, and b. Use these results to calculate
"p =
p2 # p
2
and " x =
x2 # 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 contains some formulae for integrals. The CRC Handbook of ! supplement and A.4 ! 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 = 12 h" ( n +
1 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 xm, 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 = -1. 4. 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
Assignment #5 Chemistry 314 Summer 2008 Due Thurs., July 31. 1. 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 lˆz operator that a. the spherical harmonic function Y22 is an eigen function of the lˆz operator and identify the eigenvalue, and 1 b. show that ! dxy = Y22 " Y2"2 ) is not an eigenfunction of the lˆz operator. ( i 2 ! 3. For the hydrogen atom, evaluate the following: (Hint: Use the known properties of the operators. No proof is needed.) ! ! (a) Lˆ2Y21 (", # ) (b) Lˆ zY30 (", # ) ! !
! !
!
(c) Lˆ2"1s (d) Lˆ z" 21#1 (e) Lˆ2" 21#1 (f) Lˆ z"
3dz 2
(g) Hˆ = " 2 p Hˆ " 2 p z
z
! (h) Lˆ2 = " 2 p Lˆ2 " 2 p z
z
!
(i) L for " 2 p
z
! 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.)
Assignment #6 Chemistry 314 Summer 2008 Due Tues., Aug 5. 1. Read Engel, Quantum Chemistry and Spectroscopy Chapter 10 2. 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 P10.12 3. (a) Use the provided table of atomic numbers, to write down the ground state electron configuration of the following atoms and ions. (Example Li = 1s22s1) 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.)
Assignment #7 Chemistry 314 Summer 2008 Due Thurs., Aug. 7. 1. Read Engel, Quantum Chemistry and Spectroscopy Chapter 10.4. 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 " = c1 f1 ( x ) + c 2 f 2 ( x ) 2
where f1 ( x ) = x ( a " x ) and f 2 ( x ) = x 2 ( a " x ) .
!
are
(a) For the case ! when the length of the box is a=1, show that the required integrals ! 2 h h2 h2 , H12 = H 21 = , H 22 = , H11 = 6m 30m 105m 1 1 1 , S12 = S21 = , and S22 = S11 = 30 140 630
! (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.!
Assignment #8 Chemistry 314 Summer 2008 Due Tues., Aug. 12. 1. Read Engel, Quantum Chemistry and Spectroscopy Chapter 12 2. Problems from Engel, Quantum Chemistry and Spectroscopy Q12.6 Q12.12 P12.1 P12.2 P12.6
Assignment #9 Chemistry 314 Summer 2008 Due Thurs., Aug. 14. 1. Read Engel, Quantum Chemistry and Spectroscopy Chapter 13 2. 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.)