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Homework Statement
Part (a):Find the first order shift in energy
Part(b): What is the degeneracy after perturbation? Find Show average shift in energy is zero.
Homework Equations
The Attempt at a Solution
I've shown part (a), the troubling part is part (b).
Part (b)
With the perturbation, the degeneracy is lifted. Hence degeneracy = 0.
For ##j=l+\frac{1}{2}##, shift is ##\Delta E_+ = \frac{1}{4}mc^2 \alpha^4\frac{(l+\frac{1}{2})(l+\frac{3}{2}) - l(l+1) - \frac{3}{4}}{n^3 l(l+\frac{1}{2})(l+1)}##
For ##j=l-\frac{1}{2}##, shift is ##\Delta E_- = \frac{1}{4}mc^2 \alpha^4\frac{(l-\frac{1}{2})(l+\frac{1}{2}) - l(l+1) - \frac{3}{4}}{n^3 l(l+\frac{1}{2})(l+1)}##.
The total perturbation is given by the sum of positive perturbation and negative perturbation:
[tex]\delta E= \Delta E_+ + \Delta E_-[/tex]
[tex]\delta E = \frac{mc^2 \alpha^4}{2n^3 l}[/tex]
Clearly, the average perturbation for a given n and l is not zero. How does it even become zero then?