(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have two questions. Both involve derivations from textbooks, not end-of-the-chapter problems. The two textbooks are Quantum Mechanics by Griffiths, and A Modern Approach to Quantum Mechanics by John Townsend.

My first question is about the discussion of Larmor precession found on page 180 of Griffiths. On that page, he calculates

[tex] \langle S_{x} \rangle = \frac{\hbar}{2}\mbox{sin}(\alpha)\mbox{cos}(\gamma B_{0}t) [/tex]

[tex] \langle S_{y} \rangle = -\frac{\hbar}{2}\mbox{sin}(\alpha)\mbox{sin}(\gamma B_{0}t) [/tex]

[tex] \langle S_{z} \rangle = \frac{\hbar}{2}\mbox{cos}(\alpha) [/tex]

and then states, "Evidently

[tex] \langle \bf{S} \rangle [/tex] is tilted at a constant angle [tex] \alpha [/tex] to the z-axis, and precesses about the field the field at the Larmor frequency [tex] \omega = \gamma B_{0} [/tex]..."

My question is how is does one know something about the expectation value of S from the expectation values of the components of S? Isn't it correct that

[tex] \langle {\bf S} \rangle \neq \langle S_{x} \rangle + \langle S_{y} \rangle + \langle S_{z} \rangle [/tex]

Is one supposed to just "see/know" that because the expectation values of the components of the spin have a trigonometric function of alpha, the expectation value of the spin is tilted at an angle alpha to the z-axis? Or, is this understanding similar to transforming/decomposing a vector from spherical coordinates to Cartesian coordinates (looking at figure 4.10 and decomposing S)? If that is the case, then I understand.

My second question involves the force on a magnetic dipole. On page 3 of Townsend, he states that, "If we call the direction in which the inhomogeneous magnetic field is large the z direction, we see that

[tex] F_{z} = \vec{\mu}\cdot \frac{\partial \vec{B}}{\partial z}\simeq \mu_{z}\frac{\partial B_{z}}{\partial z}" [/tex]

Here's my question. I know that [tex] \vec{F}=\nabla(\vec{\mu}\cdot\vec{B}) [/tex]

How does he pull the partial derivative into the dot product before first applying the dot product? Is it because the magnetic dipole moment is a constant in the z direction?

2. Relevant equations

[tex] \vec{F}=\nabla(\vec{\mu}\cdot\vec{B}) [/tex]

[tex] \vec{S}=S_{x}\hat{x}+S_{y}\hat{y}+S_{z}\hat{z} [/tex]

[tex] \langle S^{2} \rangle = \langle S_{x}^{2} \rangle + \langle S_{y}^{2} \rangle + \langle S_{z}^{2} \rangle [/tex]

3. The attempt at a solution

Refer to the problem description. Thank you for any help.

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# Larmor Precession and Force on a Magnetic Dipole

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