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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=ehB2m

Step by step solution

01

total energy of magnetic field

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

E=ehB2m(ml+ms)+Eo...................(1)

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

02

find level splits

The possible values for ml arel=1

ml=-1,0,1.............2

Since ms can have values of ±12, we conclude that the 2p level splits into 3+2=5 levels, since there are 3 possible values for ml and 2 possible values for ms.

The quantityml+2ms can have values of

role="math" localid="1658468430759" ml+2ms=-2,-1,0,1,2............3

For example, let us take the following two levels:ml+2ms=1 and ml+2ms=2.

03

energy difference ∆E

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

∆E=ehB2m2-1∆E=ehB2m

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Most popular questions from this chapter

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The 21cm Line: One of the most important windows to the mysteries of the cosmos is the 21cm line. With it astronomers map hydrogen throughout the universe. An important trait is that it involves a highly forbidden transition that is, accordingly, quite long-lived. But it is also an excellent example of the coupling of angular momentum. Hydrogen's ground state has no spin-orbit interaction—forl=0there is no orbit. However, the proton and electron magnetic moments do interact. Consider the following simple model.

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