1. Physics 451 Quantum mechanics I Fall 2012 Oct 12, 2012 Karine Chesnel

2. Announcements

3. Homework next week: HW # 13 due Tuesday Oct 16 Pb 3.3, 3.5, A18, A19, A23, A25 HW #14 due Thursday Oct 18 Pb 3.7, 3.9, 3.10, 3.11, A26 Announcements Quantum mechanics

4. 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

5. Quantum mechanics Hilbert space Inner product Norm Schwarz inequality Orthonormality

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

7. Quantum mechanics Determinate states Stationary states – determinate energy Generalization of Determinate state: Standard deviation: For determinate state: operator eigenstateeigenvalue

8. Quantum mechanics Quiz 16 Since any wave function can be written as a linear combination of determinate states (stationary states), for which we can write The wave function is itself a determinate state and we can write A. True B. False

9. Quantum mechanics Eigenvectors & eigenvalues For a given transformation T, there are “special” vectors for which: is transformed into a scalar multiple of itself is an eigenvector of T is an eigenvalue of T

10. Quantum mechanics Eigenvectors & eigenvalues To find the eigenvalues: We get a N th polynomial in : characteristic equation Find the N roots Spectrum

11. Quantum mechanics Hermitian transformations Hermitian operator: 1. The eigenvalues are real 2. The eigenvectors corresponding to distinct eigenvalues are orthogonal 3. The eigenvectors span the space