課程名稱︰量子力學一
課程性質︰必修
課程教師︰高涌泉
開課學院：理學院
開課系所︰物研所
考試日期（年月日）︰2012.11.06
考試時限（分鐘）：110分
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Suppose we have determined that (for a spin 1/2 system)
1 1
|+>x = --- [|+> + |->] and |->x = --- [|+> - |->].
√2 √2
(a) What further information does one need to determined |+>y and |->y ?
(How does one determine |+>y and |->y ?)
(b) Can these two states be determined uniquely?
2. The spin component Sn of a spin 1/2 particle along the direction
^ ^ ^ ^ → ^
n=sinθcosφi+sinθsinφj+cosθk as indicated in Fig.1 is Sn=S‧n.
Obtain the eigenvalues and eigenvectors of Sn.
Fig.1 http://ppt.cc/G4Vb (θ,φ圖示)
3. Prove the following theorem for the non-degenerate case:
If A and B are 2 commuting hermitian operators, there exists (at least) a
basis of common eigenvectors that diagonalizes them both.
4. Consider a spin 1/2 particle with a magnetic moment. At time t=0, the
state of the particle is |Ψ(t=0)>=|+>. The system is allowed to evolve
→ 1 ^ ^
in a uniform magnetic field B = --- (B_0i+B_0k). What is the probability
√2
that the particle will be measured to have spin down in the z direction
after time t?
5. Explain
(a) How did EPR (Einstein, Podolsky and Rosen) define "an element of
physical reality" in their 1935 paper?
(b) Why EPR consider the Quantum Mechanical description to be incomplete?
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