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Magnetic moment of a current loop:
iA
currentarea
enclosed by current
loop
Orbiting electrons form a current loop which give rise to a magnetic field.
Since the current is defined as the direction of flow of positive charge, the orientation of the magnetic moment will be antiparallel to the
angular momentume of the electron and can be found using the right hand rule.
The current loop of the orbiting electron sets up a magnetic dipole which behaves like a bar magnet with the north-south axis directed along .
Kepler’s Law of areas: A line that connects a planet to the sun sweeps out
equal areas in equal times.
m
L
T
A
2
T
Amr
T
mr
T
rmmvrL 2
22 2
vA
T
qi
Lm
eL
m
qiA
22
e
Remember that the z component of angular momentum is quantized in units of so the magnetic dipole
moment is quantized as well:
J/T10274.92
24m
eB
The Bohr magneton
mmm
eL
m
eB
ez
ez
22
where
22
2
md
d
t
L
B
Torque exerted:
Magnetic dipole tends to want to align itself with the magnetic field but it can never align due to the
uncertainty principle!
Here, the gravitational force provides the torque in place of a magnetic field and the angular momentum comes from the spinning of the top.
LddL
sin
dtLBm
qdtLd
e
sin2
Bm
e
dt
Ld
Ldt
d
eL 2sin
1
No magnetic field Magnetic field applied
allowed transitions
forbidden transition
angular momentum must be conserved
…photons carry angular momentum.
Remember that not all transitions are allowed. “Satellite” lines appear at the plus or minus the Larmor frequency only and not at multiples of that frequency.
1,0,1
1
m
You expect a number of equally spaced “satellite lines” displaced
from the emission lines by multiples of the Larmor frequency.
So we have seen that the current loop created by an electron orbiting in an atomic creates a dipole moment that interacts like a bar magnet with a magnetic field.
For many atoms, the number and spacing of the satellite lines are not what we would expect just from the orbital magnetic moment….there must be some other contribution to the magnetic moment.
dq Classically: you could imagine a scenario where the electron had some volume and the charge were distributed uniformly throughout that volume such that if the electron spun on its axis, it would give rise to current loops.
The electron has its own magnetic
moment, and acts as a little bar
magnet as well.
dq
Lm
eL
m
qiA
22
In analogy to the orbital magnetic moment:
the magnetic moments contributed by a differential elements of charge can be summed to be:
Sm
eL
m
q
ei
es
22
the “spin” angular momentum
the “spin” magnetic moment
More generally, if the charge is not uniform:
Sm
eg
es
2
the “g” factor
Beam split into two discrete parts! Outer electron in silver is in an s state (l=0), magnetic moment comes from the spin of the outer electron.
In addition to the orbital magnetic moment, we must take into account the spin.
The spin orientation:
2
1
2
1where ormmS ssz
Electrons come in “spin up” and “spin down” states.
2
3)1( ssS
The magnitude of the spin angular momentum is:
SgLm
e
es
20
-e
nucleus
spin
spin
Magnetic field, B, seen by the electron due to the orbit of the nucleus
We have learned about how an external magnetic field interacts with the magnetic moments in the atom, but if we look at this from the point of view of an electron, we realize that the electron “sees” a magnetic field from the apparent orbit of the
positively charged nucleus.
sssj
jjjmmJ
jjJ
SLJ
jjz
,,1,
:number quantum momentumangular total
,,1,with
)1(
??
a
a
a
aab
b
b
The first two pictures give the same outcome. Even though a and b are identical, you can tell them apart by following them along their unique paths.
Quantum mechanically, each particle has some probability of being somewhere at a particular time, which overlaps greatly at the collision point.
Which particle emerges where? In wave terms, they interfered.
bb
A Bx1 x2 A Bx2 x1
valley
vall
ey
hil
l
hill
Wavefunction not generally symmetric under exchange of identical particles!!
Consider two particles in a box, one in the n=1 state, the other in the n=2 state.
)()()()(
2
1),( 122121 xxxxxx BABAs )()()()(
2
1),( 122121 xxxxxx BABAs
21 xx 21 xx
Symmetric: probability generally highest when particles are closest together. “Huddling”.
Antisymmetric: probability generally highest when particles are furthest apart. “avoiding one another”.