18. For a certain harmonic oscillator of effective mass = 4.21 x 10-24kg. the difference in energy levels is given by 2.97 x 1021J. Calculate the force constant of the oscillator.

Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 2.33 × 10−26 kg and force constant 155 N m−1.

A hydrogen molecule can be approximated a simple harmonic oscillator with force constant k = 1.1 x103 N/m. Calculate the frequency of the emmissin produced from the transition between the third level to ground state. m = 1.673 x 10-27 Kg.

Vibrations in the diatomic molecule CO can be approximated as a harmonic oscillator, where the angular frequency w = 6.505 x 1013 Hz and the reduced mass is equal to u = 1.14 × 10-27 kg. Assume the molecule is in its fırst excited vibrational state. Its vibrational wavefunction can then be written as V1 (x) = (4) /2a xe where a = . If we were to measure the bond length of the molecule, what is the most likely displacement from the equilibrium bond distance in the first excited vibrational state? Give your answer in Angstroms [Note: The equilibrium displacement in the Quantum harmonic oscillator corresponds to r = 0, ie the coordinate x measures displacement from equilibrium]

Calculate the zero-point energy of a harmonic oscillator consisting of a particle of mass 5.16 x 10-26 kg and force constant 285 N/m. What is the energy per mole?

In a molecule of hydrogen iodide HI (HI is used in organic and inorganic synthesis as one of the main sources of iodine and as a reducing agent) the vibrational frequency of the molecule is 6.69x10^13 Hz. Iodine is much more heavier than hydrogen, so I can be considered immobile compared to H. Determine the expected value of the potential energy for the hydrogen atom in this molecule in the ground state. Use this to calculate the expected value of the kinetic energy.

Calculate the energy of the quantum involved in the excitation of (i) an electronic oscillation of period 1.0 fs, (ii) a molecular vibration of period 10 fs, (iii) a pendulum of period 1.0 s. Express the results in joules and kilojoules per mole.

Calculate the energy of the quantum involved in the excitation of (i) an electronic oscillation of period 2.50 fs, (ii) a molecular vibration of period 2.21 fs, (iii) a balance wheel of period 1.0 ms. Express the results in joules and kilojoules per mole.

11. ( Draw the potential energy as a function of x. On top of the potential function, draw horizontal lines corresponding to the ground and first two excited quantum mechanical state energies of this system. Label the lines with their respective "v labels". Finally, using your energy state lines as individual axis, draw the wave functions for the ground and first two excited states. Consider the standard one-dimensional Harmonic oscillator centered about x = 0.

Evaluate Δx = (⟨x2⟩ − ⟨x⟩2)1/2 and Δpx = (⟨px2⟩ − ⟨px⟩2)1/2 for the ground state of (a) a particle in a box of length L and (b) a harmonic oscillator. Discuss these quantities with reference to the uncertainty principle.