Download CHAPTER 8: Atomic Physics

Total Angular Momentum
Orbital angular momentum
Spin angular momentum
Total angular momentum
L, Lz, S, SzJ and Jz are quantized
Total Angular Momentum

If j and mj are quantum numbers for the single electron
(hydrogen atom)

Quantization of the magnitudes

The total angular momentum quantum number for the single
electron can only have the values
The Total Angular Momentum
Diagram
Figure 8.5 When forming the total angular
momentum from the orbital and spin angular
momenta, the addition must be done
vectorially,
.
Spin-Orbit Coupling
An effect of the spins of the electron and the orbital angular
momentum interaction is called spin-orbit coupling.

• The dipole potential energy
• The spin magnetic moment 
•

.
is th·e magnetic field due to the proton
where cos a is the angle between
Total Angular Momentum
No external magnetic field: J can have a fixed valuein only one
direction. This direction can be chosen randomly
 Only Jz can be known because the uncertainty principle forbids Jx
or Jy from being known at the same time as Jz
Total Angular Momentum
With an external magnetic field:

will precess about
Total Angular Momentum

Now the selection rules for a single-electron atom become



Δn = anything
Δmj = 0, ±1
Δℓ = ±1
Δj = 0, ±1
Hydrogen energy-level diagram for n = 2 and n = 3 with the spinorbit splitting
The Energy-Level
Diagram of Sodium
Fine structure splitting is too
small on this scale(not shown)
Many-Electron Atoms
Hund’s rules:
1) The total spin angular momentum S should be maximized to the
extent possible without violating the Pauli exclusion principle.
2) Insofar as rule 1 is not violated, L should also be maximized.
3) For atoms having subshells less than half full, J should be
minimized.

We label with 1 and 2 the two-electron atom

There are LS coupling and jj coupling to combine four angular
momenta J.
LS Coupling

This is used for most atoms when the magnetic field is weak.

If two electrons are in a single subshell, S = 0 or 1 depending on
whether the spins are antiparallel or parallel.

For given L, there are 2S + 1 values of J
For L > S, J goes from L − S to L + S
For L < S, there are fewer than 2S + 1 possible J values
The value of 2S + 1 is the multiplicity of the state



Table 8-2 p287
LS Coupling



The notation for a single-electron atom becomes
n2S+1 LJ
The letters and numbers are called spectroscopic symbols.
There are singlet states (S = 0) and triplet states (S = 1) for two
electrons.
LS Coupling
Magnesium(0ne electron in 3s
subshell and the other excited to the nl subshell)

There are separated energy
levels according to whether
they are S = 0 or 1

Allowed transitions must
have ΔS = 0

No allowed (forbidden)
transitions are possible
between singlet and triplet
states with much lower
probability
LS Coupling

The allowed transitions for the LS coupling scheme are




ΔL = ±1
ΔJ = 0, ±1
ΔS = 0
(J = 0 → J = 0 is forbidden)
A magnesium atom excited to the 3s3p triplet state has no lower
triplet state to which it can decay.
It is called metastable, because it lives for such a long time on
the atomic scale.
jj Coupling

It is for the heavier elements, where the nuclear charge causes the
spin-orbit interactions to be as strong as the force between the
individual and .
Mercury
Figure 8-11 p290
Figure 8-12 p290
Clicker - Questions
8.3: Anomalous Zeeman Effect

More than three closely spaced optical lines were observed.

The interaction that splits the energy levels in an external magnetic
field
is caused by
interaction.
Orbital contribution
and
Spin magnetic moment

The magnetic moment depends on

The 2J + 1 degeneracy for a given total angular momentum state J is
removed by the effect of the
.
If the
is small compared to internal magnetic field, then and
precess about while precesses slowly about
.

Total Angular Momentum
With an external magnetic field:

will precess about
Anomalous Zeeman Effect

The total magnetic moment is
μB is the Bohr magneton and
it is called the Landé g factor



