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PROBLEM 2. The Fine Structure of One-Electron Systems [13 pnts] a) The fine structure levels are characterized by the quantum number $j$. What are the $j$ values of a 5p electron. [1 pnt] b) Briefly describe the physical mechanism that leads to the coupling of $l$ and $s$. [2 pnts] c) Consider the hypothetical case that $l = 3$ and $s = 9/2$. What are the possible values of $j$? [2 pnts] d) Back to the 5p electron. Calculate the energy of the fine structure levels w.r.t. to the unperturbed 5p energy. The fine structure constant $A$=60 [cm$^{-1}$]. [1 pnt] Hint: $V_{so} = \frac{A}{2}(j(j + 1) - l(l + 1) - s(s + 1))$. e) The shift of the $j$ levels is asymmetric w.r.t. the \"unperturbed\" 5d binding energy. Show that conservation of energy is not violated. [2 pnts] f) Consider the upper, least bound $j$ level. This system is put in an external magnetic field $B$, sketch the behavior of the binding energies of the relevant $m_j$ states as a function of $B$. [2 pnt] g) What does happen if the external magnetic field becomes much stronger than the internal magnetic field. [1 pnt] h) Estimate the order of magnitude of the external magnetic field for which the fine structure of the atom breaks down. [2 pnt] Hint: $g_j = 1 + \frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}$ and $\mu_B = 0.47$ [cm$^1$/T]

          PROBLEM 2. The Fine Structure of One-Electron Systems [13 pnts]
a) The fine structure levels are characterized by the quantum number $j$. What are the $j$ values of a 5p electron. [1 pnt]
b) Briefly describe the physical mechanism that leads to the coupling of $l$ and $s$. [2 pnts]
c) Consider the hypothetical case that $l = 3$ and $s = 9/2$.
What are the possible values of $j$? [2 pnts]
d) Back to the 5p electron. Calculate the energy of the fine structure levels w.r.t. to the unperturbed 5p energy. The fine structure constant $A$=60 [cm$^{-1}$]. [1 pnt]
Hint: $V_{so} = \frac{A}{2}(j(j + 1) - l(l + 1) - s(s + 1))$.
e) The shift of the $j$ levels is asymmetric w.r.t. the \"unperturbed\" 5d binding energy. Show that conservation of energy is not violated. [2 pnts]
f) Consider the upper, least bound $j$ level. This system is put in an external magnetic field $B$, sketch the behavior of the binding energies of the relevant $m_j$ states as a function of $B$. [2 pnt]
g) What does happen if the external magnetic field becomes much stronger than the internal magnetic field. [1 pnt]
h) Estimate the order of magnitude of the external magnetic field for which the fine structure of the atom breaks down. [2 pnt]
Hint: $g_j = 1 + \frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}$ and $\mu_B = 0.47$ [cm$^1$/T]

PROBLEM 2. The Fine Structure of One-Electron Systems [13 pnts]
a) The fine structure levels are characterized by the quantum number j. What are the j values of a 5p electron. [1 pnt]
b) Briefly describe the physical mechanism that leads to the coupling of l and s. [2 pnts]
c) Consider the hypothetical case that l = 3 and s = 9/2.
What are the possible values of j? [2 pnts]
d) Back to the 5p electron. Calculate the energy of the fine structure levels w.r.t. to the unperturbed 5p energy. The fine structure constant A=60 [cm^-1]. [1 pnt]
Hint: Vso = (A)/(2)(j(j + 1) - l(l + 1) - s(s + 1)).
e) The shift of the j levels is asymmetric w.r.t. the ünperturbed5̈d binding energy. Show that conservation of energy is not violated. [2 pnts]
f) Consider the upper, least bound j level. This system is put in an external magnetic field B, sketch the behavior of the binding energies of the relevant mj states as a function of B. [2 pnt]
g) What does happen if the external magnetic field becomes much stronger than the internal magnetic field. [1 pnt]
h) Estimate the order of magnitude of the external magnetic field for which the fine structure of the atom breaks down. [2 pnt]
Hint: gj = 1 + (j(j+1)-l(l+1)+s(s+1))/(2j(j+1)) and = 0.47 [cm^1/T]

Added by Michael R.

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