5. Nucleon- Nucleon Potential, Shell Structure

Question 1

How do we obtain the shell model for atoms?

Answer 1

Solve the SE for a single electron in the EM potential of a nucleus

Question 2

What is the solution for the shell model?

Answer 2

• Set of discrete energy levels, numbered by n
• For each n, have l = …n-1 degenerate E levels
• For each l, have m_l = -l … l
• For each n, l, m_l we have two states with opposite electron spin

Question 3

What is the behaviour of the nucleon-nucleon potential?

Answer 3

• Short ranged on order of nucleus size
• Stronger than EM repulsion of p’s but repulsive at small separations
• Charge symmetric and independent (pp) = (nn) = (pn)

Question 4

Why does SoC imply m_l and m_s are not good quantum numbers?

Answer 4

They are not eigenstates of the orbital momentum and spin operators J = L + S

Question 5

What is the expectation value for < l s > ?

Answer 5

h bar^2 / 2 [ j(j+1) - l(l+1) - s(s+1) ]
= h_bar ^2 l/2 if j = l+1/2
= h_bar ^2 (-l+1)/2 if j = l - 1/2

Question 6

How does SoC affect the energy levels?

Answer 6

It breaks the degeneracy e.g. 1p -> 1p_1/2 1p_3/2

Question 7

How many degenerate energy states are there at each energy level?

Answer 7

2j + 1