Transcript ppt

Harris Chapter 7
- Atomic Structure
•
7.1
– Orbital Magnetic Moments, discovery of intrinsic spin
•
7.2 & 7.3
– Identical Particles (warning: examples in book all inf-squ well)
– Exclusion Principle
•
7.4 & 7.5
– Multielectron Atoms, effective charges
– Hartree Treatment
•
7.6
– Spin-Orbit Effect
•
7.7
– Adding QM Angular Momenta
•
7.9 & 7.8
– Multielectron Spectroscopic Notation
– Zeeman Effect

Summary So Far

http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi
 2 2


 2m

 V   E 

2

1
2

r
r
r
r2


 2   1
2m r 2
 
 
 
1
 sin  
r 2 sin 
r   Rnl r  Ym  ,  
 2 1
r r 2 r

2
 2m r


 V

R  ER

1
2


r 2 sin 2 
7.1 Orbital Magnetic Moments and
Discovery of Intrinsic Spin
Two kinds of Angular Momentum
• Classical Angular Momentum
–
–
–
–
L=rxp
r vector, p vector  L vector
L obeys vector math
Any L possible, no contraints on Lx Ly Lz
• Quantum
–
–
–
–
–
–
Quantum Mechanical Angular Momentum
L=rxp
r vector, p vector operator  L 3 component operator
L obeys …… got to be careful
L described by two labels l , m
L and Lz can be known, Lx and Ly cannot
Bohr Model of Ang Momentum
Classical or
Semi-classical
description
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11
Vector Model of QM Ang. Momentum
quantum numbers

m
E&R Fig 7-12
Edmonds
“A.M. in QM”
pg 19: “We might imagine the vector moving in an
unobservable way about the z-axis...”
pg 29: “The QM probability density, not being time dependent,
gives us no information about the motion of the
particle in it’s orbit.”
*(r,t) (r,t)
(r,t)=(r) eiwt
Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991
Otto Stern & Walther Gerlach
~1922

L

 
rp
 n
 
p
  dq  n
Bohr’s Q hypothesis
Sommerfeld’s Q hypothesis
3
L2
   1 2
2
Lz
 m 
1
Assigned by advisor Max Born to demonstrate existence of the l, ml quantum numbers
Orbital Magnetic Moment
  i A
i
Q
q

t 2 r / v
L  r p  r mv
E&R Fig 7-11
g
 1 proton

0 neutron
 1 electron

 
q 
L
2m
e 
  g 
L
2m

Orbital Magnetic Moment
E&R Fig 7-11
  i A
e 
  g 
L
2m



g

e
   g 
  1 
2m
 1 proton

0 neutron
 1 electron

z
  z  g 
e
m 
2m
E&R Fig 7-11
e 
  g 
L
2m

Bohr magneton
bohr

  g  bohr

L

e

2m

0.927  10
 23
Am
2
B

Potential Energy of Orientatio n  U

 B

U   z B

F

  U

  z B  0
e 
  g 
L
2m

B

Potential Energy of Orientatio n  U

F

  U
Fz
 z

 B

U   z B

  z B
dB
dz
Different ml states experience different forces
Use B as z-axis.
Potential Energy of Orientatio n  U

