Practice with matrices and operators. 1. Find the eigenvalues and corresponding normalized eigenvectors of the following matrices 200 2. Replace the following classical mechanical expressions with their corresponding my in three-dimensional space 6
6 Practice with matrices and operators. 1. Find the eigenvalues and corresponding normalized eigenvectors of the following matrices: ë ê é û ú ù -1 2 2 2 ë ê ê é û ú ú ù -2 0 0 0 -1 2 0 2 2 2. Replace the following classical mechanical expressions with their corresponding quantum mechanical operators: K.E. = mv2 2 in three-dimensional space
p=mv, a three-dimensional Cartesian vector y-component of angular momentum: Ly=ZPx-Xpz 3. Transform the following operators into the specified coordinates h a Lx=11yaz-z8y1 from Cartesian to spherical polar coordinates i ao from spherical polar to Cartesian coordinates 4. Match the eigenfunctions in column B to their operators in column A. What is the eigenvalue for each eigenfunction? Column A Column B d2 d Ix2-x dx 4x4-12x2+3 lll
7 p = mv, a three-dimensional Cartesian vector. y-component of angular momentum: Ly = zpx - xpz. 3. Transform the following operators into the specified coordinates: Lx = -h i îï í ïì þï ý ïü y ¶ ¶z - z ¶ ¶y from Cartesian to spherical polar coordinates. Lz = h - i ¶ ¶f from spherical polar to Cartesian coordinates. 4. Match the eigenfunctions in column B to their operators in column A. What is the eigenvalue for each eigenfunction? Column A Column B i. (1-x2) d2 dx2 - x d dx 4x4 - 12x2 + 3 ii. d2 dx2 5x4 iii. x d dx e 3x + e-3x
d2 d 4x+2 d2 dx2 4x3-3x Review of shapes of orbitals 5. Draw qualitative shapes of the(1)s, (3)p and(5)d atomic orbitals(note that these orbitals represent only the angular portion and do not contain the radial portion of the hydrogen like atomic wave functions) Indicate with the relative signs of the wave functions and the position(s)(if any) of any nodes 6. Plot the radial portions of the 4s, 4p, 4d, and 4f hydrogen like atomic wave functions 7. Plot the radial portions of the 1s, 2s, 2p, 3S, and 3p hydrogen like atomic wave functions for the Si atom using screening concepts for any inner electrons 8
8 iv. d2 dx2 - 2x d dx x2 - 4x + 2 v. x d2 dx2 + (1-x) d dx 4x3 - 3x Review of shapes of orbitals 5. Draw qualitative shapes of the (1) s, (3) p and (5) d atomic orbitals (note that these orbitals represent only the angular portion and do not contain the radial portion of the hydrogen like atomic wave functions) Indicate with ± the relative signs of the wave functions and the position(s) (if any) of any nodes. 6. Plot the radial portions of the 4s, 4p, 4d, and 4f hydrogen like atomic wave functions. 7. Plot the radial portions of the 1s, 2s, 2p, 3s, and 3p hydrogen like atomic wave functions for the Si atom using screening concepts for any inner electrons
Labeling orbitals using point group symmetry 8. Define the symmetry adapted"core"and"valence "atomic orbitals of the following systems NH3 in the C3v point group, H2O in the C2v point group H2O2(cis)in the C2 point group Nin Dooh, D2h, C2v, and Cs point groups N2 in Dooh, D2h, C2v and Cs point groups A problem to practice the basic tools of the Schrodinger equation 9. A particle of mass m moves in a one-dimensional box of length L, with boundaries at x 0 and x=L. Thus, V(x)=0 for sXSL, and V(x)=oo elsewhere. The normalize
9 Labeling orbitals using point group symmetry 8. Define the symmetry adapted "core" and "valence" atomic orbitals of the following systems: NH3 in the C3v point group, H2O in the C2v point group, H2O2 (cis) in the C2 point group N in D¥h, D2h, C2v, and Cs point groups N2 in D¥h, D2h, C2v, and Cs point groups. A problem to practice the basic tools of the Schrödinger equation. 9. A particle of mass m moves in a one-dimensional box of length L, with boundaries at x = 0 and x = L. Thus, V(x) = 0 for 0 £ x £ L, and V(x) = ¥ elsewhere. The normalized
1/2 nTx eigenfunctions of the Hamiltonian for this system are given by pn(x) n22h2 with En=2mL2, where the quantum number n can take on the values n=1, 2, 3 a. Assuming that the particle is in an eigenstate, yn(x), calculate the probability that the particle is found somewhere in the region O sXs4. Show how this probability depends on n. b. For what value of n is there the largest probability of finding the particle in 0 c. Now assume that Y is a superposition of two eigenstates, p=an+bm, at time t=0 What is Y at time t? What energy expectation value does p have at time t and how does this relate to its value at t=0? d. For an experimental measurement which is capable of distinguishing systems in state Yn from those in Ym, what fraction of a large number of systems each described by
10 eigenfunctions of the Hamiltonian for this system are given by Yn(x) = è ç æ ø ÷ 2ö L 1/2 Sin npx L , with En = n2p2-h2 2mL2 , where the quantum number n can take on the values n=1,2,3,.... a. Assuming that the particle is in an eigenstate, Yn(x), calculate the probability that the particle is found somewhere in the region 0 £ x £ L 4 . Show how this probability depends on n. b. For what value of n is there the largest probability of finding the particle in 0 £ x £ L 4 ? c. Now assume that Y is a superposition of two eigenstates, Y = aYn + bYm, at time t = 0. What is Y at time t? What energy expectation value does Y have at time t and how does this relate to its value at t = 0? d. For an experimental measurement which is capable of distinguishing systems in state Yn from those in Ym, what fraction of a large number of systems each described by