magnetic moment of a current loop: current area enclosed by current loop orbiting electrons form a...

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”.