Post on 17-Jan-2016
1 February 2012 Modern Physics IV Lecture 3 1
Modern Physics for Frommies IVThe Universe - Small to Large
Lecture 3
Fromm Institute for Lifelong Learning University of San Francisco
Agenda• Administrative Matters• Atomic Physics• Molecules
Administrative Matters• This is Lecture 3, Lecture 8 will be Wed. 7 March
• Wikis from previous course
http://modphysicsfrommiies.wiki.usfca.edu/Note the ii in frommiis
http://modphysfromm2.wiki.usfca.edu
http://modphysfromm3.wiki.usfca.edu
The current course wiki, http://modphysfromm4.wiki.usfca.edu , now includes a Glossary of Mathematical Symbols,
Other glossaries can be found on Google by searching “mathematics symbols”
1 February 2012 Modern Physics IV Lecture 3 4
Quantum Mechanical View of AtomsBohr model discarded as an accurate description of nature
Certain aspects have however been retained
e.g. Electrons in an atom exist only in discrete states of definite energy, the stationary states
Transitions between these states require the emission (or absorption of a photon.
According to wave mechanics, electrons do not travel in well defined circular orbits ala Bohr. The electron, because of its wave nature, is better thought of as spread out in space as a “cloud”.
The size and shape of the electron cloud can be found by solving the Schrödinger equation for the atom and forming the probability distribution, | |2.
1 February 2012 Modern Physics IV Lecture 3 5
Ground state of hydrogen
1 February 2012 Modern Physics IV Lecture 3 6
Schrödinger’s Equation in Spherical Coordinates
x
2
2
2 2 22
2 2 2
( , , ) , ,2
where
x y z E x y zm
x y z
Separation of variables:
Assume a solution of the form , ,x y z X x Y y Z z
y
zr
In Cartesian coordinates
1 February 2012 Modern Physics IV Lecture 3 7
Want to do the same thing with spherical symmetry
1 February 2012 Modern Physics IV Lecture 3 8
Spherical Time Independent Schrödinger Equation
2
22 2 2 2 2 2
2
0
1 1 1 2sin 0
sin sin
1where for the hydrogen atom
4
mr E V
r r r r r
eV
r
Separation of variables:
Try solution of form , ,r R r
( , , ) , l mn lr R r Y
: , cos
where the are
l lm imml l
ml
Y P e
P
spherical harmonics
associated Legendre polynomials
: lnR r associated LaGuerre functions
1 February 2012 Modern Physics IV Lecture 3 9
Quantum NumbersIf we do QM in for a particle confined in a 1-D and 3-D potential well or rigid box. (See Course II Lecture 4)
The solutions are characterized by a single quantum number (n) in the 1-D case and by three numbers (nx, ny and nz) in 3-D.
These quantum numbers arise from the imposition of boundary conditions on the solutions. We might expect that in the 3-D problem of the hydrogen atom the solutions will be characterized by numbers corresponding to Boundary conditions applied in 3-D. Restrictions on the values of these quantum numbers arise from the mathematics of the LaGuerre functions and the spherical harmonics.
Actually, we need a fourth number. There is an additional degree of freedom which I will treat in a few minutes.
1 February 2012 Modern Physics IV Lecture 3 10
202 2
0
1,2,3,
1
2
4
n
n
E m eE
n n
principle quantum number
Results from boundary conditions on solution of the R part of the separated Schrödinger eqn.
Bohr result
R part contains the potential energy
n alone determines the energy levels (actually there is a slight deviation from this)
Consequence of central inverse square force.
1 February 2012 Modern Physics IV Lecture 3 11
0,1,2,3, , 1
l n orbital angular momentum quantum number
associated with and parts of Sch. eq
n.
R r
tangential
Quantu
Classically
m boundary conditions
or
1 2
L r
hL l l
p L rmv
Note disageement with Bohr quantization where
2
in particular, the ground state has 0 0
hL n
l L
The semiclassical planetary model with electrons in orbits is not a good one
1 February 2012 Modern Physics IV Lecture 3 12
Note: All these transitions have l = 1
1 February 2012 Modern Physics IV Lecture 3 13
s p d f g h etc.
l=0 1 2 3 4 5 etc.
Notation for states: nl, e.g. 4d is n=4, l=2
, 1, ,0,1, , 1, lm l l l l magnetic quantum number
is a vector quantity, conserved in a central potential
The solution for specifies that is an integer related to
's -component.
l
L
m
L z
2z l
hL m
1 February 2012 Modern Physics IV Lecture 3 14
Aside on Angular Momentum
r
v
Particle of mass m moving with circular speed v around an axis at radius r.
