Instructors: Kurur/Pandey

1. The maximum potential energy that a diatomic molecule can store is 1/2 k x², where x is the amplitude of the vibration. If the force constant is 1.86 x 10³ N m⁻¹, calculate the maximum amplitude of vibration for the CO molecule in the ground vibrational state.
2. It can be proved generally that for a harmonic oscillator

   <x2> = (n +

   1 2

   )

   ℏ √ µ k

   and that

   <x4> = (n2 + 2n + 1)

   3ℏ2 4µ k

   .

   Verify these formulas for the first two states of the harmonic oscillator.
3. An analytic expression that is a good approximation to the potential energy curve of a diatomic molecule is V(x) = D(1 − exp(−β x))² where D and β are parameters that depend on the molecule. Derive a relation between the force constant and the parameters D and β. Now show that

   β = 2π c ν

   ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

   √

   µ 2D

   ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

   ,

   where ν is the vibrational frequency expressed in cm⁻¹.
4. Verify the recursion relation

   Hn+1(z) − 2zHn(z) + 2nHn−1(z) = 0

   using the first few Hermite polynomials.
5. In the infrared spectrum of H⁷⁹Br, there is an intense line at 2559 cm⁻¹. Calculate the force constant of H⁷⁹Br and the period of vibration of H⁷⁹Br.
6. In the vibrational motion of HI, the iodine atom remains stationary because of its large mass. Assume that the hydrogen atom undergoes harmonic motion and that the force constant is 317 N m⁻¹, what is the vibrational frequency ν₀? What is the zero point energy if H is replaced by D? Assume that there is no change in the force constant.
7. Calculate the moment of inertia of H³⁵Cl, H³⁷Cl, and D³⁵Cl all of which have an equilibrium bond length of 1.275 Å. Determine the energies of the first three rotational states for H³⁵Cl. Use this information to predict these values for D³⁵Cl.
8. Show by direct operation that the functions sinθ exp(iφ), sinθ exp(−iφ), and cosθ are eigen functions of L_z. What are the eigen values?
9. Use the operator for L^2 in polar coordinates to show that the function (3cos²θ − 1) is an eigen function of this operator. What is the eigen value? What is the quantum number l for this function?
10. The normalization of Y_l^m(θ,φ) = N P_l^m(cosθ)exp(imφ) is performed as follows:

    N2

    ∫

    π θ = 0

    (Plm)* Plm sinθ dθ

    ∫

    2 π φ = 0

    exp(−imφ)exp(imφ) dφ = 1,

    while its orthogonality to Y_l′^m′ implies that

    ∫

    π θ = 0

    ∫

    2π φ=0

    Ylm (Yl′m′)* sinθ dθ dφ = 0.

    1. Show that Y₁⁻¹(θ, φ) is normalized and it is orthogonal to Y₂¹(θ, φ).
    2. Determine the normalization constant for the function given in problem 2.
11. For angular momentum with quantum number l=3, how many m-values are there? What is the semi-angle of the cone subtended by the angular momentum vector if its z-projection is 2ℏ?
12. From the definition of angular momentum, L^→ = r^→ × p^→, and following the procedure outlined in the class obtain the Cartesian form of the y-component of the angular momentum operator.
13. Show that [L_y, L_z]=iℏ L_x. (Hint: Use the information from the previous problem.)
14. Show that [L_x, y]=i ℏ z.
15. Determine the positions of the first three rotational transitions for H³⁵Cl and D³⁵Cl.
16. In the far infrared spectrum of H⁷⁹Br, there is a series of lines separated by 16.72 cm⁻¹. Calculate the values of the moment of inertia and the internuclear separation in H⁷⁹Br.
17. The J = 0 → J = 1 line in the microwave absorption spectrum of ¹²C¹⁶O and of ¹³C¹⁶O was found to be at 3.84235 cm⁻¹ and 3.67337 cm⁻¹. Calculate (a) the bond length of ¹²C¹⁶O, (b) the relative atomic mass of ¹³C.
18. A hydrogen-like atom can be formed from a proton and a negative muon whose mass is approximately 206 times that of the electron. What are the energies and most probable radius for the 1s and 2p levels of this atom?
19. Using the uncertainty principle argue that free electrons cannot exist in the nucleus. The diameter of a typical nucleus is 10⁻¹⁴ m.
20. For a hydrogen atom in the ground state find the classically forbidden region and calculate the probability of finding the electron in this region.
21. Compute the average value of r, the most probable value of r, and the root-mean-square value of r for the 1s and 2p levels of the hydrogen atom. Compare the three kinds of values and explain the origin of their differences.
22. Show that the hydrogenlike atomic wave function ψ₂₁₀ is normalized and that it is orthogonal to ψ₂₀₀.
23. Calculate the probability that an electron described by a hydrogen 1s wave function will be found within one Bohr radius of the nucleus.
24. Prove that <V> = 2<E> and, consequently, that <K> = −<E>, for a 2s electron.
25. Compute <r> in the 2s, 2p states of the hydrogen atom. Compare your result with the general formula

    <rnl> =

    a0 2

    ⎡ ⎣

    3n2 − l(l+1)

    ⎤ ⎦

    .

26. Where do the maxima in r²ψ_2s²(r) occur?
27. What combinations of the d (l = 2) atomic orbitals will produce the Cartesian function d_xz = xzR_nl(r) and d_xy = xyR_nl(r).
28. If we were to ignore the inter-electronic repulsion in helium, what would be its ground state energy and wave function? The experimental ground state energy of He is −79.0 eV.

This document was translated from L^AT_EX by H^EV^EA.