# 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. – AHusain Oct 6 '18 at 4:03
• Ah okay that is fine. Thanks for the edit! – ahelwer Oct 6 '18 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. – AHusain Oct 6 '18 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 – user1271772 Oct 6 '18 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. – user1271772 Oct 6 '18 at 22:35

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