Discussion “Challenge
” 8D
P212, Week 8
Magnetic Dipoles in Atomic Physics
Here is one reason that we are studying magnetic dipole moments: nature is
filled
with current loops at
the atomic and subatomic levels!
Magnetic dipoles appear constantly in those fields of physics.
Consider the hydrogen atom = one proton (charge +
e
) with an electron (charge –
e
) orbiting around it.
As you know, the electron can live in a number of “orbitals”, which are states with different orbital
angular momentum
L
.
From quantum mechanics, we know that
L
of an electron in such an orbital is
L
=
l
=
where
l
is a “quantum number” that takes only integer
values: 0, 1, etc.
Further, the electron has spin angular
momentum
S
, where
S
=
s
=
. For an electron,
s
is always 1/2.
You may think of spin this way: the electron is constantly
revolving around its own internal axis, producing a constant
angular momentum
=
/2.
The diagram shows a hydrogen atom in a “d-state” with
l
= 2.
Both the spin
S
and orbital angular
momentum
L
are pointing upwards (+
z
) in the figure, but that doesn’t have to be the case.
We’ll
consider 4 states:
↑↑
,
↓↓
,
↑↓
,
and
↓↑
.
We’ll use the first, thick arrow to refer to
L
(since it is bigger
than
S
), so
↑↑
means
l
z
= +2,
s
z
= +1/2, while
↓↑
means
l
z
= -2,
s
z
= +1/2, etc.
1
As it happens, any rigid charged body that is either orbiting or spinning has a
magnetic dipole moment
that is proportional to its angular momentum.
The electron’s intrinsic magnetic moment (due to its spin)
is
µ
s
=
µ
B
, where
µ
B
= 9.3 x 10
-24
J/T
is a physical constant called the “Bohr magneton”. The orbital
motion of the electron also produces a magnetic moment:
µ
l
=
l
µ
B
(a)
Given the spin and orbital directions shown in the figure, in which directions
do the magnetic
moments
µ
s
and
µ
l
point?
Don’t forget the electron has a negative charge. Draw them on the figure.