 B

U   z B

F

  U
Fz

  z B
 z
dB
dz
Different ml states experience different forces
Stern & Gerlach
~1922
Harris Fig 7.3, 7.4
Stern & Gerlach
~1922
Intended to demonstrate space quantization (l), & therefore expected odd number of spots,
but observed an even number.
http://upload.wikimedia.org/wikipedia/en/2/29/Stern-Gerlach_experiment.PNG
Despite Stern's careful design and feasibility
calculations, the experiment took more than a year to
accomplish. In the final form of the apparatus, a beam of silver
atoms (produced by effusion of metallic vapor from an oven
heated to 1000°C) was collimated by two narrow slits (0.03 mm
wide) and traversed a deflecting magnet 3.5 cm long with field
strength about 0.1 tesla and gradient 10 tesla/cm. The splitting of
the silver beam achieved was only 0.2 mm.
Accordingly, misalignments of collimating slits or the
magnet by more than 0.01 mm were enough to spoil an
experimental run. The attainable operating time was usually only
a few hours between breakdowns of the apparatus. Thus, only a
meager film of silver atoms, too thin to be visible to an unaided
eye, was deposited on the collector plate.
Stern described an early episode:
http://www.physicstoday.org/pt/vol-56/iss-12/p53.html
Stern described an early episode:
After venting to release the vacuum, Gerlach removed the
detector flange. But he could see no trace of the silver atom beam
and handed the flange to me. With Gerlach looking over my
shoulder as I peered closely at the plate, we were surprised to see
gradually emerge the trace of the beam. . . . Finally we realized what
[had happened]. I was then the equivalent of an assistant professor.
My salary was too low to afford good cigars, so I smoked bad cigars.
These had a lot of sulfur in them, so my breath on the plate turned
the silver into silver sulfide, which is jet black, so easily visible. It
was like developing a photographic film.7
http://www.physicstoday.org/pt/vol-56/iss-12/p53.html
Wolfgang Pauli ~ 1924
• Pauli Exclusion Principle
• No two electrons can have the
same quantum number
• Postulated an additional
quantum number (i.e. label)
• Believed it came from the
interaction between electrons.
Ralph Kronig ~1925
• Spinning Electron Idea
w
Goudsmit & Ulhenbeck ~ 1925
• Studied high
resolution spectra of
alkali elements
Ocean Optics - Helium
Ocean Optics - Neon
Giancoli – fig 36.21
The old and the new term scheme of hydrogen [5]. The scheme
shows the multiplet splitting of the excited states of the
hydrogen atom with principal quantum number n=3, presented
by Goudsmit in the form in which it appeared in the original
publications of1926. The assignment in the current notation has
been added at the right. With the development of quantum
mechanics the notation changed. The quantum numbers L and J
now usedfor the orbital and total angular momentum,
respectively, correspond to K-1/2 and J-1/2 in the figure. The
"forbidden component" referred to by Goudsmit is of the type 3
2P
2
1/2 --> 2 S in which the total angular momentum is conserved
and L changes by plus or minus 1.
[5] S. Goudsmit and G.E. Uhlenbeck, Physica 6 (1926) 273.
Uhlenbeck & Goudsmit
~ 1925
The discovery note in
Naturwissenschaften is dated 17 October
1925. One day earlier Ehrenfest had written
to Lorentz to make an appointment and
discuss a "very witty idea" of two of his
graduate students. When Lorentz pointed out
that the idea of a spinning electron would be
incompatible with classical electrodynamics,
Uhlenbeck asked Ehrenfest not to submit the
paper. Ehrenfest replied that he had already
sent off their note, and he added: "You are
both young enough to be able to afford a
stupidity!"
http://www.lorentz.leidenuniv.nl/history/spin/spin.html
Uhlenbeck & Goudsmit
~ 1925
Ehrenfest's encouraging response to
his students ideas contrasted sharply with that
of Wolfgang Pauli. As it turned out, Ralph
Kronig, a young Columbia University PhD
who had spent two years studying in Europe,
had come up with the idea of electron spin
several months before Uhlenbeck and
Goudsmit. He had put it before Pauli for his
reactions, who had ridiculed it, saying that "it
is indeed very clever but of course has
nothing to do with reality". Kronig did not
publish his ideas on spin. No wonder that
Uhlenbeck would later refer to the "luck and
privilege to be students of Paul Ehrenfest".
http://www.lorentz.leidenuniv.nl/history/spin/spin.html
“This isn't right. This isn't even wrong.”
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html
His ability to make experiments self
destruct simply by being in the
same room was legendary, and has
been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
There were some people thinking about
electron spin in those days, but there was a lot of basic
opposition to such an idea. One of the first was Ralph
de Laer Kronig. He got the idea that the electron
should have a spin in addition to its orbital motion. He
was working with Wolfgang Pauli at the time, and he
told his idea to Pauli. Pauli said, "No, it's quite
impossible." Pauli completely crushed Kronig.
Then the idea occurred quite independently
to two Young Dutch physicists, George Uhlenbeck
and Samuel Goudsmit. They were working in Leiden
with Professor Paul Ehrenfest, and they wrote up a
little paper about it and took it to Ehrenfest. Ehrenfest
liked the idea very much. He suggested to Uhlenbeck
and Goudsmit that they should go and talk it over with
Hendrik Lorentz, who lived close by in Haarlem.
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian
Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field
Theory."
“This isn't right. This isn't even wrong.”
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html
They did go and talk it over with Lorentz.
Lorentz said, "No, it's quite impossible for the
electron to have a spin. I have thought of that myself,
and if the electron did have a spin, the speed of the
surface of the electron would be greater than the
velocity of light. So, it's quite impossible." Uhlenbeck
and Goudsmit went back to Ehrenfest and said they
would like to withdraw the paper that they had given
to him. Ehrenfest said, "No, it's too late; I have
already sent it in for publication "
His ability to make experiments self
destruct simply by being in the
same room was legendary, and has
been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian
Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field
Theory."
The calculation
(using current values)
LIw
w
SIw
2
ss  1   m r 2 w
5
v  rw
r < 2.8 E-19 m
b > 3 * 10 + 6
value from Bhabha scattering at CERN
“This isn't right. This isn't even wrong.”
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html
That is how the idea of electron spin got
publicized to the world. We really owe it to
Ehrenfest's impetuosity and to his not allowing the
younger people to be put off by the older ones. The
idea of the electron having two states of spin provided
a perfect answer to the duplexity.
His ability to make experiments self
destruct simply by being in the
same room was legendary, and has
been dubbed the "Pauli effect"
(Frisch 1991, p. 48; Gamow 1985).
"The Birth of Particle Physics," edited by Laurie M. Brown and Lillian
Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field
Theory."
Letter fm Thomas to Goudsmit
Part of a letter by L.H.
Thomas to Goudsmit (25
March 1926). Reproduced
from a transparency
shown by Goudsmit during
his 1971 lecture. The
original is presumably in
the Goudsmit archive kept
by the AIP Center for
History of Physics.
http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html
intrinsic spin
• Fundamental objects
– electron
– neutrino
– photon
spin – ½
spin – ½ , but LH only
spin – 1
• Composite objects
– proton
– neutron
–  delta
spin – ½
spin – ½
spin – 3/2
How to Denote Wavefunctions
(version 1)
n lm sms
 Rn r  Yml  
 sm
s
the spinor has no ‘functional form’
because spin
is not a spatial feature
n lm sms

