y will be observed to be in pn? What energies will these experimental measurements find and with what probabilities? e. For those systems originally in p =apn+ bYm which were observed to be in Yn at time t, what state (Yn, pm, or whatever )will they be found in if a second experimental measurement is made at a time t later than t? f. Suppose by some method(which need not concern us at this time)the system has been prepared in a nonstationary state(that is, it is not an eigenfunction of H). At the time of a measurement of the particle's energy, this state is specified by the normalized wave function y=/30)1/2 3) X(L-x)for O<XsL, and Y=0 elsewhere What is the probability that a measurement of the energy of the particle will give the n2兀2h value En 2mL2 for any given value of n? g. What is the expectation value of H, i.e. the average energy of the system, for the wave function Y given in part f? A problem on the properties of non-stationary states
11 Y will be observed to be in Yn? What energies will these experimental measurements find and with what probabilities? e. For those systems originally in Y = aYn + bYm which were observed to be in Yn at time t, what state (Yn, Ym, or whatever) will they be found in if a second experimental measurement is made at a time t' later than t? f. Suppose by some method (which need not concern us at this time) the system has been prepared in a nonstationary state (that is, it is not an eigenfunction of H). At the time of a measurement of the particle's energy, this state is specified by the normalized wave function Y = è ç æ ø ÷ 30ö L5 1/2 x(L-x) for 0 £ x £ L, and Y = 0 elsewhere. What is the probability that a measurement of the energy of the particle will give the value En = n2p2-h2 2mL2 for any given value of n? g. What is the expectation value of H, i.e. the average energy of the system, for the wave function Y given in part f? A problem on the properties of non-stationary states
10. Show that for a system in a non-stationary state Eit/h the average value of the energy does not vary with time but the expectation values of other properties do vary with time A problem about Jahn-Teller distortion 11. The energy states and wave functions for a particle in a 3-dimensional box whose lengths are L1, L2, and L3 are given by hani kn2 2n3 2 E(n1, n2, n3)=8m L +D)+L/ and nIx H(n1n2n3)= y Sin(isin( L2 Sin ngIz L3 These wave functions and energy levels are sometimes used to model the motion of electrons in a central metal atom(or ion) which is surrounded by six ligands in ar octahedral manner
12 10. Show that for a system in a non-stationary state, Y = å j CjYje -iEj t/h - , the average value of the energy does not vary with time but the expectation values of other properties do vary with time. A problem about Jahn-Teller distortion 11. The energy states and wave functions for a particle in a 3-dimensional box whose lengths are L1, L2, and L3 are given by E(n1,n2,n3) = h2 8m ë ê é û ú ù è ç æ ø ÷ ö n1 L1 2 + è ç æ ø ÷ ö n2 L2 2 + è ç æ ø ÷ ö n3 L3 2 and Y(n1,n2,n3) = è ç æ ø ÷ 2 ö L1 1 2 è ç æ ø ÷ 2 ö L2 1 2 è ç æ ø ÷ 2 ö L3 1 2 Sinè ç æ ø ÷ ö n1px L1 Sinè ç æ ø ÷ ö n2py L2 Sinè ç æ ø ÷ ö n3pz L3 . These wave functions and energy levels are sometimes used to model the motion of electrons in a central metal atom (or ion) which is surrounded by six ligands in an octahedral manner
a. Show that the lowest energy level is nondegenerate and the second energy level is triply degenerate if L1=L2=L3. What values of n1 n2, and n3 characterize the states elonging to the triply degenerate level? b. For a box of volume V=LiL2L3. show that for three electrons in the box(two in the nondegenerate lowest"orbital", and one in the next), a lower total energy will result if the box undergoes a rectangular distortion(L|=L2*L3). which preserves the total volume than if the box remains undistorted(hint if V is fixed and LI=L2, then L3 L,2 and Li is the only"variable") c. Show that the degree of distortion(ratio of l3 to Li) which will minimize the total energy is L3=v2 LI. How does this problem relate to Jahn-Teller distortions? Why (in terms of the property of the central atom or ion) do we do the calculation with fixed volume? d. By how much(in eV)will distortion lower the energy (from its value for a cube, LI h2 L,=L3)ifv=8 and 8m=6.01 x 10-27 erg cm2. 1eV=1.6x 10-erg
13 a. Show that the lowest energy level is nondegenerate and the second energy level is triply degenerate if L1 = L2 = L3. What values of n1, n2, and n3 characterize the states belonging to the triply degenerate level? b. For a box of volume V = L1L2L3, show that for three electrons in the box (two in the nondegenerate lowest "orbital", and one in the next), a lower total energy will result if the box undergoes a rectangular distortion (L1 = L2 ¹ L3). which preserves the total volume than if the box remains undistorted (hint: if V is fixed and L1 = L2, then L3 = V L1 2 and L1 is the only "variable"). c. Show that the degree of distortion (ratio of L3 to L1) which will minimize the total energy is L3 = 2 L1. How does this problem relate to Jahn-Teller distortions? Why (in terms of the property of the central atom or ion) do we do the calculation with fixed volume? d. By how much (in eV) will distortion lower the energy (from its value for a cube, L1 = L2 = L3) if V = 8 Å3 and h2 8m = 6.01 x 10-27 erg cm2. 1 eV = 1.6 x 10-12 erg
