1

More on Magnetic MomentsRecall when a magnetic moment is placed in an external magnetic moment it will experience a torque (like a compass)

If the orientational potential energy cannot be dissipated, the magnetic moment will precessabout the magnetic field (like a spinning top)This is because

Bl

rrr×= μτ

precess. will Thus . lar toperpendicu be also will

change the Since . lar toperpendicu and

planey -x thein be willdirection The

LLLddtLdL

Bl

rrr

rrr

rrr

=

×=

τ

μτ

2

More on Magnetic Moments

3

More on Magnetic MomentsThis is called Larmor precession and we can use the figure to calculate ωL

frequency precession Larmor theis 2

sin

sin2sin1

meB

BdtdL

dtdL

me

dtdL

Ldtd

L

l

lL

=

==

−===

ω

θμτ

θμθϕω

4

More on Magnetic MomentsIf the magnetic field is uniform there will be no net force on the magnetic moment (but there will be a torque)If the magnetic field is nonuniform there will be a net force on the dipole

( )

Lg

mgzB

zBB

zF

Bl

lBlz

lz

l Z

r

h

r

rr

μμ

μμμ

−=

∂∂

−=∂∂

=⋅−∂∂

−=

used wewhere

5

More on Magnetic MomentsForce on magnetic moments in a nonuniformB field

6

Stern-Gerlach Experiment

What would be expected classically?

7

Stern-Gerlach ExperimentStern-Gerlach’s experiment with silver atoms resulted in two distinct lines, not three

8

Stern-Gerlach ExperimentPostcard sent to Bohr from Gerlach

9

SpinPhipps and Taylor repeated the Stern-Gerlach experiment using hydrogen and found the same resultTo explain these results Goudsmit and Uhlenbeck (grad students) proposed that in addition to orbital angular momentum, electrons had intrinsic angular momentum, or spin

In spite of its name, there is no classical analog for spin (even though we draw electrons like spinning tops)

10

Spin

Spin of electrons = 1/2

11

SpinThe great thing about spin from our standpoint is that the spin angular momentum algebra is identical to that of orbital angular momentum

[ ] [ ] [ ][ ] [ ] [ ] 0,,,

, ,, ,,

)1(

,...,

,...2,23,1,

21,0

222

2222

,,

,2

,2

===

===

++=

=

+=

−=

=

zyx

yxzxzyzyx

zyx

mssmsz

msms

s

SSSSSS

SiSSSiSSSiSSSSSS

mS

ssS

ssm

s

ss

ss

hhh

h

h

χχ

χχ

12

SpinSpin is an intrinsic property of a particle or atomDifferent particles and atoms have different spins

Pions, Higgs bosons have spin 0Electrons, positrons, muons, protons, neutrons, quarks have spin 1/2Photons and W and Z-bosons have spin 1Delta particles have spin 3/2Gravitons (may have) spin 2

13

SpinThis also means our complete hydrogen wave function must include a spin wave function

And we’ll need to specify s, ms as additional quantum numbers

( ) ( ) χψ h

iEt

extx−

=Ψ ,

14

SpinFor the most part we will focus on electrons

Spin s = 1/2

2)only are (there ionseigenfunct spin theare and

or or or down spinor 2

or or or up spinor 2

431

21

21 2

−+

−

+

↓↓−=

↑↑+=

=⎟⎠⎞

⎜⎝⎛ +=

χχ

χ

χ

h

h

hh

z

z

S

S

S

15

SpinJust as there was an orbital magnetic dipole moment, so will there be a spin magnetic dipole moment

BμΔE

mμgSμgμ

Sμgμ

BμΔE

mμgLμgμ

gLμgμ

s

sBlzBs

s

Bss

l

lBlzBl

l

lBl

l

z

z

rrh

r

h

r

rrh

r

h

r

⋅−=

−=−=

−=

⋅−=

−=−=

=−=

moment magnetic Spin

1 where

moment magnetic Orbital

16

SpinThis explains the results from Stern-Gerlachand Phipps-Taylor

17

Spin

Aside, even in the absence of an external magnetic field, the electron will still experience a magnetic fieldThus we would expect a doublet splitting of the hydrogen atom spectral lines

18

SpinMagnetic field produced by orbiting proton (nuclear dipole moment)

19

Spin

A polarized beam is one where all electrons have spin up or spin down

What is the spin wave function for a polarized beam with spin up?

An unpolarized beam has equal amounts of spin up and spin down

What is the wave function for an unpolarized beam?

20

SpinStern-Gerlach experiments

21

SpinThe Stern-Gerlach apparatus can be used to measure the direction of the electron spinBecause only one component of spin (Sx, Sy, Sz) can be measured at a time measurement of a spin component in an orthogonal direction will result in 50/50 probability of spin up/down along the new direction

Any previous knowledge of the original spin direction is destroyed