n lm sms
n lm sms

nl
lm sms
Two types of Magnetic Moments
L
S
e 
  g 
L
2m

bohr 
g

 1 proton

0 neutron
 1 electron
  g  bohr

L

e
2m

e 
s  g s
S
2m

0.927  10  23 Am 2
 5.59
gs

proton
 3.83 neutron
 2.00 electron

 s  g s bohr

S

interesting fundamental constants
-2.002 319 304 3622 (15)
1.602 176 487
(40) x 10-19 C
7.2 & 7.3 Complications from
having Identical Particles
Exchange Symmetry
7.4 & 7.5 Multielectron Atoms
En = ( 13.6 eV ) (Z2/n2)
r   Rnl r  Ym  ,  
r = n2 ao / Z
ao = 0.529 Å
r   Rnl r  Ym  ,  
Rn   G e

 / 2
Prob = r2 R* R
  0 orbitals get sucked down the most
Crossings occur for the upper orbitals
4s 3d
3s
2s
1s
4p
3p
2p
Note: This shows how the
orbitals shift as viewed
from the perspective
of an s-orbital.
Hartree-Fock Method
Hartree-Fock Methods
Choose initial shape
For Coulomb Potl V(r)
Loop until V(r) doesn’t change much
Solve Schro Eqn
for En n
Insert fine structure
corrections
Build atom according to
This set of orbital energies En
Use the collection of n*n to
Get new electron charge distrib
Use Gauss’ Law to
get new V(r) shape
V (r )  k
Z eff e
r
Using effective charge is a very crude approximation.
2
r2 ~ n2 ao / Zeff
En ~ (Zeff2/n2) ( -13.6 eV )
Hartree-Fock
Effective Charge Effects
7.6 Spin-Orbit Effect
Corrections to the Coulomb Potl
for H-atom
•
•
•
•
•
•
•
Central Potential
Spin-Orbit (electron viewpoint)
Relativistic Spin (Thomas precession)
Relativistic Kinetic Energy
Spin-Orbit (nucleus viewpoint)
Spin-Spin
Impact of External Fields
– Zeeman Effect (applied B-field)
– Stark Effect (applied E-field)
Spin-Orbit Interaction
L
s
s
L
Note: L.H. Thomas showed that in the x-form between
non-inertial reference frames a factor of ½ appears.
Goal: find expression for the orientational potential energy
of electron intrinsic mag moment (s)
in terms of orbital motion (L) and forces (~ dV/dr).
U

  s  B

s
L
U
1  
  s  B
2
Note: L.H. Thomas showed that in the x-form between
non-inertial reference frames a factor of ½ appears.
U
1  
  s  B
2
Q
l
I dl 
l  Q
 Ze  v 
t
t

