[Q]Degeneracy and Commutability

  1. Hi

    Liboff said that with two operator [A,B] = 0, If some eigenstate of operator A are degenerate, they are not necessarily completely common eigenstate of B.

    How is it proved or where can i find proof of this theorem?

    Please answer me.
     
  2. jcsd
  3. vanesch

    vanesch 6,236
    Staff Emeritus
    Science Advisor
    Gold Member

    Simple example: consider in C^3, the unity operator, 1, and the projector on the first coordinate Px (Px(x,y,z) = (x,0,0) )

    Clearly they commute: [1,Px] = 0 (as the unity operator commutes with all).
    now, consider the vector (1,2,0). This is an eigenstate of 1 (all vectors are eigenstates of 1). However, it is not an eigenstate of Px. Only vectors of the kind (x,0,0) are eigenstates of Px with eigenvalue 1, and vectors of the kind (0,y,z) are eigenstates of Px with eigenvalue 0. But (1,2,0) is neither.
     
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