Physics Lecture 13 3/23/ Andrew Brandt Monday March 23, 2009 Dr. Andrew Brandt 1.Loose ends from Ch. 4 Nuclear Motion+Lasers 2.QM Particle in a box 3.Finite Potential Well

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 p e +p N =0 so p e =-p N with KE N = p 2 /2M where the reduced mass is defined as For M=m get but in the case of Hydrogen so What about heavier atom? 3/23/ Andrew Brandt2

Electron Transition Example An electron makes a transition from n=2 state to n=1, find,  of emitted photon 3/23/ Andrew Brandt3

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] 3/23/ Andrew Brandt4

3/23/ Andrew Brandt5 Three Level Laser

Particle in a Box Again 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 3/23/ Andrew Brandt6

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= Does this make sense? 3/23/ Andrew Brandt7 Probability Wave Function

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 3/23/ Andrew Brandt8

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) 3/23/ Andrew Brandt9

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 3/23/ Andrew Brandt10