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PH 301

PH 301. Dr. Cecilia Vogel Lecture 11. Review. Probability uncertainty. matter waves Schroedinger eqn requirements. Outline. Heisenberg Uncertainty Principle. What it does not mean: It does not mean you can’t measure position ( or momentum) very precisely.

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PH 301

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  1. PH 301 Dr. Cecilia Vogel Lecture 11

  2. Review • Probability • uncertainty • matter waves • Schroedinger eqn • requirements Outline

  3. Heisenberg Uncertainty Principle What it does not mean: • It does not mean you can’t measure position (or momentum) very precisely. • It does not mean you need better measuring instruments. • It does NOT just a matter of not knowing: If Dx is large enough, an electron will pass thru both of two slits and interfere with itself

  4. Another Uncertainty Principle • What it means • If you only have a small time Dt to measure energy, you can’t accurately measure energy. • If a particle only lives for a short timeDt, you can’t accurately measure its energy. • Since E=mc2, you can’t accurately measure its mass! • For a short enough period of time Dt, you can violate conservation of energy by DE.

  5. Example Suppose the rest mass of a particle is 1200 MeV/c2, and its lifetime at rest is 410 ns. A) Find the uncertainty in its rest energy, due to the fact that you only have 410 ns to measure it. B) Find the uncertainty in its rest mass. C) Is this a substantial fraction of its rest mass? No

  6. Wavefunction requirements • Uncertainty principle • is a mathematical certainty. • Physical requirements on Y: • Schroed eq • new law, like Newton’s Law • continuous • prob not depend on infinitesimal diff • continuous dY/dx • finite dY2/dx2 in Schroed Eq • finite and must go to zero at + and – infinity • square integrable — finite prob

  7. Free Particle • A free particle feels no forces • V=0 everywhere. • not ever precisely true, but useful. • The Schroedinger eqn is a wave eqn • Very much like wave equation for sound, • except for the i.

  8. Free Particle by Analogy • Solutions to wave eqn for sound are sin and cos • Will that work here? Try in • Won’t work: • Left side will be cos, right side will be sin, won’t be equal for all x and t. • Left side will be imaginary, right side will be real, can’t be equal ever.

  9. Free Particle with Momentum • What will work here? Try in • Works if E = p2/2m • E = K, since V=0. This eqn is not relativistic.

  10. Momentum Direction • What is the difference between and • Phase Velocity: • One on the left has vph = w/k = fl. • One on right has vph = -w/k = -fl. • it moves in the negative direction. • kinetic energy is the same

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