課程性質︰選修
Mathematics for Chemistry II Midterm Exam
2003. 4.18
1.The Pauli spin matrices are σx=(0 1),σy=(0 -i),and σz=(1 0)
1 0 i 0 0 -1
Show that σi^2 = (1 0).
0 1
Define the commutator of two n×n matrices, A and B, by [A,B]=AB-BA. In the
case that AB=BA, the commutator is zero, and the two matrices are said to
commute. Find [σx,σy],[σy,σz],and [σz,σx] in terms of the Pauli spin
matrices. (10%)
2.a)How many N-letter words can be constructed with just the latters α and β
? (5%)
b)Start from the left, treat α and β as noncommuting, and show that the
sum of all these words is equal to the matrix product
(α β)(α β)^(N-1) (1)
β α 1 (5%)
c)There is no significance to summing all these words, but in the Ising
model of ferromagnetism there arises the analogous problem of constructing
the partition functions Q for a sequence of N spins in a row. Each of
these spins(of magnetic moment μ) can be either up or down(wavefunction
α or β), and Q is a sum over all posible configurations(words):
Q = Σe^(-Ei/kT)
where Ei is the total energy of the ith configuration. According to the
Ising model, the energy of a configuration in a magnetic field H may be
calculated as follow: An α spin contributes μH to Ei and a β spin
contributes -μH. A spin following an identical spin contributes J to Ei,
and a spin following the opposite spin contribute -J. Build up the
sequence of spins one at a time, calculate the contribution to Ei at each
step, and show that
Q = (e^(-μH/kT) e^(μH/kT)) (e^(-(μH+J)/kT) e^(-(-μH-J)/kT))^(N-1) (1)
e^(-(μH-J)/kT) e^(-(-μH+J)/kT) 1
(5%)
3.Local Minimum
A function F(x,y) has a local minimum at any point where its first
derivatives δF/δx and δF/δy are zero and the matrix of second derivatives
δ^2 F/δx^2 δ^2 F/δxδy
A = [ δ^2 F/δyδx δ^2 F/δy^2 ] is positive definitive. Is this true
for F1 = x^2 - x^2 y^2 + y^2 + y^3 and F2 = cosx cosy at x=y=0? Does F1 have
a global minimum or can it approach -∞? (10%)
4.Show that for any matrix A, the matrix A﹢A is positive definitive. Then
apply this result to A = F + iλG, with F and G Hermitian, and λ real, to
show that for all x
(x﹢F^2 x)(x﹢G^2 x) ≧ (1/4)(x﹢[F,G]x)^2, where equality holds at
λ=(x﹢[F,G]x)/(2i(x﹢G^2 x)). (10%)
5.Skew-Symmetric and Unitary Matrices
a)Show that the eigenvalues of skew-Hermitian matrix are imaginary, and that
eigenvectors belonging to different eigenvalues are orthogonal. (5%)
b)Show that the eigenvalues of a unitary matrix are of magnitude unity, and
that eigenvectors belonging to different eigenvalues are orthogonal. (5%)
6.Simultaneous Diagonalization
Find the eigenvalues and corresponding eigenvectors of the Huckel molecular
orbital bond matrix B for the following molecule as shown in the figure.
4
Choose a new basis, {χi}i=1, with each χi symmetric or antisymmetric to
the vertical plane perpendicular to the plane of the molecule. Then construct
the matrix β=(∫χi* B χj dV), which is the direct sum of two 2×2
matrices and two 1×1 of β, if you indicate clearly how (χi) is defined.
｜
╱╲ (15%)
7.Equilibrium of Springs and Masses in the Presence of An External Force
In a system with three springs, three masses and one force exerted on the
mass m3 as shown in the following figure, white out the equation e=Ax, y=Ce
, and AT y=fext. What are the displacements?
｜ ├x1→| ├x2→| ├x3→| fext
├spring─mass─spring─mass─spring─mass─→ (8%)
8.Normal Modes:Springs and Masses
｜ c1 c2 c3 │
├spring─m1─spring─m2─spring┤
With masses m1 = m2 = 1 and spring constants c1 = c3 = 4, c2 = 6, show that
the equation of motion is u + Ku = 0 with K=[10 -6].
tt -6 10
Find its natural frequencies ω1 and ω2, and from the eigenvectors find its
two pure oscillations ui = (a cosωi t + b sinωi t)xi, where i=1,2. (10%)
9.If K and M are positive definite and Kx=λMx, prove that λ is positive.
This is the generalized eigenvalue problem, with two matrices.
a)Show that Kx=λMx can be reduced to normal eigenvalue problem by
1/2
transforming to the mass-weighted coordinate, q = M x. (4%)
b)Find the two eigenvalues when M=[1 0] and K=[ 2 -1].
0 2 -1 2 (4%)
c)Show that the product of two symmetric matrices A and B is not a symmetric
matrix in general. What is the condition for the product AB to be
symmetric? (4%)
d)If M=[1 0] and spring constants c1 = c2 = c3 = 1, find the two pure
0 2 -1
oscillations of the system Mu + Ku = 0. Since M K is no longer symmetric
tt
, its eigenvectors x1 and x2 are no longer orthogonol; verify that now
T T
x Mx = q q = 0 (4%)
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