11.7 Calculate the vibrational frequency for a proton if k=600Nm−^1.
11.8 For a^12 C^16 O bond, if k=500Nm−^1 calculate (a) the reduced mass and (b) the vibrational
frequency.
11.9 For a^13 C^16 O bond, if k=500Nm−^1 calculate (a) the reduced mass and (b) the vibrational
frequency.
11.10 If the mass is doubled, what is the change in vibrational frequency?
11.11Write (but do not solve) the probability of the ground state being in the classically forbidden
region.
11.12 Water (H 2 O) absorbs infrared radiation at 3μm (3400 cm−^1 ) due to the O–H stretch. What
wavelength would you predict is absorbed by D 2 O?
11.13 Write (but do not solve) the average squared position of the ground state.
11.14 Write (but do not solve) the average squared position of the 9 =1 state.
11.15 Calculate the energy of the ground state by putting the ground-state wavefunction into
Schrödinger’s equation.
11.16 For a simple^16 O 2 molecule, calculate (a) the reduced mass and (b) the frequency change
when the isotope^15 O is present.
11.17 Write Schrödinger’s equation for the potential V(x) =Ax^2 +Be−x. If B <<Awhat will the
wavefunctions be approximately?
11.18 Suppose that the mass is attached to two springs with spring constants k 1 and k 2 , one on
each side of the mass. Write Schrödinger’s equation for this system.
11.19For a simple harmonic oscillator show (but do not solve) in integral form the probability of
the ground state being in the classically forbidden region.
11.20 Calculate the energy of the 9 =1 state of the simple harmonic oscillator by substituting
the wavefunction into Schrödinger’s equation.
11.21 For the following questions, assume that a particle of mass mis in a valley between two
hills. You model the potential as:
|x|≤L/2|x|≥L/2where cL^2 is larger than the total energy of the particle.
(a) Write Schrödinger’s equation for these potentials.
(b) How are aand brelated to c?
(c) If Lis very small, with a value of 0.1 Å and mis very small, with a value of 1.054 ×
10 −^34 kg, what can be said about the uncertainity in velocity?
(d) Estimate the probability of finding the particle in the |x|≤L/2 and |x|≥L/2 regions.
Be sure to explain your reason for the estimates.
(e) If bis very small and c=0, what can be said about the wavefunctions in the region
|x|≤L/2?
11.22 For the following, assume that the molecules are linear and that all bonds are equal and
can be described as a spring with a constant k.
(a) Consider an ethene molecule, H 2 C 7 CH 2. Calculate the reducedmass, assuming that each
CH 2 behaves as a rigid object.
Vx()=cL^2Vx() (=+ax^23 bx)