L r mv
Magnitude: sin
Here, 90 sin 1
L mvr
L mvr
Direction: to plane of and with sense
determined by right hand rule.
r v
1 February 2012 Modern Physics IV Lecture 3 15
C A B
sinC AB
A
C
To the plane of AB
Right Hand Rule
Direction of advance of a right hand screw
1 February 2012 Modern Physics IV Lecture 3 16
Note that the vector product is not commutative
A B B A
Again look at Right Hand Rule
C A B
‛ A
C A × B
B × A
1 February 2012 Modern Physics IV Lecture 3 17
1 fixed
restrictedz l
L l l
L m
Space quantization
Note choice of z axis is arbitrary.
1 February 2012 Modern Physics IV Lecture 3 18
Energy is dependent solely on n.
Presence of multiple ls and ms for a given n states are degenerate
This degeneracy is removed if directional symmetry is broken by say a B or E field.
What about Lx and Ly?
1 February 2012 Modern Physics IV Lecture 3 19
2 2
z
2 2
If and are known, knowledge of 2nd component
3rd is
consequence of
also known.
x y zL
L L
L L L
If we know exactly, we know nothing of
Uncertainty pr
we know
inciple:
nothing of and
z
z
x y
L
L
L L
1 February 2012 Modern Physics IV Lecture 3 20
x
y
z
Ly
Lx
Liz
L
1 February 2012 Modern Physics IV Lecture 3 21
Magnetic effects
Normal Zeeman effect: Transition between 1s and 2p
Spectral lines broaden and split into 3 lines as B is applied and increased. 3 lines = “normal Zeeman effect”
Consider the electron “orbit” to be a current loop with
IA
2
2 2 2
e rq erv eIA A L
T r v m
Vector: 2
eL
m
1 February 2012 Modern Physics IV Lecture 3 22
Apply external magnetic field
Torque:
Potential energy: B
N B
U B
Bohr magneton
2
2
l
B
z l B
e
m
m
em
m
1 February 2012 Modern Physics IV Lecture 3 23
quantized quantizedL
Additional potential energy term:
B z B lU B m B
Each degenerate energy level, l, is split into 2l+1 separate energy levels, ml.
B has specified a direction in space (z axis) and the symmetry responsible for the degeneracy has been broken.
1 February 2012 Modern Physics IV Lecture 3 24
The Stern-Gerlach experiment:
If B is inhomogeneous there will be a net force as well as torque on the atom
1 February 2012 Modern Physics IV Lecture 3 25
For l 0 the states should separate according to ml
2 lines seen instead of the expected 3 (or 2l+1 = odd)
Haven’t seen the whole picture yet.
1 February 2012 Modern Physics IV Lecture 3 26
Electron Spin
Wolfgang Pauli:
Relativity besides n, l, ml need 4th quantum number
G. Hollenbeck and S. Goudsmit:
Propose intrinsic “spin” angular momentum for the electron
s = ½ħ
Another magnetic quantum number: ms = ½
1928, P. A. M. Dirac justifies this from relativity.
i m
1 February 2012 Modern Physics IV Lecture 3 27
Gives magnetic effects like orbital angular momentum.
Intrinsic spin → intrinsic magnetic dipole moment
New magnetic quantum numbers:
ms = ± 1/2
Doubles number of states for a given n
States are degenerate unless a spatial direction is specified, e.g. external E or B field
Quantum state now specified by {n, l, ml, ms}
1 February 2012 Modern Physics IV Lecture 3 28
Return to the Stern-Gerlach experiment
l = 0 state will give 2 lines for ms = ± 1/2
Fine structure:
Even in the absence of external fields, very high resolution spectroscopy reveals splitting of spectral lines.
Rest frame of electron: nucleus orbits and appears as a current loop. Interacts with spin magnetic moment and breaks degeneracy
Line separation is about 5 x 10-5 eV compared to the 2p 1s transition energy of 10.2 eV
Hyperfine structure:Arises from the spin angular momentum and consequent spin magnetism of the nucleon(s)
1 February 2012 Modern Physics IV Lecture 3 29
s-wave states are spherically symmetric, not so for l 0
1 February 2012 Modern Physics IV Lecture 3 30
Quantum StatisticsConsider a system of 2 particles, say electrons
1 2 1 2 1 2( , ) *P r r dv dv dv dv
1 2
1 2
Wave function for the system is ,
we observe a probability of finding the particles
in volume elements at and
r r
r r
It is easy to show that
* *
i.e. no change in observable for
1 2 1 2 2 1 2 1
identical particles no observable change if they are interchange d
* , , * , ,r r r r r r r r
1 February 2012 Modern Physics IV Lecture 3 31
So, under interchange 2 possibilities
±
If 2 identical particles interchange = +they are said to obey Bose-Einstein statistics and are called bosons.
