Physics 451

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

2. Announcements Homecoming

3. Quantum mechanics Announcements • 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

4. Infinite- dimensional space Wave function are normalized: Hilbert space: functions f(x) such as Quantum mechanics Hilbert space N-dimensional space Wave functions live in Hilbert space

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

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

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

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 • True • B. False

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

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

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