Problems The following are some problems that will help you refresh your memory about material you should have learned in undergraduate chemistry classes and that allow you to exercise the material taught in this text Suggestions about what you should be able to do relative to the background material in the Chapters of Part 1 1. You should be able to set up and solve the one - and two-dimensional particle in a box Schrodinger equations. I suggest you now try this and make sure you see a. How the second order differential equations have two independent solutions, so the most general solution is a sum of these two b. How the two boundary conditions reduce the number of acceptable solutions from two to one and limit the values of e that can be"allowed c. How the wave function is continuous even at the box boundaries but d/dx is not. In general d yp/dx, which relates to the momentum because-i-h d/dx is the momentum operator, is continuous except at points where the potential v(x)undergoes an infinite jump as it does at the box boundaries. The infinite jump in V, when viewed classically, means that the particle would undergo an instantaneous reversal in
1 Problems The following are some problems that will help you refresh your memory about material you should have learned in undergraduate chemistry classes and that allow you to exercise the material taught in this text. Suggestions about what you should be able to do relative to the background material in the Chapters of Part 1 1. You should be able to set up and solve the one- and two-dimensional particle in a box Schrödinger equations. I suggest you now try this and make sure you see: a. How the second order differential equations have two independent solutions, so the most general solution is a sum of these two. b. How the two boundary conditions reduce the number of acceptable solutions from two to one and limit the values of E that can be “allowed”. c. How the wave function is continuous even at the box boundaries, but dY/dx is not. In general dY/dx, which relates to the momentum because – i h d/dx is the momentum operator, is continuous except at points where the potential V(x) undergoes an infinite jump as it does at the box boundaries. The infinite jump in V, when viewed classically, means that the particle would undergo an instantaneous reversal in
momentum at this point, so its momentum would not be continuous. Of course, in any realistic system, v does not have infinite jumps, so momentum will vary smoothly and thus d ' p/dx will be continuous d. How the energy levels grow with quantum number n as n- e. What the wave functions look like when plotted 2. You should go through the various wave functions treated in the Part 1(e.g, particles in boxes, rigid rotor, harmonic oscillator )and make sure you see how the probability plots of such functions are not at all like the classical probability distributions except when the quantum number is very large 3. You should make sure you understand how the time evolution of an eigenstate y produces a simple exp(i tE/-h)multiple of p so that p does not depend on time However, when p is not an eigenstate(e.g, when it is a combination of such states), its time propagation produces a p whose probability distribution changes with time 4. You should notice that the densities of states appropriate to the 1-, 2, and 3- dimensional particle in a box problem( which relate to translations in these dimensions) depend of different powers of E for the different dimensions 5. You should be able to solve 2x2 and 3x3 Huckel matrix eigenvalue problems both to obtain the orbital energies and the normalized eigenvectors. For practice, try to do so for
2 momentum at this point, so its momentum would not be continuous. Of course, in any realistic system, V does not have infinite jumps, so momentum will vary smoothly and thus dY/dx will be continuous. d. How the energy levels grow with quantum number n as n2 . e. What the wave functions look like when plotted. 2. You should go through the various wave functions treated in the Part 1 (e.g., particles in boxes, rigid rotor, harmonic oscillator) and make sure you see how the |Y| 2 probability plots of such functions are not at all like the classical probability distributions except when the quantum number is very large. 3. You should make sure you understand how the time evolution of an eigenstate Y produces a simple exp(-i tE/ h) multiple of Y so that |Y| 2 does not depend on time. However, when Y is not an eigenstate (e.g., when it is a combination of such states), its time propagation produces a Y whose |Y| 2 probability distribution changes with time. 4. You should notice that the densities of states appropriate to the 1-, 2-, and 3- dimensional particle in a box problem (which relate to translations in these dimensions) depend of different powers of E for the different dimensions. 5. You should be able to solve 2x2 and 3x3 Hückel matrix eigenvalue problems both to obtain the orbital energies and the normalized eigenvectors. For practice, try to do so for
