Question

One can assume a quantum mechanical harmonic oscillator model for the N-H stretching vibrations of the peptide bonds. For the harmonic oscillator the energy levels are given by: E, = (V+})ħw where: W= /k/ u In the above express k is the force constant and u is the reduced mass. (a) Write the Schrödinger equation in terms of the reduced mass u, being sure to define all symbols. (b) Calculate the frequency of the infrared radiation absorbed by the N-H bonds in B-sheet regions of a protein in wavenumbers. Assume that the force constant is k = 650 Nm". (c) Calculate the frequency of the infrared radiation absorbed by the N-H bonds in non- hydrogen bonded regions of a protein in wavenumbers. Assume that the force constant is k = 690 N m.

Answer 1

Schrodinger eqn is given by, - t^2/2 m partial. differential^2 psi/ partial. differential x^2 + v(x) psi = E psi Now taking t = h/2 pi, m = M_N middot M_H/M_N + M_H partial. differential^2 psi/ partial. differential x^2 + 8 pi^2 mu/h^2 [E - v (x)] psi = 0 here mu reduced of the system [Here N- H] & v(x) = 1/2 k x^2 Now substituting alpha = 8 pi^2 mu E/h^2, beta = squareroot 4 pi^2 mu k/h^2 we have partial. differential^2 psi/d x^2 + (alpha - beta^2 x^2) psi = 0 required eqn (ii) w = squareroot k/mu = rose = 1/2 pi squareroot k/mu Now mu = m_H middot m_N/m_H + m_n = 01 times 14/15 = 14/15 times 1.57 times 10^-27 kg = 1.558 times 10^-27 kg (i) for k = 650 Nm^-1