If 2 identical particles interchange = -, they are said to obey Fermi-Dirac statistics and are called fermions.
Bosons have integral spin, e.g. photons, mesons, some atoms and nuclei, ………
1 February 2012 Modern Physics IV Lecture 3 32
Fermions have ½ integral spin, e.g. leptons, nucleons, some atoms and nuclei,………..
For fermions
Cannot have 2 identical particles with the same set of quantum numbers.
Pauli Exclusion Principle
You can stick as many bosons into a quantum state as you want.
1 February 2012 Modern Physics IV Lecture 3 33
Electrons are fermions.
Build some elements. As electrons are added the exclusion principle will have an effect.
Hydrogen: 1 e in the 1s state 1s1
He: 2 e in the 1s state, ms =1 and -1, 1s2
No more e can be added to the 1s state without violating the exclusion principle !
The K shell is filled
1 February 2012 Modern Physics IV Lecture 3 34
Li: 3rd e has to go in 2s state 2s1
Be: 4th e in the 2s state, ms =1 and -1, 2s2
2s state (subshell) is now filled
B: 5th e has to go in the 2p state 2p1
p state has 2l+1 = 3 values of ml, each with 2 values of ms, accommodating C, N, O, F and Ne as 2p2 – 2p6.
2p subshell is now filled, as is the L shell
Na: 11th e has to go in 3s state 3s1 etc.
1 February 2012 Modern Physics IV Lecture 3 35
3s and 3p each with 2l+1 ml values each having 2 values of ms.
Weirdoes:
3d, 4d and 5d subshells fill up the transition metals followed by the lanthanides and the actinides
Complicated inter electron interactions mess things up
If electrons were bosons, they would all sit in the ground state, 1s, and chemistry would be very different.
1 February 2012 Modern Physics IV Lecture 3 36
17 February 2010 Modern Physics II Lecture 6 37
Bonding in molecules
Ionic bonding:
NaCl
P for Na outer electron
17 February 2010 Modern Physics II Lecture 6 38
Na
Na has 11 e-
10 reside in inner closed shells Last e- spends most of its time outside these shells.
The last e- feels net attraction due to +1e, not all that strong
Cl
Cl has 17 e-
12 are in closed shells 1s22s22p63s2
Others are in non spherically symmetric p states
17 February 2012 Modern Physics II Lecture 6 39
H
ml=0, unpaired 4 states ml=±1, ms=±1
Exclusion principle allows one more e- in ml = 0 with spin oriented opposite to that of the last Cl e-
If an extra electron happens to be in the vicinity it can be in this state and could see an attraction due to Cl nucleus as much as +5e.
Stronger than the +1e attraction between Na nucleus and its outer electron charge distribution in slide #36 and an ionic bond between Na and Cl.
17 February 2010 Modern Physics II Lecture 6 40
Covalent bonding:
H+H H2If H atoms are close together, e- clouds overlap and e- “orbit” both nuclei.
Both H s in ground state. Electron spins can be either parallel (S = 1) or antiparallel (S = 0)
1st consider S = 1:Exclusion principle 2 e- with same quantum numbers must be in different places, i.e. belong to different atoms.
(+) nuclei repel, no bond is formed.
17 February 2012 Modern Physics II Lecture 6 41
S = 0:
e- have different values for ms, spend a lot of time in the internuclear region
(+) nuclei are attracted to the internuclear e- and a bond is formed.
In a wave picture, exclusion principle destructive interference when S=1 and constructive when S=0.
17 February 2012 Modern Physics II Lecture 6 42
Energetics point of view:
For S = 0, e- can occupy same space, space of 2 atoms rather than 1 x is increased.
H. U. P p can be less energy is less
Molecule has lower energy than the 2 separate atoms
H2 is stable
Binding energy is 4.5 eV for H2
17 February 2010 Modern Physics II Lecture 6 43
0.074 nm
In the vicinity of r0 we may
approximate
, constants for attractive,
repulsive parts of
, are small integers
m n
A BU
r rA B
U
m n
17 February 2012 Modern Physics II Lecture 6 44
Activation energy often need to break earlier bonds
2 2 2
2 2
2 H + O 2H O
H and O must 1st be broken into H and O atoms
spark
UA = 0 for hypergolic materials, don’t need spark
17 February 2012 Modern Physics II Lecture 6 45
Energy storage in biological systems
adenosine triphosphate
ATP ADP + (phosphate group) + ENERGY