Discussion Fr 17.6.2005  10:30 - 11:15 am lecture hall SN 20.2

Problem Set 8   (angular momentum, rotation)
 

Problem 1

The energy of a rotational state of a (linear) molecule is E_rot = Bl(l+1), where l is the rotational quantum number and B =  ^h²/_2I . Formally B is measured in energy units, i.e. in Joule. Commonly, for B the unit of wavenumbers [cm^-1] is used, with the conversion relation: B [cm^-1] = ^h²/_2Ihc = ^h/_8π²cI , where c is the speed of light (in cm/s).

1. Calculate the moment of inertia I = µr² for the isotopes H³⁵Cl, D³⁵Cl and H³⁷Cl. Use the reduced masses of the isotopes m_HCl = m_Hm_{Cl /}(m_H+m_Cl) where m_H = 1 u, m_D = 2 u, m_Cl = 35 u and m_Cl = 37 u. The equilibrium bond length is r = 127,5 pm.
2. Calculate B in wavenumbers for all isotopes.
3. What is the energy gap (in wavenumbers) between the states l=10 and l=11?
4. The mean rotational energy at room temperature is approximately 200 cm^-1. Which l state is closest in energy?
5. The lowest l state of HCl is l=0. What is the corresponding rotational energy?

Problem 2

What is the frequency of rotation of molecular oxygen in the state J=11 (B = 1,446 cm^-1; r = 120,7 pm)? Use the classical relationship E_rot = ¹/₂Iw² for the determination of the rotational energy. What is the speed of the rotating oxygen atoms?

Problem 3

1. Normalize the spherical harmonic function Y_2,0 = C (3 cos²θ - 1).
2. Create the (non-normalized) spherical harmonicsY_2,+1 and Y_2,+2 by applying the operator L₊.

Problem 4

Are the following spherical harmonics orthogonal to each other: Y_2,0, Y_1,0, Y_0,0? Consider all possible constellations.

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