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Physics 451

Physics 451. Quantum mechanics I Fall 2012. Sep 10, 2012 Karine Chesnel. Homework. Homework 4: T Sep 11 by 7pm Pb 1.9, 1.14, 2.1, 2.2 Homework 5: Th Sep 13 by 7pm Pb 2.4, 2.5, 2.7, 2.8. Quantum mechanics. Announcements. Quantum mechanics.

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Physics 451

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  1. Physics 451 Quantum mechanics I Fall 2012 Sep 10, 2012 Karine Chesnel

  2. Homework • Homework 4: T Sep 11 by 7pm • Pb 1.9, 1.14, 2.1, 2.2 • Homework 5: Th Sep 13 by 7pm • Pb 2.4, 2.5, 2.7, 2.8 Quantum mechanics Announcements

  3. Quantum mechanics No student assigned to the following transmitters: 2214B68 17A79020 1E5C6E2C 1E71A9C6 Please register your i-clicker at the class website!

  4. Quantum mechanics Ch 1.6 Quiz 4a Uncertainty principle Which statement is accurate for these electronic wave functions? • Both the position x and the momentum are fairly well defined • The position of the particle is fairly well defined but the momentum is poorly defined • The momentum of the particle is fairly well defined but the position is poorly defined • Both the position and the momentum are poorly defined.

  5. Quantum mechanics Ch 1.6 Momentum De Broglie formula 1924 particle wave Heisenberg’s uncertainty Principle 1927 Uncertainty principle Position x

  6. Quantum mechanics Ch 1.6 • Calculate and • Calculate and • Estimate the product • Compare to Pb 1.9 Uncertainty principle How to check the uncertainty principle?

  7. Quantum mechanics Ch 1 Density of probability Probability between two points where Pb 1.14 Probability current

  8. Quantum mechanics Ch 2.1 In general Here function of x only The potential is independent of time “Stationary state” General solution: Time-independent Schrödinger equation

  9. Quantum mechanics Ch 2.1 Function of time only Function of space only Time-independent Schrödinger equation Plugging the general solution: in the Schrödinger equation

  10. Quantum mechanics Ch 2.1 General solution: Time-independent Schrödinger equation • Time dependent part:

  11. Quantum mechanics Ch 2.1 Stationary state Global solution: Time-independent Schrödinger equation • Space dependent part: Solution y(x) depends on the potential function V(x).

  12. Quantum mechanics Quiz 3b “If the particle is in one stationary state, its expectation value for position is not changing in time.” A. True B. False

  13. Quantum mechanics Ch 2.1 with is independent of time The expectation value for the momentum is always zero In a stationary state! are zero!) (Side note: does not mean that and Stationary states Properties: • Expectation values are not changing in time (“stationary”):

  14. Quantum mechanics Ch 2.1 Stationary states Properties: • Hamiltonian operator - energy

  15. Quantum mechanics Ch 2.1 where • Associated expectation value for energy Stationary states • General solution

  16. Quantum mechanics Ch 2.1 Pb 2.1 Pb 2.2 Stationary states a) En must be real b) yn(x) can always be real c) yn(x) is either real or odd, when V(x) is even Classical analogy: The kinetic energy is always positive! However, in QM, it is possible that at some locations x

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