Editing Zeeman effect (section)

== Intermediate field for j = 1/2 ==
In the magnetic dipole approximation, the Hamiltonian which includes both the [[Hyperfine structure|hyperfine]] and Zeeman interactions is{{cn|date=March 2025}} 
:<math>	H = h A \vec I \cdot \vec J  - \vec \mu \cdot \vec B </math>
:<math>	H = h A \vec I \cdot\vec J + ( \mu_{\rm B} g_J\vec J  + \mu_{\rm N} g_I\vec I ) \cdot \vec {\rm B} </math>
where <math>A</math> is the hyperfine splitting at zero applied magnetic field, <math>\mu_{\rm B}</math> and <math>\mu_{\rm N}</math> are the [[Bohr magneton]] and [[nuclear magneton]], respectively (note that the last term in the expression above describes the <em>nuclear</em> Zeeman effect), <math>\vec J</math> and <math>\vec I</math> are the electron and nuclear angular momentum operators and <math>g_J</math> is the [[Landé g-factor]]:
<math display="block"> g_J = g_L\frac{J(J+1) + L(L+1) - S(S+1)}{2J(J+1)} + g_S\frac{J(J+1) - L(L+1) + S(S+1)}{2J(J+1)}.</math>

In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the <math>|F,m_f \rangle</math> basis. In the high field regime, the magnetic field becomes so strong that the Zeeman effect will dominate, and one must use a more complete basis of <math>|I,J,m_I,m_J\rangle</math> or just <math>|m_I,m_J \rangle</math> since <math>I</math> and <math>J</math> will be constant within a given level. 
	
To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the <math>|F,m_F \rangle </math> and <math>|m_I,m_J \rangle </math> basis states. For <math>J = 1/2</math>, the Hamiltonian can be solved analytically, resulting in the '''Breit–Rabi formula''' (named after [[Gregory Breit]] and [[Isidor Isaac Rabi]]). Notably, the electric quadrupole interaction is zero for <math>L = 0</math> (<math>J = 1/2</math>), so this formula is fairly accurate.

We now utilize quantum mechanical [[ladder operator]]s, which are defined for a general angular momentum operator <math>L</math> as

:<math> L_{\pm} \equiv L_x \pm iL_y </math>

These ladder operators have the property
:<math> L_{\pm}|L_,m_L \rangle = \sqrt{(L \mp m_L)(L \pm m_L +1)} |L,m_L \pm 1 \rangle</math>

as long as <math>m_L</math> lies in the range <math>{-L, \dots ... ,L}</math> (otherwise, they return zero). Using ladder operators <math>J_{\pm}</math> and <math>I_{\pm}</math> 
We can rewrite the Hamiltonian as

:<math> H = h A I_z J_z + \frac{hA}{2}(J_+ I_- + J_- I_+) + \mu_{\rm B} B g_J J_z + \mu_{\rm N} B g_I I_z</math>

We can now see that at all times, the total angular momentum projection <math>m_F = m_J + m_I</math> will be conserved. This is because both <math>J_z</math> and <math>I_z</math> leave states with definite <math> m_J </math>  and <math> m_I </math> unchanged, while <math> J_+ I_- </math> and <math> J_- I_+ </math> either increase <math> m_J </math> and decrease <math> m_I </math> or vice versa, so the sum is always unaffected. Furthermore, since <math>J = 1/2</math> there are only two possible values of  <math>m_J</math> which are <math>\pm 1/2</math>. Therefore, for every value of <math> m_F </math> there are only two possible states, and we can define them as the basis:

:<math>|\pm\rangle \equiv |m_J = \pm 1/2, m_I = m_F \mp 1/2 \rangle </math>

This pair of states is a [[two-level quantum mechanical system]]. Now we can determine the matrix elements of the Hamiltonian:

:<math> \langle \pm |H|\pm \rangle = -\frac{1}{4} hA + \mu_{\rm N} B g_I m_F \pm \frac{1}{2} (hAm_F + \mu_{\rm B} B g_J- \mu_{\rm N} B g_I))</math>
:<math> \langle \pm |H| \mp \rangle = \frac{1}{2} hA \sqrt{(I + 1/2)^2 - m_F^2}</math>

Solving for the eigenvalues of this matrix – as can be done by hand (see [[two-level quantum mechanical system]]), or more easily, with a computer algebra system – we arrive at the energy shifts:

:<math> \Delta E_{F=I\pm1/2} =  -\frac{h \Delta W }{2(2I+1)} + \mu_{\rm N} g_I m_F B \pm \frac{h \Delta W}{2}\sqrt{1 + \frac{2m_F x }{I+1/2}+ x^2 }</math>
:<math>x \equiv \frac{B(\mu_{\rm B} g_J - \mu_{\rm N} g_I)}{h \Delta W} \quad \quad \Delta W= A \left(I+\frac{1}{2}\right)</math>

where <math>\Delta W</math> is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field <math>B</math>,
<math>x</math> is referred to as the 'field strength parameter' (Note: for <math>m_F = \pm(I+1/2)</math> the expression under the square root is an exact square, and so the last term should be replaced by <math>+\frac{h\Delta W}{2}(1\pm x)</math>). This equation is known as the '''Breit–Rabi formula''' and is useful for systems with one valence electron in an <math>s</math> (<math>J = 1/2</math>) level.<ref>{{cite book |last1=Woodgate |first1=Gordon Kemble |title=Elementary Atomic Structure |date=1980 |publisher=Oxford University Press |location=Oxford, England |pages=193–194 |edition=2nd}}</ref><ref>First appeared in:  {{cite journal |last1=Breit |first1=G. |last2=Rabi |first2=I.I. |title=Measurement of nuclear spin |journal=Physical Review |date=1931 |volume=38 |issue=11 |pages=2082–2083 |doi=10.1103/PhysRev.38.2082.2|bibcode=1931PhRv...38.2082B }}</ref>

Note that index <math>F</math> in <math>\Delta E_{F=I\pm1/2}</math> should be considered not as total angular momentum of the atom but as ''asymptotic total angular momentum''. It is equal to total angular momentum only if <math>B=0</math>
otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different <math>F</math> but equal <math>m_F</math> (the only exceptions are <math>|F=I+1/2,m_F=\pm F \rangle</math>).