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Can someone help me understand the Hamiltonian on the attached picture. What does the notation with the annihilation and creating operators written in a row vector exactly mean? Does it mean I should just take the dot product as written on the picture? Evidently it doesn't since this just gives two anticommutators each equaling 1.

Also I would like it if someone can explain a bit about the Hamiltonian on the picture or can link to a place, where they expalin it intuitively :)

Also I would like it if someone can explain a bit about the Hamiltonian on the picture or can link to a place, where they expalin it intuitively :)