課程名稱︰量子力學（二） 課程性質︰【必修】物理學研究所；【選修】天文物理所、應用物理所、物理學系 課程教師︰管希聖 開課學院：理學院 開課系所︰物理學研究所 考試日期（年月日）︰2017/05/02 考試時限（分鐘）：210 min 試題 : Midterm Examination PHYS M1420: Quantum Mechanics(II) Tuesday; May 2, 2017 Problem 1.(22 points) Consider the problem of a spinless particle of mass m in an attractive spherical square well potential, ┌ -V_0 ,r≦a V(r) = │ , └ 0 ,r＞a (a) write down the radial wave functions in the two regins of 0≦r≦a and r＞a, respectively, in terms of the spherical Bessel functions j_l(ρ), spherical Neumann functions n_l(ρ), and/or spherical Hankel functions h_l^(1) (ρ), h_l^(2) (ρ) for (i) the positive energy case (E＞0), (4 points) (ii) the negative energy case (E＜0). (4 points) (b) Show that the quantization condition for zero angular momentum (l=0) bound state (E＜0) is κ＝－k' cot(k'a), where k' is the wave number inside the well and iκ is the complex wave number for the exponential tail outside. (8 points) (c) From part (b), show that a bound state exists only if the magnitude of the depth of the well has at least a certain minimum value of π^2 (hbar)^2 V_0≧──────── 8ma^2 (6 points) Problem 2.(25 points) (a) For a spin 1/2 electron, find the state ∣\hat{n}, ±〉 with spin up/down the direction of the nuit vector \hat{n}=(n_x, n_y, n_z)=(sinθ cosφ, sinθ sinφ , cosθ), i.e., find the eigenvecotrs of the spin operator \hat{n}‧\vector{S} along the direction of \hat{n}. (7 points) (b) Suppose that an spin 1/2 electron is in a uniform magnetic field \vector{B_0} = B_0 \hat{z}. At time t=0, the electron spin is in the state of ∣\hat{n}, -〉, claculate the expectation value of 〈\vector{S}(t)〉 at time t. (8 points) (c) An additional magnetic field \vector{B_1} = B_1 cos(ωt) \hat{x} + B_1 sin(ωt) \hat{y} is now applied. If an electron in the combined field \vector{B_0} + \vector{B_1} has spin pointing along the +\hat{z} axis at time t=0, what is the probability that will have flipped to -\hat{z} axis at time t. (10 points) Problem 3.(22 points) If an electron bound to a proton through Coulomb potential (hydrogen atom) is in a state of orbital angular momentum l and is subjected to an additional spin-orbital Hamiltonian of η\vector{L}‧\vector{S}, where \vector{L} and \vector{S} are the orbital angular momentum and spin operators, respectively, and η is a constant. (a) Find the total angular momentum j-states(eigenstate) in terms of the product state of ∣l m_l〉 and ∣s m_s〉 [i.e., find the Clebsch-Gordon (C-G) coefficients for the states of thr total angular momentum \vector{J} = \vector{L} + \vector{S} which can have values of j = l±l/2], where l and s are the values of the orbital and spin angular momenta of the electron, respectively, and m_l hbar and m_s hbar are their corresponding z-component values (If you don't know how to find the answer for the general case of any value l, you may try to solve the problem for a special case of l=1 to get partial credits). (12 points) (b) Find the eigenenergies of the electron and the corresponding eigenfunctions in the spherical corrdinate basis and spinor (representation). (10 points) Problem 4.(16 points) (a) Using Wigner-Eckert theorem to derive angular momentum selection rules for an irreducible (i) first-rank tensor operator (vector operator) and (4 points) (ii) second-rank (e.g. quadrupole) tensor operator. (4 points) (b) Explain (i) what the accidental degeneracy is, (4 points) (ii) why the accidental degeneracy observed in hydrogen atom is lost in multielectron atoms. (4 points) Problem 5.(20 points) Estimate the approximate values of the two lowest energy levels of a partical of mass m in a one-dimensional potential ┌ ＋∞, x≦0 V(x)＝│ 1 └ ─ mω^2 x^2, x＞0 2 by means of (a) the variational method (10 points) (b) the WKB method (10 points). ------------------------------------------------------------------------------- sinρ cosρ spherical Bessel and Neumann functions(l=0): j_0(ρ)＝───, n_0(ρ)＝－─── ρ ρ Possible useful Gaussian integrals: \int_{-∞}^{∞} exp(-αx^2) dx = (π/α)^{1/2} for α＞0, \int_{-∞}^{∞} x^2 exp(-αx^2) dx = 1/(2α) (π/α)^{1/2}, \int_{-∞}^{∞} exp(-αx^2 + βx) dx = exp(β^2/{4α})(π/α)^{1/2}. --

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