Question

9. Write the Schrödinger equation for the Beryllium (Be). (Hint: The beryllium atom has a nucleus with 4+ charge and four electrons.) (10 pts)

Answer 1

For a single particle the time-independent Schrodinger Equation stands as:

= EU

18=8A+$2,44

wc-=H Atz 24

But in case of Be atom, there are four elecrons with +4 charge in the center.

Let us specify the electrons as
1.2.3.4
having a distance
1: 2:13:4
from the nuclues respectively. Suppose the relative separation of the electrons be
Tij = lri - ri
, where
i, j = 1,2,3,4
but $i \neq j$ .

The net wavefunction will be the resultant of that of each of the electrons ,i.e.,

(14)ý II = (14) ¢(£4) ¢( 1 ) ¢( L1) ¢ = 4
, where $\phi_i(r_i)$ is the wavefunction of $i^{th}$ electron.

For the potential energy, each of them will face a coulombic force from the nucleus, as well as, there will be coulombic force among each other.

Thus Hamiltonian operator stands as:

#?"I=!?
, where $V_{ij}$ is the coulomb potential between $i^{th}$ and $j^{th}$ electron.

= ( ܙ * ܕ

For Be atom, . So the Schrodinger equation will be:

Z = 4

( )+ t e - Ev

21 + 1 - r2 + 1 - 73 + 2 1 - - TTEO \ri = EU 1 - 74/ +- 2m 4760 r12 1 - 7 13 7 + + V 1 1 1 +- +- +- ) 123 124 134/ 1 - 114 +

This is the time-independent Schrodinger equation of Be atom.