Question

For an electron in the l = 2 state:

(a) enumerate all the possible values of quantum numbers j and mj ;

(b) draw the corresponding vector diagrams;

(c) estimate the maximum value of the spin-orbit coupling energy E. 

Answer 1

$l=2\\\therefore n=3$

(a) $s=1/2$

$j=(n+s),(n-s)$

$j=(5/2),(3/2)$

For j = 5/2

$jm=j(j+1)\hbar^2$

$jm=\dfrac{35}{4}\hbar^2$

For j = 3/2

$jm=\dfrac{15}{4}\hbar^2$

(c) $\Delta E_{so}=\alpha\dfrac{Z\hbar^3}{4m^2c}\dfrac{1}{r^3}[j(j+1)-l(l+1)-s(s+1)]$

$s=1/2,\space l=2,\space j=5/2$

$\Delta E_{so}=\alpha\dfrac{Z\hbar^3}{4m^2c}\dfrac{1}{r^3}\bigg[\dfrac{5}{2}\bigg(\dfrac{5}{2}+1\bigg)-2(2+1)-\dfrac{1}{2}\bigg(\dfrac{1}{2}+1\bigg)\bigg]$

$\Delta E_{so}=2\alpha\dfrac{Z\hbar^3}{4m^2c}\dfrac{1}{r^3}$