The magnetic total angular momentum numbers mJ from −J to J in integral
steps.
splits each state J into 2J + 1 equally spaced levels separated ΔE = V.
For photon transitions between energy levels
ΔmJ = ±1, 0 but
is forbidden when ΔJ = 0.
Figure 8.15 Examples of transitions for the normal Zeeman effect.
The nine possible transitions are labeled, but there are only three
distinctly different energies because the split energy levels are
equally spaced (∆E) for both the 1D2 and 1P1 states
Figure 8-15 p293
Anomalous Zeeman effect for Na
Figure 8-16 p294
Rydberg Atom
An atom that is highly excited with the outermost electron in a high energy level near ionization
Properties of Rydberg atoms:
Useful Combinations of Constants
Solution: In the first excited state, go to the next
higher level. In neon one of the 2p electrons is
promoted to 3s, so the configuration is 2p^5 3s^1
. By the same reasoning the first excited state of
xenon is 5p^5 6s^1.
The 3s state of Na has energy of -5.14eV.Determine the
effective nuclear charge.
From Figure 8.4 we see that the radius of Na is about
0.16 nm. We know that for single-electron atoms it
holds
Problem 8.21
1
1


3
 1240eV  nm  

  7.334 10 eV.
1 2
 766.41 nm 769.90 nm 
As in Example 8.8 the internal magnetic field is
1.
E 
hc

hc
9.109 1031 kg  7.334 103 eV 

mE
B

 63.4 T
19
16
e
1.602 10 C  6.582 10 eV  s 
Problem 8.31
 
1.
av  

B
B

 L  2S  and J  L  S
 L  2S    L  S  J
Using the equation in the problem, we have
J2

Now L  2S  L  S  L2  2S 2  3L  S and using J  L  S we have J  J  L2  S 2  2L  S so
J 2  L2  S 2
and
L·S 
2

B 3J 2  S 2  L2
3J 2  S 2  L2
. Therefore av  
L  2S  L  S 
J
2
2J 2


. With B defined to be in the +z-direction, V   av  B 
B B 3J 2  S 2  L2
2J 2
J z . As usual
3J 2  S 2  L2
J z  mJ , so V  B BmJ
. Now the vector magnitudes are J 2  j ( j  1)
2
2J
S 2  s( s  1) 2 , and L2  (  1) 2 so we obtain V  B BmJ g where
g
3 j ( j  1)  s( s  1)  (  1)
j ( j  1)  s( s  1)  (  1)
.
 1
2 j ( j  1)
2 j ( j  1)
2
,
Problem 8.33
1/ 2  3 / 2   1/ 2  3 / 2   1  1  2
2 1/ 2  3 / 2 
 3 / 2  5 / 2   1/ 2  3 / 2   1(2)  1  1  4
g  1
2  3 / 2  5 / 2 
3 3
1. a) L  0 , S  J  1/ 2 ; g  1 
(b) L  1 , S  1/ 2 , and J  3 / 2 ;
(c) L  2 , S  1/ 2 , and J  5 / 2 ; g  1 
 5 / 2  7 / 2   1/ 2  3 / 2   2(3)  1  1  6
2  5 / 2  7 / 2 
5 5
Problem 8.15
Problem 8.17
Problem 8.19
Problem 10. 18
1. (a) Using dimensional analysis and the fact that the energy of each photon is
hc /   3.14 1019 J , then
N
5 103 J/s

 1.59 1016 photon/s .
19
t 3.14 10 J/photon
(b) 0.02 mole is equivalent to 0.02 N A  1.20 1022 atoms . The fraction participating is
1.59 1016
 1.33 106 .
22
1.20 10
(c) The transitions involved have a fairly low probability, even with stimulated emission. We are
saved by the large number of atoms available.
Problem 10.19
1.15 eV 1.602 1019 J
1. (a) The energy of each photon is E2  E1 
·
 1.84 1019 J/photon .
photon
1eV
Since P  E / t then we have P  1.84 1019 J/photon  5.50 1018 photons/s   1.00W .
hc 1.986 1025 J  m

 1.08 106 m  1.08 μm
(b)  
19
E
1.84 10 J
Problem 10.20
1. (a) With the given wavelength, we know that each photon has an energy of
1.986 1025 J  m
E

 5.658 1019 J .
9

35110 m
hc
Therefore the number of transitions required is N 
40 103 J
 7.07 1022 photons .
19
5.658 10 J/photon
40 103 J
(b) Average power is energy divided by time: P 
 1.14 1013 W
9
3.5 10 s
or about 11 TW
Problem 10.22
2 1 m 
 6.67 109 s
8
2.998 10 m/s
16 103 m
16 103 m
(b) For the 16-km round trip t 

