- In The Standard Free Particle Dirac Picture Hamiltonian Is Given By H A P Bm In This Picture The A Matries Are Named 1 (57.44 KiB) Viewed 37 times
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In the standard free particle Dirac picture Hamiltonian is given by H = a.p+Bm. In this picture the a matries are named odd, while B, E, I are designated as even. This is because the a matrices connect the upper two components of any bi-spinor to the lower two components of another such bi-spinor. The even matrices only connect the upper two components to one another. Due to this, we can construct a representation differing from the standard Dirac, where the picture is entirely described by the even matrix B. This representation can be obtained by Foldy-Wouthuysen (F-W) transformation U, where U is defined by: E + m +ă.pl U with E = +102 + m2 2E(E + m) and with standard ã and B. (Note that only 2nd term in the numerator of U has an operator nature.) a) Show that U is unitary and H' = UHut = BE. Is H' what we expected? b) For a particle of well-defined momentum, it is more natural to work in momentum space, where xi = i... Applying the Heisenberg equation in the F-W representation, show that the velocity др operator is Ši = Bone Pi E