B 
s
L

o I dl  rˆ
4
r2

o Ze v  rˆ
B  
4
r2


o Ze v  rˆ
B  
4
r2
E
E

 
B    o o v  E

1  
B   2 vE
c

Ze rˆ
4o r 2


e E  F

F

  V
 
dV
rˆ
dr


1 dV 
1 dV  r
B   2
v  rˆ   2
v
ec dr
ec dr
r

B 
1 1 dV 
L
2
emc r dr
 

 
L  r  mv   mv  r
U
1  
  s  B
2
2
e 
s  g s
S
2m


B 
U

1
2m 2 c 2
1 dV
r dr
 
SL
NRG shift depends on
relative orientation
of L and S
1 1 dV 
L
2
emc r dr
L
S
How to evaluate E and S·L
Etot
 U 
*
2

njmls U njmls r dr sin  d d

all space
U

1
2m 2 c 2
1 dV
r dr
involved
in radial
integrations
 
SL
depends
on A.M.
qu. no.s
L
S
E 
1
2m2c 2
1 dV
r dr
 
S L
R
J
J
2
 LS

L  S 
2
J 2  L2  S 2  2 L  S
2 L  S  J 2  L2  S 2
E 
1
2m 2 c 2
1 dV
r dr
R
1
 j j  1    1  ss  1  2
2
electronSpin-Orbit “locks” the angle between L & S
 J is now a well-defined direction.
E 
1
2m2c 2
1 dV
r dr
 
S L
R
S
NOTE
Lz
is
no longer
well-defined
ml not a
good q. no.
L
J
S
L
Revised H-atom Level Scheme
3s
3p
3d
add in
spin-orbit
correction
2s
2p
3s1/ 2
3 p3 / 2
3d 5 / 2
3 p1/ 2
3d 3 / 2
2 p3 / 2
2s1/ 2
2 p1/ 2
nlj
1s
not required to specify NRG
j mj l ml s ms
1s1/ 2
not required to specify NRG
mj ml s ms
absolutely worthless
electron Spin-Orbit is more
important in higher-Z atoms
E 
1
2m 2 c 2
1 dV
r dr
R
1
 j j  1    1  ss  1  2
2
fn’l expression only for H-atom,
for all others, must come from
Hartree procedure
Splitting
(eV)
Li
Na
K
Rb
Cs
0.42E-4
21.E-4
72.E-4
295.E-4
687.E-4
Bigger atoms
larger Z (central charge)
~ same size
1 dV
r dr
larger
7.7 QM Angular Momentum
Bohr Model of Ang Momentum
Note: s-states (l=0) have no Bohr model picture
Eisberg & Resnick: Fig 7-11
Vector Model of Ang. Momentum
quantum numbers

m
E&R Fig 7-12
Edmonds
“A.M. in QM”
pg 19: “We might imagine the vector moving in an
unobservable way about the z-axis...”
pg 29: “The QM probability density, not being time dependent,
gives us no information about the motion of the
particle in it’s orbit.”
Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991
ADDITION OF
ANGULAR MOMENTUM
L2
Ltot = L1 + L2
L1
Ltot = L1 + L2
1 m1
 tot
2
m2
mtot
Ylto tmto t  ,  
Yl1m1  ,  
Yl2 m2  ,  
Ltot = L1 + L2
1 m1
 tot
mtot
1   2
  tot
mtot
 1   2
 m1  m2
2
m2
Addition of Angular Momentum
www.bokerusa.com
aligned configuration
“aligned” does not mean straight
www.cartowning.co.za/DBNRECGC.htm
jack-knife configuration
“jack-knife” does not mean antiparallel
Detailed Example
L1
L2
Problem: Two objects each travel in a p-orbit ( l=1 ).
The total energy of each object is degenerate wrt ml, so
we have no detailed knowledge of ml.
What are the allowed values of
ltot, mtot ?
l1=1, l2=1, m’s degenerate
m1
m2
mtot
1
1
2

