molecule to form Na. In this problem we will attempt to calculate the vibrational distribution of the newly-formed N, ions, using a somewhat simplified approach a. Calculate(according to classical mechanics) the velocity(in cm/sec)of a 100 eV electron, ignoring any relativistic effects. Also calculate the amount of time required for a 100 eV electron to pass an N2 molecule, which you may estimate as having a length of 2A b. The radial Schrodinger equation for a diatomic molecule treating vibration as a harmonic oscillator can be written as h2(2(.,c平 +(-re)2y=E Substituting Y(r)=F(() this equation can be rewritten as t2 a2 -2u F()+(r-Te)2F(r)=E f(r The vibrational Hamiltonian for the ground electronic state of the N2 molecule within this approximation is given by where rn, and kn, have been measured experimentally to be 769A;kN,=2294x106
16 molecule to form N2 + . In this problem we will attempt to calculate the vibrational distribution of the newly-formed N2 + ions, using a somewhat simplified approach. a. Calculate (according to classical mechanics) the velocity (in cm/sec) of a 100 eV electron, ignoring any relativistic effects. Also calculate the amount of time required for a 100 eV electron to pass an N2 molecule, which you may estimate as having a length of 2Å. b. The radial Schrödinger equation for a diatomic molecule treating vibration as a harmonic oscillator can be written as: - -h2 2mr 2 è ç æ ø ÷ ¶ ö ¶r è ç æ ø ÷ ö r 2 ¶Y ¶r + k 2 (r - re) 2Y = E Y , Substituting Y(r) = F(r) r , this equation can be rewritten as: - -h2 2m ¶ 2 ¶r 2 F(r) + k 2 (r - re) 2F(r) = E F(r) . The vibrational Hamiltonian for the ground electronic state of the N2 molecule within this approximation is given by: H(N2) = - -h2 2m d2 dr2 + kN2 2 (r - rN2 ) 2 , where rN2 and kN2 have been measured experimentally to be: rN2 = 1.09769 Å; kN2 = 2.294 x 106 g sec2
he vibrational Hamiltonian for the N2t ion, however, is given by HN2)=2d+2(mN)2, where IN2 and kN?* have been measured experimentally to be nN=11642A,kN=2009×106sc2 In both systems the reduced mass is u=1. 1624 x 10-23 g. Use the above information to write out the ground state vibrational wave functions of the N2 and 2 molecules, giving explicit values for any constants which appear in them The v=0 harmonic oscillator function is Yo =(a/t) exp(-ax2 /2) c. During the time scale of the ionization event(which you calculated in part a) the vibrational wave function of the n, molecule has effectively no time to change. As a result, the newly-formed n ion finds itself in a vibrational state which is not an eigenfunction of the new vibrational Hamiltonian, H(N?) Assuming that the n2 molecule was originally in its v=o vibrational state calculate the probability that the ni ion will be produced in its v=0 vibrational tate
17 The vibrational Hamiltonian for the N2 + ion , however, is given by : H(N2) = - -h2 2m d2 dr2 + kN2 + 2 (r - rN2 +) 2 , where rN2 + and kN2 + have been measured experimentally to be: rN2 + = 1.11642 Å; kN2 + = 2.009 x 106 g sec2 . In both systems the reduced mass is m = 1.1624 x 10-23 g. Use the above information to write out the ground state vibrational wave functions of the N2 and N2 + molecules, giving explicit values for any constants which appear in them. The v = 0 harmonic oscillator function is Y0 = (a/p) 1/4 exp(-ax 2 /2). c. During the time scale of the ionization event (which you calculated in part a), the vibrational wave function of the N2 molecule has effectively no time to change. As a result, the newly-formed N2 + ion finds itself in a vibrational state which is not an eigenfunction of the new vibrational Hamiltonian, H(N2 + ). Assuming that the N2 molecule was originally in its v=0 vibrational state, calculate the probability that the N2 + ion will be produced in its v=0 vibrational state
