課程名稱︰化學數學
課程性質︰必修
課程教師︰陸駿逸
開課學院：理學院
開課系所︰化學系
考試日期（年月日）︰2014.11.12
考試時限（分鐘）：10:20~12:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Midterm Examination of Mathematics for Chemists
1.（40%） ┌ ┐
│ 3/2 -1/2 │
Let A＝│ │
│-1/2 3/2 │
└ ┘
（a）Find the normalized eigenvectors │e1＞,│e2＞, and the corresponding
eigenvalues λ1,λ2.
┌ ┐
│ 1 0 │
（b）Show that │e1＞＜e1│＋│e2＞＜e2│＝│ │＝I
│ 0 1 │
└ ┘
（c）Calculate A^10 and cos(πA).
（d） ┌ ┐
│ f1(t) │
Let ψ(t)＝│ │which satisfies
│ f2(t) │
└ ┘
┌ ┐ ┌ ┐
d │ f1(t) │ │ f1(t) │
─ │ │＝iA│ │
dt │ f2(t) │ │ f2(t) │
└ ┘ └ ┘
┌ ┐
│ 1 │
and the initial condition ψ(t=0)＝│ │. Find ψ(t).
│ -1 │
└ ┘
2.（15%）Let │a＞ and │b＞ are two eigenvectors of a real, symmetric matrix
B. Suppose that they have different eigenvalues λa≠λb, show that │a＞
and │b＞ must be orthogonal.
3.（15%）Given a basis a, b, c of R^3, where a＝(1,1,0), b＝(1,0,1), c＝(0,0,1)
, use the Gram-Schmidt process to construct an orthogonal basis.
4.（15%）Given f(x,y)＝x^2-xy where x and y are related by the constraint
x^2+3y^2=1. Find (x,y) and f(x,y) where f(x,y) becomes minimal or maximal.
5.（15%）Let A be a hermitian matrix. Show that e^(2iA) is unitary.