Transcript Energy in Quantum Theory - National University of Singapore

States
Describe properties of physical (biological) systems
Physics: energy (kinetic+potential) of an object
Ecology: equilibrium populations in a biom
Geometry: shape (congruence) of a figure
Transformations
are changes that happen to systems
Physics objects move
Ecology individuals die and are born
Geometry figures are rotated & reflected
SYMMETRIES
are transformations that preserve states
Physics: falling preserves energy
Ecology: evolution preserves equilibrium populations
Geometry: rotations & reflections preserve shape
Consider the following quote where operation
means transformation, thing means system, and
appearance means state:
What is Symmetry?
An operational definition:
“A thing is symmetrical if one can subject it
to a certain operation and it appears exactly
the same as before.”
…Herman Weyl
Herman Weyl
(1885-1955)
Rotation & Reflection Symmetries
are geometric transformations that preserve the
shape of regular m-gons (although they may move
vertices and edges)
admits 3 rotations and 3 reflections
admits 4 rotations and 4 reflections
How many transformations does a pentagon have ?
Construct a figure with only two symmetries ?
Permutations
can be used to describe transformations
a
b
Rotation (clockwise by 120 degrees)
c
a 
c 
b  Rot
120

a
 
 
 c 
b 
Reflection about vertical line through the vertex a
a 
a 
fa
b  Re

 c 
 
 
 c 
b 
Problem: Compute the other four permutations
Composition
of transformations gives new transformations
a 
c 
b 
b  Rot
120

a  Ref

a a 
 
 
 
 c 
b 
 c 
we observe that
a 
b 
f a  Rot 120  Re f c
b  Re
a 
 
 
 c 
 c 
Multiplication Table
Problem: complete the following table

 Rot0
 Rot120
Rot
240
 Ref
a

Ref
 b
 Refc
Rot0
Rot120
Rot240
Ref a
Ref b
Ref a
Rot120
Rot0
Rot120
Ref c
Rot0
Ref c 







The Language of Symmetry
Group Theory:
Let G be a set of elements G={a,b,c, …}. Then the following
axioms endow G with a group structure:
(i) For any two elements a, b in G, ab and ba are also in G (Closure)
(ii) Composition is associative: a(bc) =(ab)c
(iii) G contains the identity e, with ae=ea=a for all a in G.
(iv) For every a in G, it’s inverse a 1 exist: aa 1  a 1a  e
Associativity
Composition of transformations is associative
because transformations are functions and
composition of functions is always associative
Problem: use the table to show that the set of
symmetries of an equilateral triangle with
composition forms a group
Problem: use the associative property to prove
that if A and B are m x m matrices and AB =
identity matrix then B is the inverse of A
Matrix Groups
Problem: for any integer m > 0 show that the set of
m x m matrices with nonzero determinant forms a
group under matrix multiplication and describe it as
a group of transformations
Problem: show that the following set of 3 x 3
matrices forms a subgroup of this group
1 0 0 0 0 1 0 1 0
0 1 0, 1 0 0, 0 0 1




0 0 1 0 1 0 1 0 0
1 0 0 0 0 1 0 1 0
0 0 1, 0 1 0, 1 0 0




0 1 0 1 0 0 0 0 1
Homomorphisms and Representations
Definition: A homomorphism of a group G to a group
H is a function f : G  H that satisfies f(ab) =
f(a)f(b) whenever a and b are elements in G. If H is
a group of m x m matrices it is called an mdimensional representation of the group G.
Problem: Show that the set of integers Z is a group
under addition and that f is a representation of Z
1 n
f ( n)  
,

0 1
nZ
Problem: Construct 2 and 3 dimensional
representations of the group of symmetries of
an equilateral triangle
References about Groups and Symmetries
Grossman, Israel and Magnus, Wilhelm, Groups and their Graphs, New
Mathematical Library published by The Mathematical Association of
America, 1992
Rosen, Joe, Symmetry Discovered, Concepts and Applications in
Nature and Science, Dover, New York, 1998
Sternberg, Shlomo, Group Theory and Physics, Cambridge University
Press, 1997
Growth as Transformations
Consider an initial rectangle, with sides of
lengths a < b, that is repeatedly transformed
into a rectangle with sides of lengths b < a+b
Sequences of such transformations characterize
growth – of biological populations and snowflakes
This transformation changes the shape – are any
properties (states) ever preserved ?
Yes - Golden Rectangles are preserved !
Transformation Geometry
a
ab
b
2 a  3b
a
a  2b
Transformation Algebra
Represent the lengths of the sides of a rectangle by a column ‘vector’
The geometric transformation induces an algebraic transformation
y 
T