 1.6 108 s .
8
4
8
2.998 10 m/s 1  3 10  2.998 10 m/s
1. (a) t 
Because this result is larger than the desired uncertainty in timing, it is important to take
atmospheric effects into account.
Why does Bose-Einstein Condensation of Atoms Occur?
Rb atom
Wieman
Na atom
Ketterle______
Eric Cornell and Carl
Wolfgang
Nobel Price
2001
Consider boson and fermion wave functions of two identical particles labeled “1” and “2”.
For now they can be either fermions or bosons:
Solutions:
Identical
probability density the same.
The solutions to this equation are
+ symmetric =boson - antisymmetric=fermion
Proof:
= nonzero probability occupying the same state favors to be
in the lower states for Bose-Einstein Condensation
Two bosons can occupy the same state this can be generalized to many bosons and under favorable
conditions Bose- Einstein condensation occurs. Bosons have integer spins and Fermions half integer spins.
Bose –Einstein condensation in Gases
Clicker - Questions
Stimulated Emission and Lasers
Tunable laser:
• The emitted radiation wavelength can be adjusted as wide as 200 nm.
• Semi conductor lasers are replacing dye lasers.
Free-electron laser:
From: Demonstration of electron acceleration in a laser-driven dielectric
microstructure
(Byer et al.Vol 503,nature,2013)
a
b
DLA structure and experimental set-up. a, Scanning electron microscope image of the longitudinal
cross-section of a DLA structure fabricated as depicted in Extended Data Fig. 1a. Scale bar, 2 mm.b,
Experimental set-up. Inset, a diagram of the DLA structure indicating the field polarization direction and
the effective periodic phase reset, depicted as alternating red (acceleration) and black (deceleration)
arrows. A snapshot of the simulated fields in the structure shows the corresponding spatial modulation in
the vacuum channel.
Stimulated Emission and Lasers
• This laser relies on charged particles.
• A series of magnets called wigglers is used to accelerate a beam of
electrons.
• Free electrons are not tied to atoms; they aren’t dependent upon atomic
energy levels and can be tuned to wavelengths well into the UV part of
the spectrum.
Scientific Applications of Lasers
• An extremely coherent and nondivergent beam is used in making
precise determination of large and small distances. The speed of
light in a vacuum is defined. c = 299,792,458 m/s.
• Pulsed lasers are used in thin-film deposition to study the electronic
properties of different materials.
• The use of lasers in fusion research
• Inertial confinement:
A pellet of deuterium and tritium would be induced into fusion by an
intense burst of laser light coming simultaneously from many
directions.
Holography
• Consider laser light emitted by a reference source R.
• The light through a combination of mirrors and lenses can be made to
strike both a photographic plate and an object O.
• The laser light is coherent; the image on the film will be an interference
pattern.
Holography
• After exposure this interference pattern is a hologram, and when the
hologram is illuminated from the other side, a real image of O is formed.
• If the lenses and mirrors are properly situated, light from virtually every
part of the object will strike every part of the film.
Each portion of the film contains enough information to
reproduce the whole object!
Holography
Transmission hologram:
• The reference beam is on the same side of the film as the object and
the illuminating beam is on the opposite side.
Reflection hologram:
• Reverse the positions of the reference and illuminating beam.
The result will be a white light hologram in which the different colors
contained in white light provide the colors seen in the image.
Interferometry:
• Two holograms of the same object produced at different times can be
used to detect motion or growth that could not otherwise be seen.
Quantum Entanglement, Teleportation, and Information
• Schrödinger used the term “quantum entanglement” to describe a
strange correlation between two quantum systems. He considered
entanglement for quantum states acting across large distances, which
Einstein referred to as “spooky action at a distance.”
Quantum teleportation:
• No information can be transmitted through only quantum entanglement,
but transmitting information using entangled systems in conjunction with
classical information is possible.
Quantum Entanglement, Teleportation, and Information
Alice, who does not know the property of the photon, is spacially
separated from Bob and tries to transfer information about photons.
1)
A beam splitter can be used to produce two additional photons that
can be used to trigger a detector.
2)
Alice can manipulate her quantum system and send that
information over a classical information channel to Bob.
3)
Bob then arranges his part of the quantum system to detect
information.
Other Laser Applications
• Used in surgery to make precise incisions
Ex: eye operations
• We see in everyday life such as the scanning devices used by
supermarkets and other retailers
Ex. Bar code of packaged product
CD and DVD players
• Laser light is directed toward disk tracks that contain encoded
information.
The reflected light is then sampled and turned into electronic signals that
produce a digital output.