1.1 Problem 1An electron in a three-dimensional rectangular box with dimensions 5.00°A 3.00°A and 6.00°A makes a radiative transition from the lowest-lying excitedstate to the ground state. Calculate the frequency of the photon emitted.1.2 Problem 2Which of the following combinations of particle in a three-dimensional cubicbox are eigenfunctions of the three-dimensional cubic box Hamiltonian?a) 1p2 ( 138 – 381) b) 1p2 ( 212 131) and c) 12 151 -12 333 1p2 5111.3 Problem 3a) Write down the normalized harmonic oscillator wave function for v = 2.b) For the harmonic oscillator in the v = 2 state calculate the expectationvalue of the position operator.c) Explain why you would have expected the result. Would you expect asimilar result for the states with a different v.1.4 Problem 4Assume that the vibration of a 12C16O molecule can be approximated as aharmonic oscillator with reduced mass µ = m1m2 m1 m2and a transition energy E between states v = 5 and v = 6 of 4.259 10-20 J. (hint: replace m with µ also remember that omega = (kf/µ)^0.5has units of radians/s).a) Calculate:i) the zero-point energy and force constant1ii) the classical turning points for the state v = 5iii) use this information to estimate the distance the atoms traverse duringtheir vibrational motion.b) Indicate whether the following quantities will increase or decrease if 14C18Ois considered instead: Justify your answers.i) the zero-point energyii) the classical turning points1CHEM 452: Homework Problem set # 5Due Fri. Feb. 19 (10 points)Important: Please note that the points are HW points and notcourse points. 1 HW point corresponds to 1.5 course points.1.1Problem 1 [2 points]An electron in a three-dimensional rectangular box with dimensions 5.00° A3.00° and 6.00° makes a radiative transition from the lowest-lying excitedAAstate to the ground state. Calculate the frequency of the photon emitted.1.2Problem 2 [2 points]Which of the following combinations of particle in a three-dimensional cubicbox are eigenfunctions of the three-dimensional cubic box Hamiltonian?1111a) v2 (?138 – ?381 ) b) v2 (?212 ?131 ) and c) 1 ?151 – 2 ?333 v2 ?51121.3Problem 3 [2 points]a) Write down the normalized harmonic oscillator wave function for v = 2.b) For the harmonic oscillator in the v = 2 state calculate the expectationvalue of the position operator.c) Explain why you would have expected the result. Would you expect asimilar result for the states with a di?erent v.1.4Problem 4 [4 points]Assume that the vibration of a 12 C16 O molecule can be approximated as am1 *mharmonic oscillator with reduced mass µ = m1 m2 and a transition energy 2?E between states v = 5 and v = 6 of 4.259 * 10-20 J. (hint: replace m withkfhas units of radians/s).µ also remember that ? =µa) Calculate:i) the zero-point energy and force constant1 ii) the classical turning points for the state v = 5iii) use this information to estimate the distance the atoms traverse duringtheir vibrational motion.b) Indicate whether the following quantities will increase or decrease if 14 C18 Ois considered instead: Justify your answers.i) the zero-point energyii) the classical turning points2 ADDITIONAL PROBLEMS NOT GRADED1.5Problem 1a) The Hermite polynomials can be de?ned asHv (y) = (-1)v ey2dv -y2edy v(1)Using this expression to ?nd the ?rst 4 polynomials.b) The Hermite polynomials obey the following recursive relationship1yHv (y) = vHv-1 (y) Hv 1 (y)2(2)verify this explicitly for v = 0 1 2.1.6Problem 2For the harmonic oscillator calculate the mean potential energy and meankinetic energy.3
