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Physics 451

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

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Physics 451

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  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

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