Assignment #1 Chemistry 314 Summer 2008

Transcription

1 Assignment #1 Due Thursday, July 17. Hand in for grading, including especially the graphs and tables of values for question This problem develops the classical treatment of the harmonic oscillator. Consider a mass m attached to one end of a spring and the other end of the spring rigidly fixed to a wall. We will ignore gravity and only consider motion of the mass in one dimension along the coordinate x. If the mass is at position x0 when the spring is undistorted, then when the spring is stretched or compressed, Hooke s law says that the restoring force is proportional to the displacement ξ = x-x 0 from the equilibrium position: F = -k(x-x 0 ) = -kξ Newton s second law says that F = ma where a is the acceleration. The acceleration is defined as the rate of change of the velocity, v, and velocity is defined as the rate of change of the position: a = dv dt and v = d" dt m x 0 x (a) Combine these equations to obtain the following differential equation for the position of the particle: m d 2 " dt + k" = 0. 2 (b) Given the initial conditions for t = 0, v = v 0 and ξ = 0, show that the position of the mass as a function of time is given by 1 2 # m& ) "( t) = v 0 % ( sin k 1 2 # &, + % ( t. $ k ' * $ m' - Hint: Substitute the given solution into the differential equation to show that the differential equation is satisfied. Note that it is always easier to show that a given solution is correct than to find the solution from scratch! PLEASE TURN OVER. CONTINUED ON BACK.

2 2. Prove that % y(x,t) = Asin 2" & ' # ( x $ vt) ( ) * is a wave of wavelength λ and frequency ν = v/λ traveling to the right with a velocity v. (Note that the symbol v is the Roman letter vee, and the symbol ν is the Greek letter nu.) Hint: Plot the wave, y versus x for several different values of t, for example for t = 0, 0.1/ν, 0.2/ν, etc. Since λ could be anything, you will need to choose x values in terms of λ, for example use x=0, 0.05λ, 0.01λ, 2λ to get reasonable plots. I suggest using a spreadsheet for this rather than a calculator. Armed with these plots, you should be able to demonstrate what is asked. 3. Problem P2.6 from Engel, Quantum Chemistry and Spectroscopy. Delete the 2 in thhe second term of the wave amplitude function. Hint: Rather than re-expressing the sine function with trigonometric identities, it is easier to use the Euler relationship (Engel Appendix A) to express it in terms of imaginary exponentials.

3 Assignment #2 Due Tues., July 22. Problems from Engel, Quantum Chemistry and Spectroscopy. Read Chapter 1. P1.1 P1.2 P1.4. Compare this wavelength to the debroglie wavelength of a 45 g golf ball travelling 90 miles per hour. P1.6 P1.17 In what part of the electromagnetic spectrum is each line? (See Fig. 8.1 and Table 8.1 on p ) P1.19 P1.21

4 Assignment #3 Due Thurs., July Read Engel, Quantum Chemistry and Spectroscopy, Chapters 2 and 3 and Sections 6.1 and Problems from Engel, Quantum Chemistry and Spectroscopy P2.9 P2.13 P2.15 P2.17. P2.21 P2.23 P2.24 P2.28 P2.29 P6.1 P6.3

5 Assignment #4 Due Tues., July Read Engel, Quantum Chemistry and Spectroscopy a. Chapter 4 b. Sections 5.1 through 5.7 c. Section 6.4 d. Sections 7.1 and This problem will enable you to test the uncertainty principle as outlined in section 6.4. a. For the n = 1 level of the particle in a 1-D box, calculate the expectation values: <x>, <x 2 >, <p>, and <p 2 > b. Use these results to calculate " p = p 2 # p 2 and " x = x 2 # x 2 c. Show that the uncertainty principle is satisfied for the level, i.e., that " x " p # h 2. Notes: The expectation values above require you to compute integrals. Appendix A in the text is a math supplement and A.4 contains some formulae for integrals. The CRC Handbook of Chemistry and Physics has much more complete integral tables. So in general, your task in doing integrals is to manipulate the integral so that it is in the form of one of tabulated integrals and then to apply the formula in the table. Even this takes some practice if you are not used to using integral tables. On the back of this sheet is a Table from the inside cover of Physical Chemistry by McQuarrie and Simon that has some tables and also useful trigonometric identities. If you wish, you can do these problems without the use of trigonometric identities or integral tables if you use Euler s formula to express the trig functions in terms of imaginary exponentials. 3. In example problem 7.2, it is shown for the n=1 level of the harmonic oscillator that the kinetic and potential energies are equal. In this problem you will show that this is also true for the ground state n=0. That is show that E kinetic,n = E potential,n = 1 h" n ( 2) for the zero-point level where n=0. Note: This result is a special case of the Virial Theorem. This theorem also applies to electrons in molecules. If a particle moves in a potential proportional to x m, then the relationship between the kinetic and potential energies is 2 E kinetic = m E potential. For the harmonic oscillator, m = 2, but the coulomb potential in atoms and molecules, m = Problems from Engel, Quantum Chemistry and Spectroscopy P5.3. Read section 5.3. Note that butadiene has 4 pi electrons. In the context of the particle in a 1-D box model, that would put 2 electrons in the n=1 level and two in n=2. Therefore the lowest optical transition would be from n=2 to n=3. P7.1

