# Introductory resources for learning about quantum Hamiltonians

I am seeking introductory resources which will enable me to answer these questions (textbooks, lecture series, etc.):

• Given a simple quantum system, how do I derive its Hamiltonian?
• Given a Hamiltonian, what questions can I answer about the system it describes (and how?).

I am approaching this topic primarily from the computer science side. I am familiar with Newtonian classical mechanics as presented in first-year undergraduate physics courses, but never learned classical Hamiltonian mechanics.

• When you say given a system, what is your input data? A classical Hamiltonian? Potential energies and masses? The force laws? Because usually when specifying a quantum system you have already given the quantum Hamiltonian. Oct 6, 2018 at 4:03
• Ah okay that is fine. Thanks for the edit! Oct 6, 2018 at 17:21
• And I'm free to ask as many questions to get you to give sufficient specification? Like how strong is the magnetic field? Is the particle moving through? Does the magnetic field change in space or time? You have to specify a well defined question in order to get a well defined answer. Oct 6, 2018 at 17:24
• @ahelwer: Perhaps the best resource is to take some introductory QM courses at a university (if available to you) or to pick up a book like Modern Quantum Mechanics by Sakurai, or the 2-part QM book series by Cohen-Tennouji. There is really not much to it. The Wikipedia page in my answer below is really all there is. If you want a particle in a magnetic field, the Hamiltonian is the Hamiltonian of the free particle (kinietic + potential energy), plus the Zeeman Hamiltonian, which is already listed in the particle zoo. but the Zeeman Hamiltonian is a model, so as AHusain said, you need more Oct 6, 2018 at 22:28
• details for a more accurate simulation. Hamiltonians are only an approximation of the true story, just like Newton's F=ma is only an approximation to the quantum mechanical Schroedinger equation. Schroedinger's equation is not actually a 100% accurate description of matter, but is justified since you can get it from the small $\alpha$ limit of quantum electrodynamics ($\alpha$ is the fine-structure constant). So a better way to model a particle in a magnetic field is to use a quantum field theory such as QED, QFD, QCD. Oct 6, 2018 at 22:35

## 1 Answer

Given a simple quantum system, how do I derive its Hamiltonian?

For quantum systems of continuous variables, the most common way to construct the Hamiltonian is to add the kinetic energy and potential energy, as described in this resource. The kinetic energy part is explained here, and various potential energy models are given here:

which unfortunately I could only get to from this page!

For quantum systems on discrete variables, you can construct any $$2^n \times 2^n$$ Hamiltonian using the Pauli matrices, and any Hamiltonian of any dimension using generalizations of the Gell-Mann matrices.

In terms of other resources: There is an open source Hamiltonian Zoo on GitHub, but it is very incomplete. So far it tells you how to derive the Hamiltonian for two charges interacting with each other (Coulomb), for a spin system interacting with a magnetic field (Zeeman), and for a 2D p-wave Fermi superfluid, but not much else. However, since this is a resource request, I think the Hamiltonian Zoo is a good starting point, because it lists the names of almost every mainstream Hamiltonian imaginable, and the best resource for learning about each of those listed Hamiltonians is Wikipedia. For example:

In every case, the Hamiltonian is in the article, just look for the equation containing the big $$H$$!

Given a Hamiltonian, what questions can I answer about the system it describes (and how?).

There is no "resource" I know that teaches people what can be learned about a system based on looking at its Hamiltonian, and I'd be quite surprised if such a resource existed. What I can tell you is that there are things about the system that can be learned by looking at the Hamiltonian (such as number of particles or number of degrees of freedom, by looking at the number of terms in the kinetic and potential energy operators in the case of continuous variables, or the size of the matrix in discrete variable Hamiltonians). However you may want to ask this part as a separate question (and not a resource request) in case other people want to suggest other things that can be learned about a system by looking at the Hamiltonian apart from what I've already told you.

• I see some good effort here. Have an upvote to make up for the encryption-cracking downvote. Oct 12, 2018 at 17:14