SECTION 11.4 (More Quantum Formalism*) 11.16 . The Hamiltonian operator is often described as. linear operator because it has the linear property that AH (x) + BH$(x) for 11.2 H[A (x) + B$(x)] = am %3D any two functions (x) and $(x) and any two constant numbers A and B. Prove that the one-dimensional Hamiltonian (11.11) is linear. 11.17. Show that the wave functions (11.21) for the infi- nite square well satisfy the orthogonality property that m(x)*\n(x) dx = 0 for m + n. %3D SECTION 115 (T Section 11.4 More Quantum Formalism 339 nensional system for now.) The form dy as a differential equation, but it is not the form in which the equation is normally written in more advanced work, where it is ctric usually reorganized to read onal h? d? + U(x) \µ = Ep 2m dx? %3D (11.10) The quantity in the square brackets on the left is a differential operator, or just Thetor: "Operating" on a function , it gives the function -(k²/2m) 99 Or " + Uy. This operator is so important, it is given a name, the Hamiltonian operator, or just Hamiltonian, * and is denoted H f h2 d? + U(x) (11.11) %3D 2m dx2 The Hamiltonian is intimately connected to the energy of the system. For example, if is the plane wave y = e'kx (11.12) = [K+ U] + U 2m %3D h? d? %3D + U \eikx Hy = 2m dx² times giver (K U)

SECTION 11.4 (More Quantum Formalism) 11.16 . The Hamiltonian operator is often described as. linear operator because it has the linear property that AH (x) + BH$(x) for 11.2 H[A (x) + B$(x)] = am %3D any two functions (x) and $(x) and any two constant numbers A and B. Prove that the one-dimensional Hamiltonian (11.11) is linear. 11.17. Show that the wave functions (11.21) for the infi- nite square well satisfy the orthogonality property that m(x)\n(x) dx = 0 for m + n. %3D SECTION 115 (T Section 11.4 More Quantum Formalism 339 nensional system for now.) The form dy as a differential equation, but it is not the form in which the equation is normally written in more advanced work, where it is ctric usually reorganized to read onal h? d? + U(x) \µ = Ep 2m dx? %3D (11.10) The quantity in the square brackets on the left is a differential operator, or just Thetor: "Operating" on a function , it gives the function -(k²/2m) 99 Or " + Uy. This operator is so important, it is given a name, the Hamiltonian operator, or just Hamiltonian, * and is denoted H f h2 d? + U(x) (11.11) %3D 2m dx2 The Hamiltonian is intimately connected to the energy of the system. For example, if is the plane wave y = e'kx (11.12) = [K+ U] + U 2m %3D h? d? %3D + U \eikx Hy = 2m dx² times giver (K U)