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Physics 451

Physics 451. Quantum mechanics Fall 2012. Karine Chesnel. Phys 451. Announcements. Test 1 next week Mo Sep 24 – Th Sep 27. Today: Review - Monday: Practice test Be prepared to present the solution of your chosen problem during class (~ 5 to 10 min). Phys 451. Review lectures,

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Physics 451

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  1. Physics 451 Quantum mechanics Fall 2012 Karine Chesnel

  2. Phys 451 Announcements Test 1 next week Mo Sep 24 – Th Sep 27 • Today: Review - Monday: Practice test • Be prepared to present the solution of • your chosen problem during class (~ 5 to 10 min)

  3. Phys 451 Review lectures, Homework and sample test EXAM I • Time limited: 3 hours • Closed book • Closed notes • Useful formulae provided

  4. Phys 451 EXAM I • Wave function, probabilities • and expectation values • 2. Time-independent Schrödinger equation • 3. Infinite square well • 4. Harmonic oscillator • 5. Free particle

  5. Phys 451 Review I What to remember?

  6. Normalization: Quantum mechanics Review I 1. Wave function and expectation values Density of probability

  7. “Operator” x “Operator” p Quantum mechanics Review I 1. Wave function and expectation values

  8. Quiz 9a What is the correct expression for the operator T= Kinetic energy? A. B. C. D. E.

  9. Heisenberg’s Uncertainty principle Quantum mechanics Review I 1. Wave function and expectation values Variance:

  10. Here The potential is independent of time “Stationary state” General solution: Quantum mechanics Review I 2. Time-independent Schrödinger equation

  11. Function of time only Function of space only Stationary state Solution: Quantum mechanics Review I 2. Time-independent Schrödinger equation

  12. A general solution is is independent of time where Quantum mechanics Review I 2. Time-independent Schrödinger equation for each Stationary state

  13. with The particle can only exist in this region Quantum mechanics Review I 3. Infinite square well 0 a x

  14. Excited states Quantization of the energy Ground state Quantum mechanics Review I 3. Infinite square well 0 a x

  15. Quantum mechanics Review I 3. Infinite square well

  16. Quiz 9b A. 0 B. 1 C. 0.5 D. 0.3 E. The particle is in this sinusoidal state. What is the probability of measuring the energy E0 in this state? 0 a x

  17. V(x) • Operator position • Operator momentum or x Quantum mechanics Review I 4. Harmonic oscillator

  18. Raising operator: Lowering operator: Quantum mechanics Review I 4. Harmonic oscillator Ladder operators:

  19. Quantum mechanics Review I with Wave packet 5. Free particle

  20. Quantum mechanics Review I 1. Identify the initial wave function 2. Calculate the Fourier transform 3. Estimate the wave function at later times Free particle Method:

  21. Quantum mechanics Review I here Dispersion relation here Physical interpretation: • velocity of the each wave at given k: • velocity of the wave packet: 5. Free particle

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