you correctly, a similar thought occurred to me while writing the question. But I've often seen the spin state of an electron referred to as if it occupied a two-dimensional Hilbert space, whereas I've never seen the state of a quantum Harmonic oscillator referred to as if it lived in a countably-infinite-dimensional Hilbert space. If the finite-dimensional space is derived from the infinite-dimensional one in that particular way, it would still be good to see a worked example. – N. Virgo Jul 30 '12 at 11:49
  • Related readings might be Heisenbergs work on the osciallator, creation/annihilation operators in matrix form, small quantum systems and maybe even Can one hear the shape of a drum?. – Nikolaj-K Jul 30 '12 at 11:56
  • I see, so matrix mechanics basically is the representation of QHO-like systems as if they lived in countable-dimensional Hilbert spaces. That's very interesting, thanks. (It'll take some time to digest.) – N. Virgo Jul 30 '12 at 13:03
  • I don't know why you formulate it using the term "as if". I mean if you can count it, it's countable. You're not dealing with Skolem's paradox-ish rigour here. – Nikolaj-K Jul 30 '12 at 13:18
  • @Nathaniel: They do live in "countable dimensional" Hilbert spaces--- the condition of finite energy restricts you to continuous functions which are densely approximated by a countable basis. This is separability of the Hilbert space. The formalism pretends that the x-basis or the b-basis is the same thing as a countable basis, but they are distributional (although formally they aren't any different). – Ron Maimon Jul 30 '12 at 14:42
  • @NickKidman I say "as if" only because in the x basis the state is a continuous function of x, and H is a continuous-valued operator. So it looks like a continuous space in that respect, even though it seems to be spanned by a countable basis. I don't fully understand that yet, but Ron says it's because of restrictions imposed by a condition of finite energy, which makes some intuitive sense. These concepts probably shouldn't be unfamiliar to me, but they are - it's one of the perils of being self-taught. – N. Virgo Jul 30 '12 at 16:01
  • I'd still like to have a better understanding of how it works with finite state spaces like spin states, though. – N. Virgo Jul 30 '12 at 16:02
  • @Nathaniel: Is the third link with the two-level system not enough to go on? When I learned QM, I found writing a program which draws an animation of the solution of $\frac{\text d}{\text dt}\psi(t)=-i H\psi(t)$ with $H$ being the number $3$ and $\psi(t)$ being a complex number with absolute value $1$, instructive. That is you just draw a picture of an arrow in the complex plane and, most importantly, its tangent vector. Consequently you can change consider other numbers for $H$ and a second arrow and then the normed superposition of these etc. This is no five lines in Mathematica. – Nikolaj-K Jul 30 '12 at 19:37
  • @NickKidman I have a very good mathematical understanding of how $\psi$ changes over time for a given $H$. What I'm not so hot on is where any given $H$ comes from, physically. There's a bit of useful stuff about that in your third link, but it seems to just state what $H$ is without deriving it. (Still, if you want to post this stuff as an answer I'll happily accept it.) – N. Virgo Jul 30 '12 at 20:53
  • (Having said that, I don't want to sound greedy, but it would be great to have an example of a Hamiltonian for a four-(or three-)state system as well - in some ways two states is a bit too trivial to be really interesting.) – N. Virgo Jul 30 '12 at 20:57
  • $H$ comes from trial and error. "What energy operator gives me the right line spectrum and S-matrix elements?". I mean the potential $U(r)\propto\frac{1}{|r|}$ is also the energy quantity around which electrostatics is build because people made up of the concept of space, time and force and this $U$ works out. Reverse engeneering, you can argue "we live in such and such space with such and such symmetries, now what objects are legal" -> seraching representations. The harm. osc. always shows up because it's the smallest expansion in a local potential minimum and you want these for bound states. – Nikolaj-K Jul 30 '12 at 21:54
  • For not too small but also not too big systems consider other constructed representations of the rotation group and it's friends (or any compact group), i.e. theories which rely e.g. on triplet states. Chemistry stuff and isospin comes to mind. The entanglement or quantum information systems are also not big. On the other hand I don't know if they give you a time evolution without getting too complicated. I don't know how small the minimal treatment of e.g. superconductivity via electron pairs is, there are certainly people around who have an idea. – Nikolaj-K Jul 30 '12 at 21:59
  • ...and thinking about it, it sounds reasonable to me that there might be microscopic three- or four-level models of lasers, but I don't know of any. – Nikolaj-K Jul 30 '12 at 22:11
  • @NickKidman I guess a quantum logic gate has to be implemented as a four-state system (two interacting qubits), so I would guess that examples of specific physical implementations must exist within the field of quantum computing - maybe I'll have a look for one next week. If you want to compile your comments so far into an answer, I will accept it. – N. Virgo Jul 31 '12 at 08:31

2 Answers2

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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.

Emilio Pisanty
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Ron Maimon
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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.

Arnold Neumaier
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