课程名称︰量子力学一 课程性质︰必修 课程教师︰高涌泉 开课学院：理学院 开课系所︰物研所 考试日期（年月日）︰2012.11.06 考试时限（分钟）：110分 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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图示) 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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