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Three telescope images of the pinwheel galaxy. In figure a, the image is of visible light. The galaxy appears as a collection of stars, very dense in the center and with spiraling arms. Figure b is the 21 c m radiation image. The spiral nature is more distinct in this image, and the central bulge is absent. Figure c overlays both the visible and 21 c m images.
The magnetic interaction between the electron and proton in the hydrogen atom is used to map the spiral arms of the Pinwheel Galaxy (NGC 5457). (a) The galaxy seen in visible light; (b) the galaxy seen in 21-cm hydrogen radiation; (c) the composite image of (a) and (b). Notice how the hydrogen emission penetrates dust in the galaxy to show the spiral arms very clearly, whereas the galactic nucleus shows up better in visible light (credit a: modification of work by ESA&NASA; credit b: modification of work by Fabian Walter).

A complete specification of the state of an electron in a hydrogen atom requires five quantum numbers: n , l , m , s , and m s . The names, symbols, and allowed values of these quantum numbers are summarized in [link] .

Summary of quantum numbers of an electron in a hydrogen atom
Name Symbol Allowed values
Principal quantum number n 1, 2, 3, …
Angular momentum l 0, 1, 2, … n – 1
Angular momentum projection m 0 , ± 1 , ± 2 ,... ± l
Spin s 1/2 (electrons)
Spin projection m s ½ , + ½

Note that the intrinsic quantum numbers introduced in this section ( s and m s ) are valid for many particles, not just electrons. For example, quarks within an atomic nucleus are also spin-half particles. As we will see later, quantum numbers help to classify subatomic particles and enter into scientific models that attempt to explain how the universe works.

Summary

  • The state of an electron in a hydrogen atom can be expressed in terms of five quantum numbers.
  • The spin angular momentum quantum of an electron is = + ½ . The spin angular momentum projection quantum number is m s = + ½ or ½ (spin up or spin down).
  • The fine and hyperfine structures of the hydrogen spectrum are explained by magnetic interactions within the atom.

Conceptual questions

Explain how a hydrogen atom in the ground state ( l = 0 ) can interact magnetically with an external magnetic field.

Even in the ground state ( l = 0 ), a hydrogen atom has magnetic properties due the intrinsic (internal) electron spin. The magnetic moment of an electron is proportional to its spin.

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Compare orbital angular momentum with spin angular momentum of an electron in the hydrogen atom.

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List all the possible values of s and m s for an electron. Are there particles for which these values are different?

For all electrons, s = ½ and m s = ± ½ . As we will see, not all particles have the same spin quantum number. For example, a photon as a spin 1 ( s = 1 ), and a Higgs boson has spin 0 ( s = 0 ).

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Are the angular momentum vectors L and S necessarily aligned?

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What is spin-orbit coupling?

An electron has a magnetic moment associated with its intrinsic (internal) spin. Spin-orbit coupling occurs when this interacts with the magnetic field produced by the orbital angular momentum of the electron.

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Problems

What is the magnitude of the spin momentum of an electron? (Express you answer in terms of . )

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What are the possible polar orientations of the spin momentum vector for an electron?

Spin up (relative to positive z -axis):
θ = 55 ° .
Spin down (relative to positive z -axis):
θ = cos −1 ( S z S ) = cos −1 ( 1 2 3 2 ) = cos −1 ( −1 3 ) = 125 ° .

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For n = 1 , write all the possible sets of quantum numbers ( n , l , m , m s ).

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A hydrogen atom is placed in an external uniform magnetic field ( B = 200 T ). Calculate the wavelength of light produced in a transition from a spin up to spin down state.

The spin projection quantum number is m s = ± ½ , so the z- component of the magnetic moment is
μ z = ± μ B .
The potential energy associated with the interaction between the electron and the external magnetic field is
U = μ B B .
The energy difference between these states is Δ E = 2 μ B B , so the wavelength of light produced is
λ = 8.38 × 10 −5 m 84 μ m

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If the magnetic field in the preceding problem is quadrupled, what happens to the wavelength of light produced in a transition from a spin up to spin down state?

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If the magnetic moment in the preceding problem is doubled, what happens to the frequency of light produced in a transition from a spin-up to spin-down state?

It is increased by a factor of 2.

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For n = 2 , write all the possible sets of quantum numbers ( n , l , m , m s ).

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Practice Key Terms 6

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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