1 / 16

Quantum Mechanics Fundamentals and Uncertainty Principles

Explore the foundational concepts in quantum mechanics, including eigenstates, statistical interpretation, uncertainty principles, Dirac notation, and Heisenberg's equations of motion. Learn how to calculate and interpret observables and wave functions.

kaylat
Download Presentation

Quantum Mechanics Fundamentals and Uncertainty Principles

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Physics 451 Quantum mechanics I Fall 2012 Oct 19, 2012 Karine Chesnel

  2. Phys 451 • Next homework assignments: • HW # 14 due Friday Oct 19 by 7pm • Pb 3.7, 3.9, 3.10, 3.11, A26 • HW #15 due Tuesday Oct 23 • Pb 3.13, 3.14, 3.17, 3.18, 3.22, 3.23 Practice test 2 M Oct 22 Work with your group! Test 2: Tu Oct 23 – Fri Oct 26

  3. We measure an observable (Hermitian operator) • Operator’s eigenstates: Discrete spectrum Continuous spectrum eigenvector eigenvalue Phys 451 Generalized statistical interpretation • Particle in a given state Eigenvectors are complete:

  4. Phys 451 Generalized statistical interpretation Particle in a given state Fourier’s trick to find Cn • Normalization: • Expectation value

  5. Phys 451 Quiz 18 If you measure an observable Q on a particle in a certain state , what result will you get? • the expectation value • one of the eigenvalues of Q • the average of all eigenvalues • A combination of eigenvalues • with their respective probabilities

  6. Phys 451 Generalized statistical interpretation Operator ‘position’: Probability of finding the particle at x=y:

  7. Phys 451 Generalized statistical interpretation Operator ‘momentum’: Probability of measuring momentum p: Pb 3.11: probability of measuring p in a given range

  8. Phys 451 The Dirac notation Different notations to express the wave function: • Projection on the position eigenstates • Projection on the momentum eigenstates • Projection on the energy eigenstates

  9. Quantum mechanics The uncertainty principle Finding a relationship between standard deviations for a pair of observables Uncertainty applies only for incompatible observables

  10. Quantum mechanics The uncertainty principle Position - momentum

  11. Quantum mechanics The uncertainty principle Position - Energy Pb 3.14

  12. Quantum mechanics The uncertainty principle Energy - time Special meaning of Dt

  13. Quantum mechanics Quiz 19 An excited particle emits a certain radiation of energy E with a band width DE. What can we say about the characteristic lifetime of excited state? • Lifetime is a least • Lifetime is a most • Lifetime is a least • Lifetime is a most

  14. constant Ehrenfest’s theorem Quantum mechanics Heisenberg equation of motion Pb 3.17

  15. when Definition for Dt: To evaluate Dt: • choose an appropriate operator • calculate and Quantum mechanics Heisenberg equation of motion Pb 3.18 Application: use Q = x, in the case of the infinite square well

  16. “bra” “ket” Inner product: Operator: Projection operator: for orthonormal basis Quantum mechanics The Dirac notation Pb 3.22 Pb 3.23 or

More Related