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Understanding Quantum Mechanics: Hilbert Space and Eigenvalues

Explore the concept of Hilbert space in quantum mechanics, where wave functions live and are normalized. Learn about observables as Hermitian operators, determinate states, and eigenvalues. Solve for eigenvectors using spectral analysis. Discover the properties of Hermitian transformations.

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Understanding Quantum Mechanics: Hilbert Space and Eigenvalues

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  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

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