課程名稱︰化學數學一
課程性質︰系定必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試時間︰2006/01/10 10:20~12:30
是否需發放獎勵金：是。
試題 :
1.(15 pts)Solve the steady state solution of x"(t)+x'(t)+x(t)=sint
2.(20 pts)Consider the eigenvalue of problem
2
-d 2
──u(x) + x u(x) =Eu(x)
2
dx
where E is the real eigenvalue. The eigenfunction u(x) obeys the boundary
conditions lim u(x)=0 and lim u(x)=0. This equation appears in the quantum
x→∞ x→-∞
theory of harmonic oscillator. Use the operator method, calculate the wave
function u (x) which has the lowest eigenvalue E .
0 0
3.(20 pts)Consider the vibration of a linear molecule XYZ along its molecule
axis. You can model the molecule as two springs connecting the atoms X, Y,
and Z. The springs constants are given as k =500 N/m and k =700 N/m. The
XY YZ
atoms X, Y, and Z have the masses 1, 12 and 14 Dalton (g/mol) respectively.
Calculate the characteristic vibration frequencies.
4.(20 pts)One of the Pauli matrix is
┌ ┐
σy=│o -i│
│i 0│
└ ┘
(a)Show that ┌ ┐
2 4 6 │1 0│
σy =σy =σy =...=│0 1│
└ ┘
(b)Show that ┌ ┐
iθσy │cosθ sinθ│
e =│-sinθ cosθ│
└ ┘
iθσy
(c)Show that e is an orthogonal matrix.
(d)Together with the initial condition u(0)=v(0)=1/√2, solve the
differential equation
┌ ┐ ┌ ┐
d │u(t)│ │u(t)│
i──│v(t)│=ωσy│v(t)│
dt └ ┘ └ ┘
where ω is a constant.
5.(10 pts)Consider a vector space formed by all constant, linear and quadratic
ploynomials in x.
(a)Show that the differentiation d/dx is a linear operator.
2 2
(b)Take a bases {1,x,x } to express any vector a+bx+cx as
┌ ┐
│a│
│b│
│c│
└ ┘
In terms of this bases, write down the matrix which represents the
operator d/dx.
6.(15 pts)Show that if two eigenvectors of a Hermitian matrix A have different
eigenvalues, then the two eigenvectors must be orthogonal.
7.(20 pts)Show that the constant C in the Legendre's equation
2
(1-x )y"(x)-2xy'(x)+Cy(x)=0
must be some special values in order that the infinite series solution
becomes the finite term polynomials.
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