[試題] 103上陸駿逸化學數學期中考

看板NTU-Exam作者Waphiphop (瓦飛帕)時間9年前 (2014/11/12 21:26)推噓3(3推 0噓 4→)

留言7則, 6人參與討論串1/1

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. -- ※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 140.112.244.95 ※ 文章網址: http://www.ptt.cc/bbs/NTU-Exam/M.1415798790.A.515.html ※ 編輯: Waphiphop (140.112.244.95), 11/12/2014 21:27:16

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