Formalism of Quantum Mechanics 2006 Quantum MechanicsProf. Y. F. Chen Formalism of Quantum Mechanics

linear DE. ： the main foundation of QM consists in the Schrödinger eq. The formalism of QM deals with linear operators & wave functions that form a Hilbert space ch4 will focus on the Hermitian operators & the superposition properties of linear DE. in Hilbert space 2006 Quantum MechanicsProf. Y. F. Chen Formalism of Quantum Mechanics

inhomogeneous linear differential ： (1) = linear differential operator acting upon (2) = eigenvalue & = eigenfunction (3) is a weight function any function in this vector space can be expanded as, =a set of linearly indep. basis functions inner product ： 2006 Quantum MechanicsProf. Y. F. Chen Definition of Inner Product & Hilbert Space Formalism of Quantum Mechanics

orthogonal ： if, then & are orthogonal. the norm of ： a basis of orthnormal, linearly independent basis functions satisfies 2006 Quantum MechanicsProf. Y. F. Chen Definition of Inner Product & Hilbert Space Formalism of Quantum Mechanics

Gram-Schmidt orthogonalization ： = linearly independent, not orthonormal basis = orthonormal basis produced by the Gram-Schmidt orthogonalization, in which is to be normalized → 2006 Quantum MechanicsProf. Y. F. Chen Gram-Schmidt Orthogonalization Formalism of Quantum Mechanics

although the Gram-Schmidt procedure constructed an orthonormal set, are not unique. There is an infinite number of possible orthonormal sets. construct the first three orthonormal functions over the range ： 2006 Quantum MechanicsProf. Y. F. Chen Gram-Schmidt Orthogonalization Formalism of Quantum Mechanics

→ it can be shown that where is the nth-order Legendre polynomials the eq. for Gram-Schmidt orthogonalization tend to be ill-conditioned because of the subtractions. A method for avoiding this difficulty is to use the polynomial recurrence relation 2006 Quantum MechanicsProf. Y. F. Chen Gram-Schmidt Orthogonalization Formalism of Quantum Mechanics

the adjoint/Hermitian conjugate of a matrix A ： from inner product space, the definition of the adjoint ： the adjoint of an operator in inner product function ： 2006 Quantum MechanicsProf. Y. F. Chen Definition of Self-Adjoint (Hermitian Operators) Formalism of Quantum Mechanics

self-adjoint/Hermitian operator ： →(1) →(2) measurement of the physical quantity ： (1) →, is real (2) is not necessarily an eigenfunction of 2006 Quantum MechanicsProf. Y. F. Chen Definition of Self-Adjoint (Hermitian Operators) Formalism of Quantum Mechanics

(1) the eigenvalues of an hermitian operator are real (2) the eigenfunctions of an hermitian operator are orthogonal (3) the eigenfunctions of an hermitian operator form a complete set proof (1) & (2) ： × 2006 Quantum MechanicsProf. Y. F. Chen The Properties of Hermitian Operators Formalism of Quantum Mechanics integrating complex conjugate

proof (1) & (2) ： ∵ ∴ → if i=j, then → → is real if i ≠ j, then → & are orthogonal ∵ the eigenfunctions of an hermitian operator form a complete set ∴ any function 2006 Quantum MechanicsProf. Y. F. Chen The Properties of Hermitian Operators Formalism of Quantum Mechanics

general form of SL eq. ： with,where p(x), q(x), and r(x) are real functions of x Ex. Legendre’s eq. ： & eigenvalues l(l+1) linear operator that are self-adjoint can be written in the form ： linear operator=Hermitian over [a,b] satisfies BCs ： 2006 Quantum MechanicsProf. Y. F. Chen The Sturm-Liouville Eq. Formalism of Quantum Mechanics

BCs ： (1) → the wave with fixed ends (2) → the wave with free ends (3) → the periodic wave show that subject to the BCs, the SL operator is Hermitian over [a, b] ： putting into → 2006 Quantum MechanicsProf. Y. F. Chen The Sturm-Liouville Eq. Formalism of Quantum Mechanics

2006 Quantum MechanicsProf. Y. F. Chen The Sturm-Liouville Eq. Formalism of Quantum Mechanics integrating by parts for the first term & using the BCs → → the SL operators is Hermitian over the prescribed interval

2006 Quantum MechanicsProf. Y. F. Chen Transforming an Eq. into SL Form Formalism of Quantum Mechanics any eq. can be put into self-adjoin form by introducing in place of proof ： Let → → to satisfy the requirement of SL eq. form for

2006 Quantum MechanicsProf. Y. F. Chen Transforming an Eq. into SL Form Formalism of Quantum Mechanics rewrite eq. as the SL form for ： with