x 

 y 
 x  y 
b 
a

b
a

2
b
T
T
T
a  










b 
a  b 
a  2b 
2a  3b 
Problem: construct the 2 x 2 matrix that represents this transformation
Fibonacci Sequence
Leonardo of Pisa (1170-1250), commonly called Fibonacci,
proposed the following problem:
A pair of rabbits is bought into a confined place. This pair and
every other pair, begets one new pair in a month, starting in
their second year. How many pairs will there be after one, two,
…, months, assuming that no deaths occur ?
Direct Solution
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
Transformation Solution
1
2
3
5
8
T
T
T
T
T
1 

















 
1
2
3
5
8
13
Phi = Golden Ratio
1
1



T
T


The transformed rectangle is similar to the original if and only if
1


 1    (1  5 ) / 2  1.618
Problem: Compute the other root of this Golden Ratio equation
Eigenvectors, Eigenvalues and the Golden Ratio
We observe that

1
T


1  




1     
 
and using the fact that
1/ 
is also a solution of the Golden Equation
1/  
1
T

 1 



 1 / 
1  1 /  
 1 /  
 1 /  
are eigenvectors of the transformation since the transformation simply
multiplies them by eigenvalues.
Invariant Subspaces
 x
The set of column vectors R  {   : x, y  R } is a 2-dim vector space
 y
1
 1 

and the sets V1  { r   : r  R } and V2  { r 
:rR}

 1 /  
 
2
are 1-dimensional vector subspaces that are invariant under
T
since u V1  Tu V1 and u V2  Tu V2
R
Problem Show that every vector in
sum
uw
with
2
can be expressed uniquely as a
u V1 , w V2
Problem Show that the k-fold transformation satisfies
T (u  w)   u  (1/  ) w
k
k
k
Geometry of Invariant Subspaces

1 
 

 1 
  1 /  
subspace
V1
subspace
V2
Problem: Show that the eigenspaces (lines) for T are orthogonal, this
follows necessarily since the matrix that represents T is symmetric
Fundamental Theorem of Growth
Theorem Every rectangle grows, through successive transformations
by T, into a Golden Rectangle.
Proof. This follows from Corollary 1 since as k increases
 c 
c u  d (1 /  ) w 
 c u   k 1 
c 
and this vector represents a Golden Rectangle.
k
k
limit
k
k
Problem: How does the Golden Ratio describe the
shape of Nautilus shells, the placement of leaves on
Calamansi plants, and the shape of the humans?
Problem: Show that the ratios of line segments in
a Pentacle (Brown page 101) all equal PHI (inscribe
the Pentacle in a unit circle in the complex plane so
the points are 5-th roots of unity and express the
intersections of lines as convex combinations)
Pentacle
References about the Fibonacci
Sequence, Golden Ratio and Growth
Brown, Dan, The Da Vinci Code (special illustrated edition),
Doubleday, New York, 2004
Hahn, Werner, Symmetry as a Developmental Principle in Nature
and Art, World Scientific, Singapore, 1998
Lawton, Wayne, Kronecker’s theorem and rational approximation
of algebraic numbers, The Fibonacci Quarterly, volume 21, number 2,
pages 143-146, May 1983
Thompson, D’Arcy Wentworth, Growth and Form, Vol. I and II,
Cambridge University Press, Cambridge, 1952
Afrizah Bte Mohd Hassan, Tan Chun Hsuan Joyce, Jenny
Angggraini Njo, Kwok Kah Peck, Chia Ling Ling, Welcome
Fibonacci Numbers, NUS Science Foundation Project Report
Tiling and Crystals
have discrete symmetry groups, consisting of
translations, rotations, and reflections, that
transform the (infinitely extended) tiling or crystal
into itself
tilings provide patterns used by artists
and arise in tissues and beehives
crystals are common in nature and are synthesized to
determine structures of compounds and proteins
Planar Tiles, Devlin, 1997, p. 164
The Moors used all 17 Wallpaper
Patterns, first discovered by the
ancient Egyptians, to decorate
the Alhambra in Granada, Spain
http://www.clarku.edu/~djoyce/wallpaper
http://www.red2000.com/spain/alhamb.html
Spatial Tiles, Devlin, 1997, p. 165
Pomegranate seeds grow to rhombic dodecahedrons
(green) from spheres in a bicubic lattice
There are 230 three-dimensional crystal patterns
classified by crystallographers in the 19-th century
Lattices
uv
v
u
0
vu
uv
u
v
u  2v
2v
2v  u
The lattice spanned by linearly independent vectors u and v
L  {mu  nv : m, n  Z}
forms a group under the operations of vector addition.
Translation Symmetry Groups
A lattice
L
acts as a group of translations of the vector space
as follows: for
L
define the transformation
R  R
2
R
T
2
by
T (w)  w  , w  R
2
They constitute the translations in the full symmetry groups of a tile
or a crystal.
2
Reciprocal Lattice
* of a lattice
L consists of
L
w   Z,   L
R
Theorem 3. If L is spanned by the column vectors of a matrix M
*
L is spanned by the columns of the transposed inverse of M
Definition: The reciprocal lattice
2 such that
all vectors w in
a c 
M 

