课程名称︰量子力学二 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）︰2013.06.18 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 : 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 为 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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