0
1

1
0
0
1
1

0
0

1
1
1
1
0

0
1

1
2
mtot
2
1
0
1
2
Possibilities (m1,m2)
Allowed Values of
ltot mtot
Basic A.M. Math
J=L+S
s

j
J
S
 s
L
mj
 m  ms
Vector Representation of J
Annoying Pictures #1
Jeff’s Qs: i) what am I supposed to think about the S & L cones as drawn?
ii) I thought I was told earlier that L & S were about z ??
Annoying Pictures #2
Jeff: Pictures such as this confuse the vector symbols L and S
with the quantum numbers ℓ and s .
For instance, how could L and S ever point in the same direction?
TOTAL ANGULAR MOMEMTUM
J=L+S
More Detailed H-atom Level Scheme
3 p3 / 2
3s
3p
3d
3s1/ 2
3 p1/ 2
3d 5 / 2
3d 3 / 2
2 p3 / 2
2s1/ 2
2s
2 p1/ 2
2p
1s1/ 2
1s
Energies & Spectra not sensitive to
l ml
Energies & Spectra not sensitive to
j mj
l ml s ms
till next page
Ocean Optics - Helium
Because of the doublets, the states cannot be completely degenerate
 “spin-orbit effect” + …
Ocean Optics - Neon
Because of the doublets, the states cannot be completely degenerate
 “spin-orbit effect” + …
7.9 Multi-electron Spectra
Multi-e Spectroscopic Notation
QUANTUM NUMBERS
principal: n
ltot , stot
jtot .
2 sto t 1
 tot j
tot
Stot = S1 + S2 + …
Ltot = L1 + L2 + …
Jtot = Ltot + Stot
stot = 1, ltot=0, jtot=1
2S
1
Two Kinds of Notation
• Where an individ electron is at
• nlj
–
–
–
–
1s1/2
2s1/2
2p1/2
2p3/2
• A.M. for whole atom
• 2Stot+1 ltot jtot
– 1 S0
– 3 S1
– 3 P0 , 3 P1 , 3 P2
Curious Things That Happen:
Ground State of Helium
1s
Ltot = L1 + L2
 tot  0
Stot = S1 + S2
0
stot  
1
asym
sym
system = (spatial wfn) (spin wfn)
1s
sys 
sys 
1
2
1
2
1s (1)1s (2)  1s (2) 1s (1)
 1s (1) 1s (2)  1s (2) 1s (1) 
1
2





  

1
  
2

1S
0
3S
1
7.8 Atoms in
External Magnetic Fields
-- the Zeeman Effect
Corrections to the Coulomb Potl
for H-atom
•
•
•
•
•
•
•
Central Potential
Spin-Orbit (electron viewpoint)
Relativistic Spin (Thomas precession)
Relativistic Kinetic Energy
Spin-Orbit (nucleus viewpoint)
Spin-Spin
Impact of External Fields
– Zeeman Effect (applied B-field)
– Stark Effect (applied E-field)
Weak-Field Zeeman
• Hartree-Fock Coulomb & related Procedures
• Fine Structure
– spin-orbit ( jtot becomes important )
– relativistic
• Zeeman
H’Zeeman = - tot * Bext
Bext < few 0.1’s Tesla
Weak Field Zeeman
tot

tot


l tot

e 
g
Ltot
2m

 s tot

e 
gs
Stot
2m

e 

( Ltot  2 Stot )
2m
electronSpin-Orbit “locks” the angle between L & S
 J is now a well-defined direction.
E 
1
2m2c 2
1 dV
r dr
 
S L
R
S
NOTE
Lz
is
no longer
well-defined
ml not a
good q. no.
L
J
S
L
Weak-Field Zeeman
eSO makes jtot good quantum number,
Jtot
tot
mltot & mstot become ‘confused’ (near worthless).
Jtot is ‘well-defined’ direction; jtot mjtot
g
project average tot onto Jtot
 tot  tot cos g
onto
J
 tot
 
μ J
J
tot

 
e 

( L  2S )  ( L  S )
2m
tot j ( j  1)
Weak Field Zeeman
Jtot
projection of
onto J
tot onto J onto B
onto J  onto J

cos 
onto B
Bext
onto J
 onto J
onto B
Jz
J
EZeeman = - tot * Bext
e
()( )
2m


 
( L  2S )  ( L  S )
j ( j  1)
mj
j ( j  1)
Bext
stot=0
Strong-Field Zeeman
• Hartree-Fock Coulomb & related procedures
• Zeeman
• Fine Structure
– spin-orbit
– relativistic
H’Zeeman = - tot * Bext
Ltot
Strong Field Zeeman
Stot

Bext
tot

e 

( Ltot  2 Stot )
2m
H’Zeeman = - tot * Bext
EZeeman
strong

e
(m tot  2 ms tot ) Bext
2m