an external magnetic moment it will experience a torque...
Transcript of an external magnetic moment it will experience a torque...
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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
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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
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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
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More on Magnetic MomentsForce on magnetic moments in a nonuniformB field
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Stern-Gerlach Experiment
What would be expected classically?
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Stern-Gerlach ExperimentStern-Gerlach’s experiment with silver atoms resulted in two distinct lines, not three
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Stern-Gerlach ExperimentPostcard sent to Bohr from Gerlach
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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)
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Spin
Spin of electrons = 1/2
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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
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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
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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 ,
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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
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21 2
−+
−
+
↓↓−=
↑↑+=
=⎟⎠⎞
⎜⎝⎛ +=
χχ
χ
χ
h
h
hh
z
z
S
S
S
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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
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SpinThis explains the results from Stern-Gerlachand Phipps-Taylor
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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
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SpinMagnetic field produced by orbiting proton (nuclear dipole moment)
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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?
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SpinStern-Gerlach experiments
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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