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The Hamiltonian of a charged particle in a magnetic field is: $$H=\frac{1}{2m}(\frac{h}{2\pi i}\nabla-qA)^2+q\phi$$ Can anybody help me find out when, how and by whom was it first derived?

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Lorentz derived the Lagrangian and the Lorentz force formula with the magnetic term in his Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern (1895). There were precursors on the formula, Thomson, Heaviside and even Maxwell himself, but they did not use the Lagrangian. Switching from velocities to momenta and from the Lagrangian to the Hamiltonian is rather routine, it might have been done in passing by Einstein somewhere.

Getting the quantum Hamiltonian became trivial after the prescription of replacing $p$ by $\frac{h}{2\pi i}\nabla$ was introduced by Schrödinger in 1926. Kennard, for example, writes Hamiltonians for the electron in electric and magnetic fields in Zur Quantenmechanik einfacher Bewegungstypen (1927).

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