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I'm an undergraduate reading up on some quantum physics so that I can help out more in the lab that I'm working in this summer. In the book I'm reading (Shankar's "Principles of Quantum Mechanics") I just came across the term interaction Hamiltonian in describing how orbiting electrons interact with a magnetic field.

I have an idea of what it might mean, but I can't find a good explanation anywhere. What is an "interaction Hamiltonian", and how does it differ from a standard Hamiltonian?

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Any Hamiltonian contains kinetic and potential parts. Interaction means the potential part of the Hamiltonian.

Sometimes a part of interaction can be treated exactly together with the kinetic part. They form an approximative or "non perturbed" Hamiltonian. The rest of interaction is then treated perturbatively and is called "a perturbation". Interaction of the atomic electron with an external magnetic field is such part of the total Hamiltonian.

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Interaction Hamiltonian is a part of total Hamiltonian that contains time-dependent term containing function referring to some other physical system that is not part of the system being described but interacts with it. For example, in

$$ H = \frac{p^2}{2m} + U -\boldsymbol \mu \cdot \mathbf E(t) $$ the term $$ U $$ is potential energy, and $$ -\boldsymbol \mu \cdot \mathbf E(t) $$ is interaction Hamiltonian, because $\mathbf E(t)$ describes electric field which is not being described by the Hamiltonian, but interacts with the system.

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  • $\begingroup$ And the spin-orbit interaction term, it is not an interaction Hamiltonian? There is no external thing or other physical system in it, but it is still an interaction Hamiltonian. $\endgroup$
    – Vladimir Kalitvianski
    Commented Jun 18, 2014 at 8:07
  • $\begingroup$ If it does not depend on external variable (like external field), only on the variables of the system, I wouldn't call it interaction term, the same I would not call potential energy $U$ as interaction term, but I suppose others may do it differently. $\endgroup$
    – Ján Lalinský
    Commented Jun 18, 2014 at 11:16
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    $\begingroup$ Lalinsky: $U$ is the interaction because without it the kinetic terms sole describe a free motion. $\endgroup$
    – Vladimir Kalitvianski
    Commented Jun 18, 2014 at 12:24

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