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Physics / Chapter 40: All About Atoms / Section 40-3

Angular Momenta and Magnetic Dipole Moments — Question 3

Section 40-3 · Physics pages 6616–6617

Question 3 An orbital Zeeman multiplet has equal adjacent energy spacings Ignore electron spin and all quadratic field effects. An orbital state with l = 2 lies in a uniform field B along z, so its first-order shift is δE = mlµBB. ml = −2 ml = −1 ml = 0 ml = 1 ml = 2 Schematic; labels, not drawing lengths, specify the data. (a) List the distinct shifts and find their full span. (b) For B = .300 T, find the adjacent spacing and span using µB = 5.788 × 10−5 eV/T.
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Question 3 - Solution An orbital Zeeman multiplet has equal adjacent energy spacings ml = −2 ml = −1 ml = 0 ml = 1 ml = 2 (a) Method and calculation ml = −2, −1, 0, 1, 2 ⇒δE/µBB = −2, −1, 0, 1, 2 . Emax −Emin = 4µBB . (b) Interpretation and further calculation ∆Eadj = µBB = 1.7364 × 10−5 eV . ∆Espan = 6.9456 × 10−5 eV . Physical check Five orbital orientations yield five equally spaced first-order energies. Spin and spin-orbit coupling can alter the observed multiplet and are explicitly omitted here.

Question 3

An orbital Zeeman multiplet has equal adjacent energy spacings

Ignore electron spin and all quadratic field effects. An orbital state with l=2l=2 lies in a uniform field B along z, so its first-order shift is δE=mlμBB\delta E=m_l\mu_BB.

Diagram shown in the original SVG worksheet.

  1. List the distinct shifts and find their full span.

  2. For B=.300TB=.300\,\mathrm T, find the adjacent spacing and span using μB=5.788×10−5eV/T\mu_B=5.788\times10^{-5}\,\mathrm{eV/T}.

Question 3 - Solution

An orbital Zeeman multiplet has equal adjacent energy spacings

Diagram shown in the original SVG worksheet.

(a) Method and calculation

ml=−2,−1,0,1,2⇒δE/μBB=−2,−1,0,1,2.m_l=-2,-1,0,1,2\Rightarrow\boxed{\delta E/\mu_BB=-2,-1,0,1,2}.Emax−Emin=4μBB.\boxed{E_{\max}-E_{\min}=4\mu_BB}.

(b) Interpretation and further calculation

ΔEadj=μBB=1.7364×10−5eV.\Delta E_{\mathrm{adj}}=\mu_BB=\boxed{1.7364\times10^{-5}\,\mathrm{eV}}.ΔEspan=6.9456×10−5eV.\boxed{\Delta E_{\mathrm{span}}=6.9456\times10^{-5}\,\mathrm{eV}}.

Physical check

Five orbital orientations yield five equally spaced first-order energies. Spin and spin-orbit coupling can alter the observed multiplet and are explicitly omitted here.

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