Physics 451 Quantum mechanics I Fall 2012 Oct 8, 2012 Karine Chesnel

Announcements Quantum mechanics Homework this week: HW # 12 due Thursday Oct 11 by 7pm A8, A9, A11, A14, 3.1, 3.2

A friendly message from the TA to the students: I have noticed in recent homeworks that more students quit to do entire problem(s). They are either short in time or overwhelmed by the length of the problems. It is understandable that this is an intense course, and the homework is time consuming. And as it is approaching the middle of the semester, all kinds of things are coming. But please be strong and do your best to learn. If you are really out of time, do as much as you can. Anyway, we don't want students to give up. Quantum mechanics

Review- Matrices Generalization (N-space) Linear transformation Matrix Hermitian conjugate Unit matrix Inverse matrix Unitary matrix Transpose Conjugate Physical space i j k i’ j’ k’ Old basisNew basis Expressing same transformation T in different bases

Homework- algebra Pb A8manipulate matrices, commutator transpose, Hermitian conjugate inverse matrix Pb A9scalar matrix Pb A11 matrix product Pb A14 transformation: rotation by angle , rotation by angle 180º reflection through a plane matrix orthogonal

Need for a formalism Quantum mechanics Linear transformation (matrix) Operators Wave function Vector

Quantum mechanics Formalism N-dimensional space: basis Operator acting on a wave vector:Expectation value/ Inner product Norm: If T is Hermitian

Quantum mechanics Hilbert space N-dimensional space Wave function are normalized: Infinite- dimensional space Hilbert space: functions f(x) such as Wave functions live in Hilbert space Pb 3.1, 3.2

Quantum mechanics Hilbert space Inner product Norm Schwarz inequality Orthonormality

Quantum mechanics Determinate states Stationary states – determinate energy Generalization of Determinate state: Standard deviation: For determinate state For a given operator Q: operator eigenstateeigenvalue

Quantum mechanics Hermitian operators Observable - operator Expectation value since Observables are Hermitian operators Examples: For any f and g functions