A particle on a ring model for electrons moving in cyclic compounds 12. The T-orbitals of benzene, C6H6. may be modeled very crudely using the wave functions and energies of a particle on a ring. Lets first treat the particle on a ring problem and then extend it to the benzene system Suppose that a particle of mass m is constrained to move on a circle(of radius r)in the xy plane. Further assume that the particle's potential energy is constant(choose zero as this value). Write down the Schrodinger equation in the normal Cartesian coordinate representation. Transform this Schrodinger equation to cylindrical coordinates where x=coso, y =sino, andz=z(z=0 in this case). Taking r to be held constant, write down the general solution, (o), to this Schrodinger equation The" boundary" conditions for this problem require thatΦ(φ)=d(φ+2π).Aply this boundary condition to the general solution. This results in the quantization of the energy levels of this system. Write down the final expression for the normalized wave functions and quantized energies. What is the physical significance of these quantum numbers that can have both positive and negative values? Draw an energy diagram representing the first five energy levels b. Treat the six T-electrons of benzene as particles free to move on a ring of radius 1.40 A, and calculate the energy of the lowest electronic transition. Make sure the Pauli
14 A particle on a ring model for electrons moving in cyclic compounds 12. The p-orbitals of benzene, C6H6, may be modeled very crudely using the wave functions and energies of a particle on a ring. Lets first treat the particle on a ring problem and then extend it to the benzene system. a. Suppose that a particle of mass m is constrained to move on a circle (of radius r) in the xy plane. Further assume that the particle's potential energy is constant (choose zero as this value). Write down the Schrödinger equation in the normal Cartesian coordinate representation. Transform this Schrödinger equation to cylindrical coordinates where x = rcosf, y = rsinf, and z = z (z = 0 in this case). Taking r to be held constant, write down the general solution, F(f), to this Schrödinger equation. The "boundary" conditions for this problem require that F(f) = F(f + 2p). Apply this boundary condition to the general solution. This results in the quantization of the energy levels of this system. Write down the final expression for the normalized wave functions and quantized energies. What is the physical significance of these quantum numbers that can have both positive and negative values? Draw an energy diagram representing the first five energy levels. b. Treat the six p-electrons of benzene as particles free to move on a ring of radius 1.40 Å, and calculate the energy of the lowest electronic transition. Make sure the Pauli
principle is satisfied! What wavelength does this transition correspond to? Suggest some reasons why this differs from the wavelength of the lowest observed transition in benzene. which is 2600 A A non-stationary state wave function 13 a diatomic molecule constrained to rotate on a flat surface can be modeled as a planar rigid rotor(with eigenfunctions, p(), analogous to those of the particle on a ring of problem 12)with fixed bond length r. At t=0, the rotational(orientational) probability distribution is observed to be described by a wave function Y(o, O) Cos2o. What values, and with what probabilities, of the rotational angular momentum, G-ihoo, could be observed in this system? Explain whether these probabilities would be time dependent as(φ0) evolves into y(φ A problem about Franck-Condon factors 14. Consider an N2 molecule, in the ground vibrational level of the ground electronic state,which is bombarded by 100 eV electrons. This leads to ionization of the N2
15 principle is satisfied! What wavelength does this transition correspond to? Suggest some reasons why this differs from the wavelength of the lowest observed transition in benzene, which is 2600 Å. A non-stationary state wave function 13. A diatomic molecule constrained to rotate on a flat surface can be modeled as a planar rigid rotor (with eigenfunctions, F(f), analogous to those of the particle on a ring of problem 12) with fixed bond length r. At t = 0, the rotational (orientational) probability distribution is observed to be described by a wave function Y(f,0) = 4 3p Cos2f. What values, and with what probabilities, of the rotational angular momentum, è ç æ ø ÷ ö -ih- ¶ ¶f , could be observed in this system? Explain whether these probabilities would be time dependent as Y(f,0) evolves into Y(f,t). A problem about Franck-Condon factors 14. Consider an N2 molecule, in the ground vibrational level of the ground electronic state, which is bombarded by 100 eV electrons. This leads to ionization of the N2