Quantum Mechanics�Lecture 3
Overall Lecture 9
CML101
August 20, 2024
Operators are Linear in Nature
Linear
Operators
Non-
Linear
Operators
Operators have to be Hermitian
Consequences of the Hermitian Property
This is also a powerful statement. In fact the eigenfunctions of a Hermitian operator are orthogonal and complete. This means that any function can be written as a linear combination of a set of eigenfunctions of a Hermitian operator. What does this mean? Say you have an oxygen atom and a hydrogen atom. And say you have solved the Schrödinger Equation for both these atoms. This means you know the eigenfunctions of oxygen and the eigenfunctions of hydrogen. (The eigenfunctions are what we chemists call orbitals!!) Since the Hamiltonian for the oxygen atom and the Hamiltonian for the hydrogen atom are Hermitian operators, it must be true that these set of eigenfunctions form a complete set!! So now lets say we want to solve for the eigenfunctions of a water molecule (that is two hydrogens and one oxygen!!) We can use the eigenfunctions of the single hydrogen and oxygen atoms as basis-functions to solve the problem for water!!! This would be convenient, because we have already solved part of the problem. In fact such, approaches are used every day in quantum chemistry to simplify problems.
Eigenvalues of Hermitian Operators are real
Proof:
Eigenfunctions of Hermitian Operators are Orthogonal
Proof:
Eigenfunctions that correspond to different eigenvalues of a hermitian operator are orthogonal.
POSTULATE 3
In any single measurement of the observable that corresponds to the operator Ậ, the only values that will ever be measured are the eigenvalues of that operator.
The eigenvalues satisfy the equation
Ậ ψ = aψ
Example of Postulate 4:
Example of Postulate 4
Problem:
Solution on next page!