A complete specification of the state of an electron in a hydrogen atom requires five quantum numbers:
n ,
l ,
m ,
s , and
. 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
Spin
s
1/2 (electrons)
Spin projection
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
=
. The spin angular momentum projection quantum number is
m
s
(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
) can interact magnetically with an external magnetic field.
Even in the ground state (
), a hydrogen atom has magnetic properties due the intrinsic (internal) electron spin. The magnetic moment of an electron is proportional to its spin.
List all the possible values of
s and
for an electron. Are there particles for which these values are different?
For all electrons,
and
As we will see, not all particles have the same spin quantum number. For example, a photon as a spin 1 (
), and a Higgs boson has spin 0 (
).
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.
A hydrogen atom is placed in an external uniform magnetic field (
). Calculate the wavelength of light produced in a transition from a spin up to spin down state.
The spin projection quantum number is
, so the
z- component of the magnetic moment is
The potential energy associated with the interaction between the electron and the external magnetic field is
The energy difference between these states is
, so the wavelength of light produced is
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?
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?
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Jharna
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Jharna
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