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

2. Announcements Quantum mechanics Homework this week: HW # 11 due Tuesday Oct 9 by 7pm 2.38, 2.39, 2.41, A1, A2, A5, A7 HW # 12 due Thursday Oct 11 by 7pm A8, A9, A11, A14, 3.1, 3.2

3. Test 1 Quantum mechanics Class average 78 / 100 Highest score: 95/100 Pb 1 17.6 / 20 Pb 2 13.5 / 20 Pb 316 / 20 Pb 414 / 20 Pb 516.8 / 20

4. Need for a formalism Quantum mechanics

5. Vectors i j k Physical spaceGeneralization (N-space) Addition - commutative - associative Scalar multiplication linear combination zero vector basis of vectors

6. Quantum mechanics Inner Product i j k Physical space  A B “Dot product” Generalization (N-space) “Inner product” Norm Orthogonality Orthonormal basis Schwarz inequality

7. Quantum mechanics Matrices Physical space i j k  A A’ Transformations: - Multiplication - rotation - symmetric… A’’ Generalization (N-space) Linear transformation Matrix Sum Product Hermitian conjugate Unit matrix Inverse matrix Unitary matrix Transpose Conjugate

8. Quantum mechanics Changing bases Physical space i j k i’ j’ k’ Generalization (N-space) Old basisNew basis Expressing same transformation T in different bases Same determinant Same trace

9. Quantum mechanics Formalism N-dimensional space: basis Operator acting on a wave vector:Expectation value/ Inner product Norm: For Hermitian operators:

10. Homework- Appendix Pb A1manipulate vectors Check properties of vectorial space (sum, scalar multiplication, zero vector…) Pb A5 Proof of Schwarz inequality use Pb A14 Proof of triangular inequality use Pb A2 Check vectorial properties for group of polynomial functions (sum, scalar multiplication, zero vector…)

11. Homework- Appendix 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