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I am taking a Quantum Mechanics II course and we were studying the relativistic corrections to the hydrogen atoms in perturbation theory. I was looking at the assignment, and a question is as follows:

Consider the non-relativistic Hamiltonian for a charged particle $-e$, in presence of an external, constant and uniform magnetic field $$B=\hat{e}_3B_0$$ in the symmetrical gauge $A=0.5B \times r $

But I don't understand what a Gauge is in this context. If I wish to find the Hamiltonian, what information is that supposed to give me? Could you explain to me what a gauge is for someone who has not had gauge theory, or link me to some basic bibliography to learn it?

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  • $\begingroup$ Have you taken a course in electromagnetism? $\endgroup$
    – knzhou
    Jun 8, 2019 at 1:03
  • $\begingroup$ yes, i have taken a course in electromagnetism. Next semester I will take electromagnetic theory, so my understanding is relatively basic $\endgroup$ Jun 8, 2019 at 1:10
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/146585/2451 and links therein. $\endgroup$
    – Qmechanic
    Jun 8, 2019 at 3:20
  • $\begingroup$ There are many choices of $\vec A$ that lead to the same $\vec B$. Such choice is called a gauge. $\endgroup$
    – my2cts
    Jun 8, 2019 at 5:51

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A Hamiltonian of a particle in electromagnetic field is typically expressed via the 4-potential of electromagnetic field, which is defined up to a gauge transformation. The gauge in your case fixes the 4-potential.

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  • $\begingroup$ @my2cts : I do not disagree, but I don't quite see your point. This does not seem to be in contradiction with my answer. $\endgroup$
    – akhmeteli
    Jun 8, 2019 at 5:27
  • $\begingroup$ @my2cts: So why don't you post your comment as an answer? $\endgroup$
    – akhmeteli
    Jun 8, 2019 at 5:47

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