5.25 The magnetic moment vectors and associated with the intrinsic spin angular momentum S and orbital angular momentum l, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by:

${\mathit{\mu }}_{\mathit{s}}=-\left(\frac{\mathit{e}}{\mathit{m}}\right)\mathit{s}\dots \dots ..\left(1\right)$

${\mathit{\mu }}_{\mathit{l}}=-\left(\frac{\mathit{e}}{2\mathit{m}}\right)\mathit{l}$ ……..(2)

Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

<p><strong>5.25 </strong>Of the two, the relation, the relation (2) is in accordance with classical physics. It follows easily from the definitions of</p><p><span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">l</mi> <mi mathvariant="bold-italic"> </mi> </mrow> </mrow> </msub> </math> </span>= IA =<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>e</mi> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </mfrac> <mo>)</mo> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </math> </span> ( …. (i)</p><p>I = mvr = m (2<span contenteditable="false"> <math> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>/</mo> <mi>T</mi> <mo>)</mo> </math> </span> …… (ii)</p><p>where r is the radius of the circular orbit which the electron of mass m and charge (-e) completes in time T.</p><p>Dividing (i) by (ii), </p><p>Clearly, <span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>I</mi> </mrow> </mrow> </mfrac> </math> </span>= [ (e/T)<span contenteditable="false"> <math> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>]</mo> </math> </span> / [m (2 <span contenteditable="false"> <math> <mi>π</mi> <msup> <mrow> <mrow> <mi>r</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>/</mo> <mi>T</mi> <mo>)</mo> </math> </span>= - (e/2m)</p><p>Therefore<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> </msub> </math> </span>= - (e/2m)l</p><p>Since the charge of the electron is negative, it is easily seen that<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> </msub> </math> </span> and l are antiparallel, both normal to the plane of the orbit.</p><p>Note <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </mrow> <mrow> <mrow> <mi mathvariant="bold-italic">s</mi> </mrow> </mrow> </msub> <mi mathvariant="bold-italic"> </mi> </math> </span>/s in contrast to<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mi>l</mi> </mrow> </mrow> </msub> </math> </span>/l&nbsp;is e/m, i.e. twice the classically expected value. This latter result (verified experimentally) is an outstanding consequence of modern quantum theory and cannot be obtained classically.</p>

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