Review Outline matter waves Schroedinger eqn requirements Probability uncertainty
Heisenberg Uncertainty Principle What it does not mean: It does not mean you cant 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 x is large enough, an electron will pass thru both of two slits and interfere with itself
Another Uncertainty Principle What it means If you only have a small time t to measure energy, you cant accurately measure energy. If a particle only lives for a short time t, you cant accurately measure its energy. Since E=mc 2, you cant accurately measure its mass! For a short enough period of time t, you can violate conservation of energy by E.
Example Suppose the rest mass of a particle is 1200 MeV/c 2, 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
Wavefunction requirements Uncertainty principle is a mathematical certainty. Physical requirements on : Schroed eq new law, like Newtons Law continuous prob not depend on infinitesimal diff continuous d /dx finite d 2 /dx 2 in Schroed Eq finite and must go to zero at + and – infinity square integrable finite prob
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.
Free Particle by Analogy Solutions to wave eqn for sound are sin and cos Will that work here? Tryin Wont work: Left side will be cos, right side will be sin, wont be equal for all x and t. Left side will be imaginary, right side will be real, cant be equal ever.
Free Particle with Momentum What will work here? Tryin Works if E = p 2 /2m E = K, since V=0. This eqn is not relativistic.
Momentum Direction What is the difference between and Phase Velocity: One on the left has v ph = /k = f. One on right has v ph = - /k = -f. it moves in the negative direction. kinetic energy is the same