6 Assignment #5 Due Thurs., July Read Engel, Quantum Chemistry and Spectroscopy a. Sections 7.2 to 7.5, 7.7 and 7.8 b. Chapter 9 2. Show by explicitly applying the l ˆ z operator that a. the spherical harmonic function Y 2 2 is an eigen function of the l ˆ z operator and identify the eigenvalue, and b. show that d xy = 1 i 2 Y 2 "2 ( 2 "Y 2 ) is not an eigenfunction of the l ˆ z operator. 3. For the hydrogen atom, evaluate the following: (Hint: Use the known properties of the operators. No proof is needed.) (a) L ˆ 2 Y 21 ",# ( ) (b) L ˆ z Y 30 (",#) (c) ˆ L 2 " 1s (d) ˆ L z " 21#1 (e) ˆ L 2 " 21#1 (f) ˆ L z " 3dz 2 (g) ˆ H = " 2 pz ˆ H " 2 pz (h) ˆ L 2 = " 2 pz ˆ L 2 " 2 pz (i) L for " 2 pz 4. Problem from from Engel, Quantum Chemistry and Spectroscopy P7.17 P9.7 (For part a. you may integral tables if you wish rather than using integration by parts if that is easier for you.)

7 Assignment #6 Due Tues., Aug Read Engel, Quantum Chemistry and Spectroscopy Chapter Problems from from Engel, Quantum Chemistry and Spectroscopy Q10.1 Q10.11, Q10.14, Q10.15 P10.1 P10.2 P10.3 P10.6 P10.10 P (a) Use the provided table of atomic numbers, to write down the ground state electron configuration of the following atoms and ions. (Example Li = 1s 2 2s 1 ) K, Be, V, Mn, Cd, P, Ne, Se, Na+, F- (b) Use Hund s rules to predict number of unpaired electrons and hence the net spin, S, of each atom in its ground state. Give also the predicted spin multiplicity, 2S+1, of each. (Example Li has one unpaired electron, which could be spin up or spin down, so Li will have spin S=1/2 and spin multiplicity 2S+1 = 2(1/2)+1 = 2.)

8 Assignment #7 Due Thurs., Aug Read Engel, Quantum Chemistry and Spectroscopy Chapter The notes from my lecture on the variational method will be most helpful for the problem below. 2. In this problem, you will apply the variation method to obtain an approximate solution for the particle in a 1-dimensional box. Use the trial wavefunction " = c 1 f 1 x where f 1 ( ) + c 2 f 2 ( x) ( x) = x( a " x) and f 2 ( x) = x 2 ( a " x) 2. are (a) For the case when the length of the box is a=1, show that the required integrals H 11 = h2 6m, H = H = h m, H = h m, S 11 = 1 30, S = S = , and S = (b) Set up and solve the secular determinant to find the best energy E of the ground state. Compare to the known energy for the ground state of the particle in a 1-D box.

9 Assignment #8 Due Tues., Aug Read Engel, Quantum Chemistry and Spectroscopy Chapter Problems from Engel, Quantum Chemistry and Spectroscopy Q12.6 Q12.12 P12.1 P12.2 P12.6

10 Assignment #9 Due Thurs., Aug Read Engel, Quantum Chemistry and Spectroscopy Chapter Problems from Engel, Quantum Chemistry and Spectroscopy Q13.17 Q13.18 Q13.19 P13.9 P13.19 (Hint: For each molecular orbital in the table, pick out the atomic orbitals with the largest coefficients and sketch them to see how they overlap.)