IGNOU MPH 12 2026 - For the hydrogen molecule which has two hydrogen atoms each with one e

Question

For the hydrogen molecule which has two hydrogen atoms each with one electron occupying the 1s energy level, write the two particle wave functions for the singlet and triplet states. Determine the eigenvalues of the effective Hamiltonian

$$\hat{H}_{eff} = \frac{1}{4}(E_S + 3E_T) - \frac{(E_S - E_T)}{\hbar^2} \hat{S}_1 \cdot \hat{S}_2 \text{, in the singlet and triplet states}$

19 Jan 2026

Answer :

Word Count : 335

For the hydrogen molecule H₂, we have two electrons, one on each hydrogen atom, both in the 1s orbital. Let the spatial wavefunctions of the electrons be (\psi_a) (centered on atom A) and (\psi_b) (centered on atom B), and the spin wavefunctions be (\alpha) (spin up) and (\beta) (spin down). The two-particle wavefunctions are constructed as products of spatial and spin parts. For two electrons, the total wavefunction must be antisymmetric under exchange. 1. Singlet state ((S = 0)): Spin part: antisymmetric [ \chi_{\text{singlet}} = \frac{1}{\sqrt{2}} (\alpha(1)\beta(2) - \beta(1)\alpha(2)) ] Spatial part: symmetric [ \psi_{\text{sym}} = \frac{1}{\sqrt{2}} (\psi_a(1)\psi_b(2) + \psi_b(1)\psi_a(2)) ] Total singlet wavefunction: [ \Psi_{\text{singlet}} = \psi_{\text{sym}} \cdot \chi_{\text{singlet}} = \frac{1}{\sqrt{2}} (\psi_a(1)\psi_b(2) + \psi_b(1)\psi_a(2)) \cdot \frac{1}{\sqrt{2}} (\alpha(1)\beta(2) - \beta(1)\alpha(2)) ] 2. Triplet state

______ __________ ________ _____ ___ ____ _________ ___ _________ __________ ________.
_____ ____ _________ _______ _____ _____ _____ ____ ____ ___ _________ ________.
_____ _______ _______ ____ _____.
_______ __________ ________ ____ ____ _________ _____ __________ __________.
___ ___ _______ ___ ______ _____ _____ _____ ________ ____ ______.
_________ __________ __________ ________ ___ __________.
__________ ______ _________ ____ ______ ______ ____.
________ ______ _________ ___ _________ ___.
________ _______ __________ ____ ___ ________ ____ ____ ________ ____ ________ ____.
___ _________ __________ ____ _________ _______ __________.
__________ _____ ________ ________ ______.
_____ ________ ________ _________ ______ ___ __________ ___ ___ ________.
___ ______ __________ _________ ___ _______ __________.
___ ______ _________ ___ __________ ____.
_______ __________ ___ __________ ____ ________ ___ _____ _________.
_____ ________ ____ ___ _________ ____ _____ ________ ____ ________.
_______ ______ _____ ___ _______ __________ _____ _______ ___ __________.
_____ ______ ______ ________ __________ __________ __________ ________ ________ ___ _____ ________.
________ ____ _______ ________ _______ ____ ____ ________ _________ ____ __________ _____.
__________ __________ __________ _______ ________ __________ ______ _____ ___ _____.
___ _________ ________ ____ _____ _____ _______ ________.
_________ _____ _______ ____ _________ ____.
________ ____ _______ ___ ___ __________ ___ ____ _______ _______ _______.
___ _____ _________ ________ ____ __________ ______ _________ ______.

IGNOU ACADEMY

Your IGNOU journey, made easier.

Solved Assignments
Study Notes
Guess Papers
Question Papers

Call Us WhatsApp

STUDY • PREPARE • SUCCEED

Skip in 10