課程名稱︰量子力學二
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）︰2013.06.18
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 :
A → → →
1. Consider the hyperfine Hamiltonian H_hf=----- S‧I where S is the electron
→ h^2
spin, I is the proton spin, and A is a constant.
(a) Obtain the hyperfine Hamiltonian matrix in the coupled basis.
(b) Provide a reason for expecting the matrix in (a) is diagonal without
doing the calculation.
(h 為 h bar)
2. (a) Derive the Hamiltonian for the relativistic perturbation
p^4
H'_rel=－----------
8m^3 c^2
(b) Argue that the correction in energy due to H'_rel in the 1st order
4 2
perturbation theory is of the form C1*α mc where C1 is a
dimensionless parameter. (You do not need to calculate C1.)
3. Consider 2 indistinguishable non-interacting spin 1/2 fermions of mass m in
a 1-dimensional infinite square well potential of length L.
(a) What is the ground state wave function?
(b) What are the wave functions of the 1st excited state?
(c) What is the degeneracy of the 1st excited state?
(d) If the fermions are subject to a repulsive, spin-independent,
inter-particle potential V(x1-x2), what is the energy of the 1st
excited state in the 1st order perturbation theory? (Leave your result
in the form of an integral.)
4. Show that the electron g-factor , g_e, is equal to 2 in the Dirac equation.
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