0

In elementary quantum mechanics, we know that when the system possess continuous rotation symmetry, the angular momentum is conserved, and the Hamiltonian of the system commutes with angular momentum operators ( $ [H, J^2 ] = 0 $ and $[H, J_z]$ ), and vice versa (I believe). In this situation we can use their eigenvalues ($j$ and $m_j$) to refer to the states since they are preserved. However, in some papers I read (for example, this paper), these notations are used in semiconductors to refer to different bands (LH and HH), while, to me, continuous rotation symmetry obviously does not exist in crystals. Therefore I wonder what am I missing here?

Qmechanic
  • 220,844
Frank Wang
  • 111

1 Answers1

1

Thinking within the framework of tight-binding approximation, one can tie different bands to the states of isolated atoms, which are labeled by their angular momentum (i.e., s-states, p-states, d-states, etc.) This terminology thus penetrates into labeling semiconductor bands, even though the angular momentum is not conserved. If I am not mistaken, this language is used even by Kittel.

This analogy is sometimes taken even further, e.g., when discussing the total momentum of holes (i.e., their spin plus their "angular momentum"), and when discussing the selection rules for light absorption and exciton formation. As I have already said, this makes perfect sense within the tight-binding picture, but needs not be taken too literally.

Unfortunately, I am not in a position to comment about the relation of this notation and the crystal symmetries (some of which are rotational symmetries).

Roger V.
  • 68,984