Download 1.1.3 (a) Prove that (AB)` = BAt using components

1
...
,
(a) Derive general formulas for the coefficients a,, a2, a,,
a
a, in terms of the eigenvalues (eI, e,, e,,. ..,en) of
matrix M.
(b) Derive general formulas for the coefficients a,, a,, a,, .. .,
a,-,, a, in terms of the components (Mij)of matrix M. (Hint:
Use the e-tensor definition
,-,,
(c)
for determinants and expan6 the determinantal secular expression.)
Do the coefficients a j change if there is a change of basis?
[Note: The results of (a) and (b) are particularly important and will be
helpful for later theory and problems throughout the h k . 1
1.1.3
(a) Prove that (AB)' = BAt using components Aij and Bij of n X n
matrices A and B.
(b) Prove that (Alx)It = (XIAT.
(c) Expand to find (ABCD)? = ?, ((xlAly))* = ?, (Ix)(yl)t?, (el,)'?
Section 1.2
1.2.1
(Back to your roots)
For
(a)
(b)
(c)
(-62
-:)
the matrix M =
find
Eigenvalues of M.
Spectral decomposition of M.
The bra andket eigenvectors of M.
(d) All the square roots of M:
V%=
iE-3
(Multiple degeneracy) The matrix p
=p
represents a simple example of what is called a pairing operator in
nuclear and superconductivity theory. It has number of repeated or
degenerate eigenvalues.
(a) Find the secular equation, the Hamiltonian Cayley equation, and
the minimal equation of p. (Hint:Problem 1.2.3 ir useful.) Find
the eigenvalues.
(b) Compute the projection operafors far each distinct eigenvalue
and write the spectral decomposition for this matrix.
(c) Find all the square roots fi of matrix p.
13.5 (Knowing spectral decomposition backwards and forwards)
7
1.22 Do the same as in Problem 1.2.1 for the matrix M =
-9
1.2.3
12.4
-9
-9
27
(Secular behivior)
The polynomial form for the secular equation of a general n x n
matrix M is
(a). Use what we have discussed about spectral decomposition and
@ products to do an eigenvalue problem backwards. Find the
matrix M = M t which has the following eigensolutions:
(a) Use the techniques in Chapter 1 (Section 1.2Bd) to find a single
set of projection operators which spectrally decompose both
matrices:
(b)
Use the result of (a) to find a single transformation T which
diagonalizes M and N.
1.2.7 What are the conjugation relations (if any) for eigenvalues (E, a d
E;), projection operators (P,, and pij), and eigenvectors ( I E , ) and
(Ejl) in the cases that operators are (a) Hermitian: H = H?, (b)
anti-Hermitian: At = -A, or (c) unitary: Ut = U-'. Check your conclusions by spectrally decomposing and diagonalizing the following
examples:
1.2.8
CHAPTER 2
Does MtM = I imply that M M r= I, as well?
(a) Prove or disprove in the case that M is a finite n x n matrix.
(b) What if M is an infinite matrix? (Examples of infinite matrices
are the representations of creation and destruction operators a
and at in the quantum harmonic oscillator eigenstates {lo), 11)
= atlo), 12) = a'll), ...1.)
BASIC THEORY AND APPLICATIONS
OF SYMMETRY REPRESENTATIONS
(ABELIAN SYMMETRY GROUPS)
In the preceding review of matrices the ideas of projection operators and
spectral decompositions were introduced. In this chapter we shall see how
frequency spectra of physical systems are analyzed in terms of mathematical
spectral decompositions. Mathematical concepts will be introduced in this
and following chapters by analyzing the simplest physical models which
exhibit them. In this way the mathematical and physical ideas can be closely
related. It is hoped that this particular pedagogical approach to the theory of
spectra will be easy to understand.
Symmetry is a key mathematical and physical concept in the classical and
quantum theory of spectra. Symmetry analysis and group theory were first
applied by Eugene Wigner and Herman Weyl shortly after the invention of
quantum mechanics. Since then applications of symmetry analysis have been
made to virtually all types of spectroscopy. Spectra, ranging in energy from
radio frequency (- l o 3 Hz) to x ray ( - loL8HZ), have given infonation
about atoms, molecules, and solids. Higher-frequency y radiation (> 10ZO
Hz) has been used to study nuclear spectra. A most widely publicized
application of symmetry principles concerns very high energy "elementary
particle" spectra where researchers are thinking about energies in excess of
eV or loz6 Hz. (1 eV is equivalent to 2.42 X loL4Hz.)
Meanwhile the application of laser devices has reopened atomic and
molecular spectroscopy. Instead of obtaining higher and higher frequency,
laser spectroscopists are obtaining ever-increasing frequency resolution. This
means finer spectral details are seen and more detailed models of atomic and
molecukr processes are needed. This has stimulated the development of new