Unitary closed-form expressions

The structure of expansion (18) for two electrons in a uniform magnetic field was studied by Kandemir [34], In particular, he reduced the recurrence relation to a closed-form expression. Here, we present a more general analysis. In this derivation, a graphical approach, based on the ideas originally developed by Isaiah Shavitt in his graphical unitary group approach (GUGA) [44, 45], proved to be very useful. 

The historically first attempt to achieve the block-diagonalization of the Dirac Hamiltonian is due to Foldy and Wouthuysen and dates back to 1950 [609]. They derived the very important closed-form expressions for both the unitary transformation and the decoupled Hamiltonian for the case of a free particle without invoking something like the X-operator. Because of the discussion in the previous two sections, we can directly write down the final result since the free-particle X-operator of Eq. (11.10) and hence Uv=o = Uq are known. With the arbitrary phase of Eq. (11.23) being fixed to zero it is given by... 

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