b d 
M 
1 T
d

b


1
 ad bc 

 c a 
Problem: Show this using a direct computation.
Fundamental Theorem of X-Ray Crystallography
Theorem If a crystal has translational symmetry group
L
and if the crystal is illuminated with X-rays having wavelength

from an incident direction described by a unit vector u, then the
reflected X-rays ONLY occur in directions described by unit vectors
v that satisfy the condition
(1/  )(v  u)  L
*
(we observe that in physics the lattice L has units of length and the
reciprocal lattice has units of inverse length also called spatial
frequency, the vector on the left is called the scattering vector)
Proof. See the discussion on Bragg scattering in Jenkins and White,
Fundamentals of Optics, or any other serious physics book.
Representations of the Symmetric Groups
The symmetric group
Sn is the group of permutations on n-objects
A m-dimensional representation of a group describes the
group as a group of transformations on the vector space
Example the n-dimensional representation
by n x n permutation matrices
Example the 1-dim trivial representation
Example the 1-dim parity representation
R
m
f ( g )  mat( g )
f (g)  1
f ( g )  det(mat( g ))
Irreducible Representations of S 2 , S3 , S 4
S2
two 1-dim, trivial and parity
S3
two 1-dim, trivial and parity
one 2-dim, matrices that rotate and reflect an equilateral triangle
S4
two 1-dim, trivial and parity
matrices that rotate a cube (permute four diagonals)
two 3-dim
matrices that rotate and reflect a regular tetradedron
one 2-dim given by composition
S4 
 S3  { 2x2 matrices}
h
where h is determined by permutation of the three cubic axes
Theorem These are all since
2
dim(
rep
)
 | Sn |  n!
irr _ rep
Decomposition into Irreducible Representations
Example the 3-dimensional permutation representation of S3
decomposes into the sum of the 1-dimensional trivial irreducible
representation and the unique 2-dimensional irreducible representation
Example Carbon tetrachloride molecules are invariant under rotations
and reflections – described by the tetradedron representation of
and their displacement configuration states are described by a 15S
dimensional representation that decomposes (Sternberg page 103) 4
into the sum of one 1-dim irreducible representations
one 2-dim irreducible representation
one 3-dim irreducible cubic representation
three copies of the 3-dim irreducible tetrahedral representation
Vibrational eigenvalues are constrained (6 with multiplicities 1,2,3,3,3,3)
and the symmetries can therefore be observed by spectroscopy.
Problem Prove this assuming that vib. eig. are eig of a symmetric
matrix H and symmetry means that p(g)H = Hp(g) for all g in Group
Lie Groups and their Representations
In addition to the discrete groups there are groups, such as the group of
ALL translations of space or ALL 3 x 3 matrices with nonzero
determinant, that are described by continuous parameters, they are
called Lie groups and they describe important symmetries of particles
and fields in physics and even describe symmetries of differential
equations that describe fluid flow and biological growth
Irreducible representations of Lie groups determine physical
observables, such as energy, mass, charge and more exotic quantities,
in the same way that irreducible representations of discrete groups
describe the vibrational eigenvalues (spectra) of molecules
The irreducible representations of the most important Lie groups, the
unitary groups, are exactly determined by the irreducible
representations of the symmetric groups that we have discussed, this
amazing result was discovered by Isaac Schur and later rediscovered by
Herman Weyl – the father of symmetry