(17.22) x'EV"XEV" XEV"X'EV" (17.23) where T is called the transition operator. The result in Eq. (17.22) shows that the matrix U in basis Wn (call it U) can be calculated from the matrix U in basis Vn according to the transformation: U=TUT+, (17.24) which is referred to as similarity transformation. It is easily established that T+T= TT+ = I in any basis (noticing that Tyx = Tyx does not imply T=T+, which is left as an exercise to show). Eigenvalues and eigenstates Given an operator A, any state /v) such that with a being a complex number, is called an eigenstate of A. The corresponding number a is called the eigenvalue of /v). The eigenvalues are the solutions of the characteristic equation A-AI = 0, where 121 (also sometimes noted det |Q1) is the determinant of the matrix Q.2 It is customary to label the eigenstates after their eigenvalues, thus, the eigenstate (2i) implicitly satisfies Alhi) = dilhi). Two or more eigenstates may have 2 For a 2 X 2 matrix e=(a), = the determinant is |Q|=ad - bc. For a general definition see, instance, http://en.wikipedia.org/wiki/Determinant. 17.1 Dirac notation the same eigenvalue, in which case the eigenvalue is said to be degenerate (in this case the corresponding eigenstates may be labeled as ...). We also have = thus, If A is Hermitian (A = A+), we have (2|A+A|hi) = = 22(2i/2i), which shows that 22 = 12il2, or that the eigenvalues of a Hermitian operator are real. It can be shown that any linear operator has at least one eigenvalue and eigenvector (the characteristic equation having at least one root). The complete set of eigenstates defines the operator's eigenspace. A key prop- erty of the eigenstates (2i) of any operator A is that they form an orthonormal basis, i.e., (2i/2j) = Sij. Therefore, the completeness relation applies to the set of eigenstates: (17.25) Using the completeness relation, we can write the matrix elements of A in the form: (17.26) which gives (17.27) The definition in Eq. (17.27), which decomposes A into a sum of eigenstate projectors /2i)(2il, is c