Quantum Theory - Particle Physics Group

CHAPTER 3. FORMALISM AND INTERPRETATION 42add up to the completeness relation∑|e i 〉〈e i | =½. (3.19)iWhile the product 〈v|w〉 of a covector and a vector yields a complex number, the tensor product|w〉〈v| is a matrix of rank 1 that is sometimes called dyadic product.For a linear transformation v → Av the components A i j of the matrix representationv i → A i jv j can be obtained by sandwiching the operator A between basis elements. Foran orthonormal basis 〈e i |e j 〉 = δ ij we can use the Kronecker–δ to pull all indices down so thatthe entries (elements) of the matrix A i j = e i (Ae j ) in Dirac notation becomeA ij = 〈e i |A|e j 〉. (3.20)In quantum mechanics the numbers 〈v|A|w〉 are hence called matrix elements even for arbitrarybra- and ket-vectors v and w. The normalized diagonal term〈A〉 v = 〈v|A|v〉〈v|v〉(3.21)is called expectation value of the operator A in the state |v〉, where the denominator canobviously be omitted if and only if |v〉 is normalized 〈v|v〉 = 1.Hermitian conjugation. If we apply a linear transformation v → Av to a vector v andevaluate a covector w, i.e. multiply with w from the left, the resulting number is〈w,Av〉 = w i A i j v j = 〈w| · A|v〉. (3.22)But we might just as well first perform the sum over i in w i A i j and then multiply the resultingbra-vector 〈w|A, with the ket-vector |v〉 from the right. In the language of linear algebra thisdefines the transposed map A T on the dual space V dual , which can be written as a matrixmultiplication w j → (A T ) i j w i with the transposed matrix A T . Using the non-degenerate innerproduct we can define the Hermitian conjugate A † of the linear operator A by(A † v,w) ≡ (v,Aw) ∀ v,w ∈ V. (3.23)Using (ϕ,ψ) = (ψ,ϕ) ∗ we obtain the matrix elements〈v|A|w〉 = 〈A † v|w〉 = (〈w|A † v〉) ∗ ⇒ 〈w|A † |v〉 = 〈v|A|w〉 ∗ . (3.24)For an orthonormal basis |e i 〉 the compoments become(A † ) ij = 〈e i |A † |e j 〉 = 〈e j |A|e i 〉 ∗ = A ∗ ji, (3.25)