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︰物理所 考試日期（年月日）︰2013.11.07 考試時限（分鐘）：120 min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.The circular polarization states |R> and |L> of a photon are related to the linear polarization states |x> and |y> by |R> = (|x> + |y>)/sqrt(2) and |y> = (|x> - i|y>)/sqrt(2) respectively. What is the matrix representation of the angular momentum operator J_z in the |x>-|y> basis? (20%) 2.What is the matrix representation for J_x using the spin-3/2 states as a basis? (20%) 3.Consider a spin-1 particle with a magnetic moment μ=-ge/2mc‧S under the influence of a magnetic field B = B_0 k^{hat}. The Hamiltonian of the system H is given by H = ω S_z with ω=geB_0/2mc. a)Consider a state |n> defined by (n^{hat}‧S)|n> = hbar |n> and S^2|n> = 2 hbar^2|n> with n^{hat} = cosφ i^{hat} + sinφ j^{hat}. We can express the state |n> as |n> = a|1,1> + 1/sqrt(2)*|1,0> + b|1,-1>. What are a and b? (20%) b)Suppose that at time t=0, the particle is in the state |Ψ(0)> = 1/sqrt(2)*(|1,1> - |1,-1>). What is the probability of finding the particle in the state |n> at time t? (20%) 4.It has been shown in the class that the solution |Ψ(t)> of the Schrodinger equation i hbar ∂/∂t |Ψ(t)> = H(t) |Ψ(t)> is of the form t |Ψ(t)> = exp(iγ_n(t))*exp(-i/hbar∫ E_n(τ)dτ)|n> in the adiabatic 0 approximation if the system starts out in an eigenstate |n,0> of H(0) and |n,t> is the solution of the equation H(t)|n,t>=E_n(t)|n,t>. a)What is γ_n(t)? (10%) b)Show that γ_n(t) is a real quantity. (10%) --

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