Chapter 8: 76E (page 344)

A hydrogen atom is subjected to a magnetic field Bstrong enough to completely overwhelm the spin-orbit coupling. Into how many levels would the 2p level split, and what would be the spacing between them?

Short Answer

Expert verified

5 levels,

And energy difference is $∆ E = \frac{e h B}{2 m}$

Step by step solution

total energy of magnetic field

In the case of a strong magnetic field Bsuch as this one, the total energy Eis given by

$E = \frac{e h B}{2 m} (m_{l} + m_{s}) + E_{o} . . . . . . . . . . . . . . . . . . . (1)$

where E_ois the zero-field energydata-custom-editor="chemistry" $(B = 0)$ .

find level splits

The possible values for m_l are $(l = 1)$

$m_{l} = - 1, 0, 1 . . . . . . . . . . . . . (2)$

Since m_s can have values of $\pm \frac{1}{2}$ , we conclude that the 2_p level splits into $3 + 2 = 5$ levels, since there are 3 possible values for m_l and 2 possible values for m_s.

The quantity $m_{l} + 2 m_{s}$ can have values of

role="math" localid="1658468430759" $m_{l} + 2 m_{s} = - 2, - 1, 0, 1, 2 . . . . . . . . . . . . (3)$

For example, let us take the following two levels: $m_{l} + 2 m_{s} = 1$ and $m_{l} + 2 m_{s} = 2$ .

energy difference ∆E

The energy difference $∆ E$ is then (we use Eq. (l))

$∆ E = \frac{e h B}{2 m} (2 - 1) ∆ E = \frac{e h B}{2 m}$

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A hydrogen atom is subjected to a magnetic field Bstrong enough to completely overwhelm the spin-orbit coupling. Into how many levels would the 2p level split, and what would be the spacing between them?

Short Answer

Step by step solution

total energy of magnetic field

find level splits

energy difference ∆E

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