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Unformatted text preview: Problems: 1. Periodic Table. What is the electronic structure of Ne, Z = 10. Make an educated guess about its chemical reactivity. What is the electronic structure of Oxygen Z = 8. What is the electronic structure of iron, Z = 26. Explain why there is 10 boxes in the middle of the periodic table. 2. V eff practice: In a strange parallel universe the attraction between the electron and proton is not the Coulomb Law but is V ( r ) = 1 2 kr 2 On the same graph, sketch the effective potential in this case for = 0 and = 1 and = 2. 3. Inflection points and classical solutions: Consider the 3 p state of hydrogen. Graph the effective potential in this case. Sketch the 3p radial wave function. Determine the inflection points of the radial wave function u nl the table of wave functions given in class is useful. (Ans. 1 . 06 a , 6 a , 16 . 93 a o ) 4. Classical orbits: Consider the classical orbits corresponding to the 3 p orbits, i.e. those orbits with L 2 = ( + 1) ~ 2 and energy E =- [ ~ 2 / (2 ma 2 o )]1 /n 2 . What is the maximum and minimum velocities of the electron in units of the speed of light, and what is the maximum radial velocity of the orbit. (Hint: Use the results of the previous that the classical turning points are at r = 3(3- 7) a o 1 . 06 a o and r = 3(3 + 7) a o 16 . 93 a o . Also use the result that the minimum of the effective potential is at r = ( + 1) a o as we showed previously. ) Extra Practice 1. Averages of PE, Angular KE, KE,: Determine the average angular kinetic energy, and the average potential energy, and the average KE, of the 2 p state of hydrogen. (Answers: ave PE=- 27 . 2eV / 4, ave angular KE = 13 . 6eV / 6, and finally we have: ave KE = E- V =- 13 . 6eV / 4 + 27 . 2eV / 4 As a a challenge compute the averager radial KE directly from Eq. (45) and show that ave KE = ave radial KE + ave angular KE 2. Average and variance of radius Determine the variance in radius of the 2 p state. 3. Verify solution: Show that the 2 p wave function satisfies the radial schr odinger equation 4. Sketch radial wave fcns: Sketch the 3 s , 3 p , 3 d radial wave functions. Why are the wave functions qualitatively different. 5. Wave function-Taxonomy: What is the total degeracy of n = 3 states. List the states . What is the squared angular momentum and the z-component of the angular momentum for each of these states. 1 2D Shrodinger Equation 1. In two dimensions the Schr odinger equations reads " P 2 x 2 m + P 2 y 2 m + V ( x,y ) # ( x,y ) = E ( x,y ) (1)- ~ 2 2 m 2 x 2 + 2 y 2 + V ( x,y ) ( x,y ) = E ( x,y ) (2) 2. For the particle in the two dimensional box the potential is V = ( inside box- L/ 2 < x,y < L/ 2 outside box (3) We solved this equation using separation of variables making an ansatz ( x,y ) = X ( x ) Y ( y ) and solving for the functions X and Y 3. We will discuss a square box L x = L y = L but you should be able to generalize this to a rectangular box and...
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This note was uploaded on 01/25/2012 for the course PHYSICS 251 taught by Professor Staff during the Fall '11 term at SUNY Stony Brook.
- Fall '11