Vibration of a diatomic molecule 15. The force constant, k, of the C-O bond in carbon monoxide is 1.87 x 106 g/sec2 Assume that the vibrational motion of Co is purely harmonic and use the reduced mass H=6.857amu Calculate the spacing between vibrational energy levels in this molecule, in units of ergs and cm Calculate the uncertainty in the internuclear distance in this molecule assuming it is in its ground vibrational level. Use the ground state vibrational wave function (Py=0; recall that I gave you this function problem 14), and calculate <x, <x2>, and Ax=(<x2>-<>2)1n2 Under what circumstances(i.e. large or small values of k; large or small values of u) is the uncertainty in internuclear distance large? Can you think of any relationship between this observation and the fact that helium remains a liquid down to absolute zero? A Variational method problem
18 Vibration of a diatomic molecule 15. The force constant, k, of the C-O bond in carbon monoxide is 1.87 x 106 g/sec2. Assume that the vibrational motion of CO is purely harmonic and use the reduced mass m = 6.857 amu. Calculate the spacing between vibrational energy levels in this molecule, in units of ergs and cm-1. Calculate the uncertainty in the internuclear distance in this molecule, assuming it is in its ground vibrational level. Use the ground state vibrational wave function (Yv=0; recall that I gave you this function in problem 14), and calculate <x>, <x2>, and Dx = (<x2> - <x>2) 1/2. Under what circumstances (i.e. large or small values of k; large or small values of m) is the uncertainty in internuclear distance large? Can you think of any relationship between this observation and the fact that helium remains a liquid down to absolute zero? A Variational Method Problem
16. A particle of mass m moves in a one-dimensional potential whose Hamiltonian is HE ax where the absolute value function is defined by x =x if x20 and x=-x ifx <0 2b4-bx2 a. Use the normalized trial wavefunction o e to estimate the enel of the ground state of this system, using the variational principle to evaluate w(b), the variational expectation value of h b. Optimize b to obtain the best approximation to the ground state energy of this system, using a trial function of the form of o, as given above. The numerically calculated exact ground state energy is 0.808616 f m a. What is the percent error In your value Another variational method Problem 17. The harmonic oscillator is specified by the Hamiltonian H=2m dx2+2 kx
19 16. A particle of mass m moves in a one-dimensional potential whose Hamiltonian is given by H = - -h2 2m d2 dx2 + a|x| , where the absolute value function is defined by |x| = x if x ³ 0 and |x| = -x if x £ 0. a. Use the normalized trial wavefunction f = è ç æ ø ÷ 2bö p 1 4 e -bx2 to estimate the energy of the ground state of this system, using the variational principle to evaluate W(b), the variational expectation value of H. b. Optimize b to obtain the best approximation to the ground state energy of this system, using a trial function of the form of f, as given above. The numerically calculated exact ground state energy is 0.808616 h- 2 3 m - 1 3 a - 2 3 . What is the percent error in your value? Another Variational Method Problem 17. The harmonic oscillator is specified by the Hamiltonian: H = - -h2 2m d2 dx2 + 1 2 kx2
Suppose the ground state solution to this problem were unknown, and that you wish to approximate it using the variational theorem. Choose as your trial wavefunction 9=v16a(a2-x2) for-a<x<a forx≥a where a is an arbitrary parameter which specifies the range of the wavefunction. Note that o is properly normalized as given a Calculate Jp'HDdx and show it to be given by: φ'Hψdx=4 ma 14 bCalculate Jo'Hodx for a=b( c. To find the best approximation to the true wavefunction and its energy, find the minimum of Jo"Hodx by setting da jo'Hddx =0 and solving for a. Substitute this
20 Suppose the ground state solution to this problem were unknown, and that you wish to approximate it using the variational theorem. Choose as your trial wavefunction, f = 15 16 a - 5 2 (a2 - x2) for -a < x < a f = 0 for |x| ³ a where a is an arbitrary parameter which specifies the range of the wavefunction. Note that f is properly normalized as given. a. Calculate õó -¥ +¥ f*Hfdx and show it to be given by: õó -¥ +¥ f*Hfdx = 5 4 -h2 ma2 + ka2 14 . b. Calculate õó -¥ +¥ f*Hfdx for a = bè ç æ ø ÷ ö -h2 km 1 4 . c. To find the best approximation to the true wavefunction and its energy, find the minimum of õó -¥ +¥ f*Hfdx by setting d da õó -¥ +¥ f*Hfdx = 0 and solving for a. Substitute this