Question

A porphyrin ring is a molecule which is present in chlorophyll, hemoglobin, and other important compounds. Some aspects of the physics of its molecular properties can be described by representing it as a one-dimensional circular path of radius r = 4Å along which 18 electrons are constrained to move.

(a) Write down the normalized one-particle energy eigenfunctions of the system, assuming that the electrons do not interact with each other.

(b) How many electrons are there in each level in the ground state of the molecule?

(c) What is the lowest electronic excitation energy of the molecule? What is the corresponding wavelength (give a numerical value) at which the molecule can absorb radiation?

Answer 1

(a) Denote the angular coordinate of an electron by \(\theta\). The Schrodinger equation of an electron

The single-valuedness of \(\psi(\theta),\psi(\theta) = \psi(\theta + 2\pi)\) requires k = 0, \(\pm1\), \(\pm2\). . . . The energy levels are given by

\(E = \frac{h^2}{2mr^2}k^2\)

(b) Let 0,1,2,. . . denote the energy levels E₀, E₁, E₂, . . . respectively. In accordance with Pauli's exclusion principle, E₀ can accommodate two electrons of opposite spins, while E_k, k \(\ne\) 0, which is 2-fold degenerate with respect to \(\pm |k|\) can accommodate four electrons. Thus the electron configuration of the ground state of the system is

0²1⁴2⁴3⁴4⁴.

(c) The electron configuration of the first excited state is

0²1⁴2⁴3⁴4³5¹.

The energy difference between E₄ and E₅ is

where \(\frac h{mc} \) = 0.0242 Å is the Compton wavelength of electron.