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Physics 3313 - Lecture 13

Physics 3313 - Lecture 13. Monday March 23, 2009 Dr. Andrew Brandt. Loose ends from Ch. 4 Nuclear Motion+Lasers QM Particle in a box Finite Potential Well. Nuclear Motion. In Bohr atom, we have implicitly assumed nucleus is fixed, since we only considered electron KE.

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Physics 3313 - Lecture 13

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  1. Physics 3313 - Lecture 13 Monday March 23, 2009 Dr. Andrew Brandt • Loose ends from Ch. 4 Nuclear Motion+Lasers • QM Particle in a box • Finite Potential Well 3313 Andrew Brandt

  2. Nuclear Motion • In Bohr atom, we have implicitly assumed nucleus is fixed, since we only considered electron KE. • Since nucleus does not (quite) have infinite mass, it will have motion if the total momentum of the atom is zero • pe+pN=0 so pe=-pN with KEN= p2/2M • where the reduced mass is defined as • For M=m get but in the case of Hydrogen so • What about heavier atom? 3313 Andrew Brandt

  3. Electron Transition Example • An electron makes a transition from n=2 state to n=1, find ,  of emitted photon 3313 Andrew Brandt

  4. Laser • Light Amplification by Stimulated Emission of Radiation • laser light is monochromatic (one color), coherent (all in phase), can be very intense, small divergence (shine laser on mirror left on moon) [I knew I forgot something] 3313 Andrew Brandt

  5. Three Level Laser 3313 Andrew Brandt

  6. Particle in a Box Again • http://user.mc.net/~buckeroo/PODB.html • Solutions: • Would have cos also, but boundary condition at x=0 implies coefficient =0 • Use boundary condition at x=L gives • which is equivalent to classical expression • Final wave function 3313 Andrew Brandt

  7. Particle in a Box Example • What is the probability that a particle in a box is between 0.45L and 0.55L for n=1? n=2? • What is the classical answer? • 10% since this is 1/10 of the length of the box • with • Integrating gives • for n=1 P=0.198 (about twice expectation), while for n=2 P=0.0065 • Does this make sense? Wave Function Probability 3313 Andrew Brandt

  8. Particle in a 3-D Box • Need 3-D Schrodinger Equation: • Factorizes into product of 3 1-D equations so • Note 3-fold degeneracy for one dimension in n=2 state • For rectangular box 3313 Andrew Brandt

  9. Finite Potential Well • Classically if E<U than particle bounces off sides, but quantum mechanically, particle can penetrate into regions I and III • For I rewrite as • With • Solutions are and • What about in the box? Since U=0 with (this is similar to infinite potential well, aka particle in box) 3313 Andrew Brandt

  10. Finite Well BC • At x=- =0 so for D must be 0 ,so • Similarly at x=+ =0 so for F must be 0 and • Finally what about boundary conditions for ? Is it 0 at x=0? • Nope at x=0 • And at x=L • But so too many unknowns! Should we quit? • Need other constraints. Derivatives must also be continuous (match slopes) • After some math again get specific energy levels, but wavelengths a little longer than infinite well and from De Broglie, this means momentum and thus energy is smaller 3313 Andrew Brandt

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