a. the allyl radical's three T orbitals b the cyclopropenly radical's three T orbitals. Do you see that the algebra needed to find the above sets of orbitals is exactly the same was needed when we treat the linear and triangular sodium trimer? 6. You should be able to follow the derivation of the tunneling probability. Doing this offers a good test of your ability to apply the boundary conditions properly, so I suggest you do this task. You should appreciate how the tunneling probability decays exponentially with the"thickness" of the tunneling barrier and with the"height of this
3 a. the allyl radical’s three p orbitals b. the cyclopropenly radical’s three p orbitals. Do you see that the algebra needed to find the above sets of orbitals is exactly the same as was needed when we treat the linear and triangular sodium trimer? 6. You should be able to follow the derivation of the tunneling probability. Doing this offers a good test of your ability to apply the boundary conditions properly, so I suggest you do this task. You should appreciate how the tunneling probability decays exponentially with the “thickness” of the tunneling barrier and with the “height” of this
barrier and that tunneling for heavier particles is less likely than for light particles. This is why tunneling usually is considered only for electrons, protons, and neutrons 7. I do not expect that you could carry off a full solution to the Schrodinger equation for the hydrogenic atom. However, I think you need to pay attention to a. How separations of variables leads to a radial and two angular second order differential equations b. How the boundary condition that o and o 2T are equivalent points in space produces the m quantum number c. How the I quantum number arises from the e equation d. How the condition that the radial wave function not"explode"(i.e, go to infinity)as he coordinate r becomes large gives rise to the equation for the energy e e. The fact that the angular parts of the wave functions are spherical harmonics, and that these are exactly the same wave functions for the rotational motion of a linear olecule f. How the energy E depends on the n quantum number as n and on the nuclear charge Z as Z, and that the bound state energies are negative( do you understand what this means? That is, what is the zero or reference point of energy?) 8. You should make sure that you are familiar with how the rigid-rotor and harmor oscillator energies vary with quantum numbers (J, M in the former case, v in the latter) You should also know how these energies depend on the molecular geometry(in the
4 barrier and that tunneling for heavier particles is less likely than for light particles. This is why tunneling usually is considered only for electrons, protons, and neutrons. 7. I do not expect that you could carry off a full solution to the Schrödinger equation for the hydrogenic atom. However, I think you need to pay attention to a. How separations of variables leads to a radial and two angular second order differential equations. b. How the boundary condition that f and f + 2p are equivalent points in space produces the m quantum number. c. How the l quantum number arises from the q equation. d. How the condition that the radial wave function not “explode” (i.e., go to infinity) as the coordinate r becomes large gives rise to the equation for the energy E. e. The fact that the angular parts of the wave functions are spherical harmonics, and that these are exactly the same wave functions for the rotational motion of a linear molecule. f. How the energy E depends on the n quantum number as n-2 and on the nuclear charge Z as Z2 , and that the bound state energies are negative (do you understand what this means? That is, what is the zero or reference point of energy?). 8. You should make sure that you are familiar with how the rigid-rotor and harmonic oscillator energies vary with quantum numbers (J, M in the former case, v in the latter). You should also know how these energies depend on the molecular geometry (in the
former) and on the force constant and reduced mass (in the latter ) You should note that e depends quadratically on j but linearly on v 9. You should know what the Morse potential is and what its parameters mean. You should understand that the morse potential displays anharmonicity but the harmonic potential does not 10. You should be able to follow how the mass-weighted Hessian matrix can be used to approximate the vibrational motions of a polyatomic molecule. And, you should understand how the eigenvalues of this matrix produce the harmonic vibrational frequencies and the corresponding eigenvectors describe the motions of the molecule associated with these frequencies
5 former) and on the force constant and reduced mass (in the latter). You should note that E depends quadratically on J but linearly on v. 9. You should know what the Morse potential is and what its parameters mean. You should understand that the Morse potential displays anharmonicity but the harmonic potential does not. 10. You should be able to follow how the mass-weighted Hessian matrix can be used to approximate the vibrational motions of a polyatomic molecule. And, you should understand how the eigenvalues of this matrix produce the harmonic vibrational frequencies and the corresponding eigenvectors describe the motions of the molecule associated with these frequencies
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
