Q1. PART A
- a)) Determine the shift in energy caused by relativistic correction to kinetic energy for n = 2 and n = 1 levels of hydrogen atom. Using appropriate selection rules, show which transitions are allowed. (500 words)
- b)) The anti-symmetric spin function for a two electron system is given as: (1/√2)[α(1)β(2)-β(1)α(2)]. Prove that χ- (1, 2) is an eigenfunction of (Ŝ)2. (500 words)
- c)) Using Thomas-Fermi method, derive an expression for the total kinetic energy of N electrons in a cubic box of side L, and hence obtain the expression for pressure. (500 words)
- d)) Using molecular orbital method, construct all pssible trial wavefunctions (including spin) for a diatomic molecule. Of these functions, which trial function is expected to have the minimum energy? Give justification for your answer. (750 words)
- e)) Write the Hamiltonian of a multi-electron atom. Obtain an expression for the effective potential experienced by an electron of this atom under the central field approximation. (250 words)
Key Points:
- Fine structure shift for hydrogen: ΔE_fs = (Z⁴ α² m_e c² / (2n⁴)) * [ (3/4) - (n / (j + 1/2)) ]
- Allowed transitions for hydrogen n=2 to n=1: 2p_(1/2) → 1s_(1/2) and 2p_(3/2) → 1s_(1/2) (Δl=±1, Δj=0,±1).
- Anti-symmetric spin function χ-(1,2) is a singlet (S=0) eigenfunction of Ŝ² with eigenvalue 0.
- Total kinetic energy of N electrons in a cubic box: E_T = (ħ² / (10m π²)) * (3π² N)^(5/3) * V^(-2/3).
Answer: