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Group theory; the application to quantum mechanics
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REDUcnON TO MAIN AXES
Ch. 1. § 3J
13
there are three rectangular axes for which one has
Di = ai E "
(i == 1, 2, 3)
the Bl are the main dielectric constants of the medium. Simllarly in the general
case in which we are actually interested, we have to search for a unitary
system of axes ei which brings the matrix A' (the transformed of ,4..) into the
diagonal form (1.13). This means solving the following problem: find n
directions ei, e~, . .• , e~ such that every vector x parallel to one of the
directions ,; will be transformed by A into a vector y 'which is parallel to %!
y == Ax = ax,
(1~14)
where (X is a constant.
Writing the "components" of this vector equationt we obtain n linear
homogeneous equations of the form
a'lxl + ... (an-«)x,+ . ,. ,. a",x" := 0
(i == 1, 2, ... , n),
(1.14a)
which are only compatible with each other if the undetermined constant Of
is a root of the secular equation 1
(all-A)
Det IA-A.It
I
(a22 -A) · · ·
I = o.
(a.-A.) I
at2
• .. ..
ala
42,.
==
..
..
II>
•
..
..
..
II>
..
..
*
(1.15)
I
..
This equation has in general n roots.l ::: (Xl II . . . . «,. (which may be distinct
or not) to which correspond the 11 directions of the axes given by' equation
(1.141.). We know only the directions of the vectors :I but ftot their'magnitude
becaUse (1.144) determines omy the ratios between the oomponents. The
roots' «, of (I. IS) are the eigenvalues, propervalues, or characteristic con..
stallts of the matrix A. They are real in the' Hermitian case; in the unitary
case 'their absolute value is 1. The are the eiaenvector. or principal direetiollSa~ . When (1.1S) represents a multiple roOt of order P(<
er
1
This name oriainates from a PrQblom in astroD9Jll)', where a siluilar "Iuation deter. .
mines the perturbation over long time interVals.
• The p~ oftllele theorebllaabe made bJd100sinaa root CIt, 0(1.15), detem1iDiog
the correapondiDa eiaenvoctor..~ •. _complotfq~tho,~>with (II-I) vecto.l'J whidt
have to be ortholonal to the first
In order to fonn'a unitary system or axes. Asa
result of the symmetry proportiea ot the HermitiaD':and:'1Ulitary lIIatrices the coefficients
all
at". list ••• Q".l, are all zero and the matrix A. takes the form:
one
I» • •
0
..
:
0)
o au .
() .. ..::: .
<<1 }~
t.-
j
(
~:
a", .
~~
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VEcrOR SPACES -
14
UNITARY GEOMETRY
[eh. 1, § 3
1
ap = tX), the vectors which have the property 1 == Ax ==
form a subspace
of p dimensions in which the direction of the axes is not determined. (See
(X%
the second note on page 12).
3.3. JOINT DIAOONALIZATION OF A SET OF MATRICES
In order that all the matrices of an Hermitian or unitary system can be
reduced at the same time 10 their principal axes, it is necessary and sufficient
that they all commute with each other.
First we "ill show that this condition is necessary: let A and 8 be two
matrices. By a transformation of axes S we reduce them simultaneously to
diagonal form A' and Sf. Then obviously they commute with each other:
A'B'
= B'A'.
Hence S-lASS-las = S-lABS == S-l BAS i.e. AB == BA.
Now we will show that the condition is sufficient: let us suppose that AS =
BA and let us make a transformation of coordinates such that 8 is diagonal:
We have
If
{J, ~
p",
QUe
== O.
So ~ fiIld that A is a Itep-'WiIe matrlxt of which all the terms are zero,
cxce~ those which are", situated on the main diagonal, or those inside certain
squar~ that share the diagonal; they correspond with the case in which
p, -"ll"e These squares are eaGh related to a subspace \ll in which the
~pal directions of the matrix B are undetermined (the matrix posseasoa a
cirtuiar, spherical, or hyperspherical symmetry)_ One may finaDy choose
these undetermined axes in such a way that A will be completely diagonal
3.4. INVAlUANCE OF A SBCULAll EQUATION
A traDsformatiOll. of coordiiudes does not chaDse She form of the secular
equation (leIS). Let US carry··out the tra1l1formation of coordiaates 5:
A
-+
A' .. S-lAS;
The rule of multiplication of determiDants gives us
IS-1AS-lII =- IS- 1 1-IA-lII·ISI
== IS- 11• lSI • IA- AII == fA -;. II.
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(1.16)
Ch. 1, I 3, 4]
15
FUNCI10N SPACE
3.S. TRACE OR. SPUR OF A MATRIX
Let A = (aft) be a matrix, the trace is then the sum of all the diagonal
elements
,.
Tr A ==
,=2: ail •
(1.17)
1
To prove that the trace is invariant we shall write equation (I. t S) in the form:
(-l)-+ (_l)'l-l(al1 +a22+ •.. a,.,.)+ ... = (-~r+ (_A)',-t Tr A + ....
Because this equation keeps its form under a transformation of coordinates
S" all the coefficients are invariant, in particular the second:
Tr A == invariant.
(1.17a)
4. Function Space. Complete Sets of Orthogonal FDDetlo..
Every function of continuous variables and particularly the wave function
in SchrOdinger mecbanics can be represented by vectors in a /UllCtion space
in which the "umber of dimensions is infinite. The operators acting on the
wave runctions ~l transform these functions into other functions producing
a transformation of this space irlto itself. On the other hand quantum mechanics can also be formulated with the help of certain relations among matrices. The matrices in this so-called matrix mechanics are matrices with an
infinite number of rows and columns. They can again be considered as
operators that transform a space with an infinite number of dimensions into
its~f. This analogy, brought forward by Hilbert, made it possible for
Schradinger and later for Dirac to show the equivalence between wave
mechanics and matrix mechanics.
There exists between the two spaces just mentioned a difference which
appears to be essential: the matrices operate on a space in which the number
of dimensions it denumerably infinite; on the contrary the Dumber of dimen..
sions in function space is of the order of a continuum. We will see, in a
moment that this difference is more apparent than real (see § 4.3).
4.1. FUNCTION SPACE
"'(x) defines a function of one variable x. The simplest case is that of a
discontinuous function where the value is given only for a finite number of
~·alues/or the variable x : Xl, %2, ••• , x., i.c. the domaia of the variable x
consists of a set of discrete points. The " correspondina values of the
function "'1' tit2 · t/J. ("', == ';
f
•
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