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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