1. Calculate the energy of the n = 5 to n = 4 transition in the Li²⁺ ion, in units of cm^-1.

2. For the 2s wavefunction of a hydrogenic atom with nuclear charge Z:

a. What is the position of the innermost radial node? Give your answer in terms of the nuclear charge Z and the Bohr radius a_o.

b. What is the probability of finding the electron within the innermost radial node? Give your answer in terms of the nuclear charge Z and the Bohr radius a_o.

3. Consider the following wavefunction for a hydrogenic atom with nuclear charge Z, where R_n,l is the hydrogenic atom radial wavefunction and Y_l,m?, is the spherical harmonic:

Ψ = 1/√2 R_3,1(r)[Y_1,1(θ , φ) - Y_{1, -1}(θ , φ)]

a. Sketch the R 3,1 radial wavefunction and radial distribution function. Determine the position of the radial node(s), if any. Give your answer in terms of the nuclear charge Z and the Bohr radius a_o.

b. Simplify the angular factor using Euler's formula. Determine the position of the angular node(s), if any. Sketch a polar plot of the angular factor as a function of T, for fixed θ = Π/2.

c. Sketch the R_3,1 contour plot of the full wavefunction zp in Cartesian coordinates, for a cut along the xy-plane at z = 0. Indicate the positions of any angular and radial nodes. The plot does not have to be quantitatively accurate, just make sure the shape, symmetry, and nodes are correct.

4. The first ionization energy, I₁, is the energy needed to remove an electron (to infinity) from the ground state of a neutral atom. For lithium, the experimental value of /1 is 5.39 electron volts (eV).

a. Estimate the effective nuclear charge experienced by the Li 2s electron.

b. For an electron in the 2s orbital of a hydrogenic atom, the most probable distance from the nucleus is:

r*_2s = (3 + √5)/Z

Explain in a few words how you would derive this expression (but no need to actually work out the math).

c. Using the effective nuclear charge from part a above, determine the most probable distance of the Li 2s electron in units of the Bohr radius ao. How does this compare with the most probable distance of the 2s electron in the Li²⁺ ion? Give a brief physical explanation for the difference in the most probable 2s distances in Li and Li²⁺.

d. For the second row elements, i.e., the row in the periodic table starting with Li, the value of h increases as one goes from left to right, ending with a value of 21.56 eV for Ne. Explain the physical reason for this trend.

5. Consider the singlet and triplet states of the excited He atom in the 1s¹25¹ configuration. The wavefunctions have the form:

ψ_singlet = [ ψ_1s (r₁) ψ_2s (r₂)+ (r₁)ψ_1s, (r₂)][α (σ₁) β (σ₂) - β (σ₁) α (σ 2)]

ψ_triplet = [ψ_1s (r₁) ψ_2s (r₂)-  ψ_2s (r₁) ψ_1s (r₂ )] α(σ₁) α (σ₂)

where ψ_1s ands are the orthonormal 1s and 2s orbitals of the hydrogenic atom, and α and β are the orthonormal spin functions.

a. Normalize the singlet wavefunction.

b. Normalize the triplet wavefunction.

c. Calculate the integral ∫ψ_singlet ψ_triplet dΤ