What is the value of the Bohr magneton? Use that value to calculate the energy difference between the ml = 0 and ml = +1 components in the 2p state of atomic hydrogen placed in an external field of 2.00 T. Strategy To find the Bohr magneton we insert the known values of e, h, and m into the

Question

What is the value of the Bohr magneton? Use that value to calculate the energy difference between the [latex]m_{\ell}=0[/latex] and [latex]m_{\ell}=+1[/latex] components in the 2p state of atomic hydrogen placed in an external field of 2.00 T.

Strategy To find the Bohr magneton we insert the known values of [latex]e, \hbar, ext { and } m[/latex] into the equation for [latex]\mu_{ B }[/latex] [see text after Equation (7.28)]. The energy difference is determined from Equation (7.31).

[latex]\mu_{z}=\frac{e \hbar}{2 m} m_{\ell}=-\mu_{ B } m_{\ell}[/latex] (7.28)

[latex]V_{B}=-\mu_{z} B=+\mu_{ B } m_{\ell} B[/latex] (7.31)

Accepted Answer

The Bohr magneton is determined to be

[latex]\begin{aligned}\mu_{ B } &=\frac{e \hbar}{2 m} \&=\frac{\left(1.602 imes 10^{-19} C ight)\left(1.055 imes 10^{-34} J \cdot s ight)}{2\left(9.11 imes 10^{-31} kg ight)}\end{aligned}[/latex]

[latex]\mu_{ B }=9.27 imes 10^{-24} J / T[/latex] (7.32)

The international system of units has been used (T = tesla for magnetic field). The energy splitting is determined from Equation (7.31) (see also Figure 7.6a):

[latex]\Delta E=\mu_{ B } B \Delta m_{\ell}[/latex] (7.33)

where [latex]\Delta m_{\ell}=1-0=1[/latex]. Hence, we have

[latex]\begin{aligned}\Delta E=\left(9.27 imes 10^{-24} J / T ight)(2.00 T ) &=1.85 imes 10^{-23} J \&=1.16 imes 10^{-4} eV\end{aligned}[/latex]

An energy difference of [latex]10^{-4}[/latex] eV is easily observed by optical means.

Question 7.7: What is the value of the Bohr magneton? Use that value to ca...

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