Hermiticityand orthogonalityNow consider two eigenfunctions [m)and |n).We have0[m)=qm[m)|n)=q,[n)and(nl0|m)=qm(n|m)(m|0|n)=q.(m|n)(m[0|n)=qm (m|n)wherethelastlineontheleftcomesfromtakingthecomplexconjugateSubtracting,wefind0=(q, -qm )(m|n)andfromthiswecandeduce(a) If m = n, then (m|n) = (mm) ± 0, so qm = qm * and qm is real.(b) If qm ± qn, then since both are real, qn - qm *+ 0 and (m|n) = 0Hermitian operator ensures that the eigenvalue of the operator is a real number
Hermiticity and orthogonality Now consider two eigenfunctions |𝑚ۧ and |𝑛ۧ . We have and where the last line on the left comes from taking the complex conjugate. Subtracting, we find and from this we can deduce (a) If m = n, then 𝑚 𝑛 = 𝑚 𝑚 ≠ 0, so 𝑞𝑚 = 𝑞𝑚 ∗ and 𝑞𝑚 is real. (b) If 𝑞𝑚 ≠ 𝑞𝑛 , then since both are real, 𝑞𝑛 − 𝑞𝑚 ∗ ≠ 0 and 𝑚 𝑛 = 0. * ˆ ˆ ˆ m m m Q m q m n Q m q n m m Q n q m n ˆ ˆ n n Q n q n m Q n q m n * 0 n m q q m n Hermitian operator ensures that the eigenvalue of the operator is a real number
The eigenvalue of aHermitian operator is a real numberProof:Ay = ay[y'Aydt=Jy(Ay)'dtajlyPdt=aflyP dtIyP≥0but :?0a,=a,Quantum mechanical operators have to have real eigenvalues
The eigenvalue of a Hermitian operator is a real number Proof: Quantum mechanical operators have to have real eigenvalues * * ˆ ˆ A d A d ( ) but : A a ˆ 2 * 2 a d a d | | | | 0 2 | | 0 * i i a a