課程名稱︰量子力學（一） Quantum Mechanics (I) 課程性質︰[必修]物理學研究所 [選修]物理學系、天文物理研究所、應用物理學研究所 課程教師︰蔣正偉 開課學院：理學院 開課系所︰物理學研究所 考試日期（年月日）︰2016/11/9 考試時限（分鐘）：120 min. 試題 : ------------------------------------------------------------------------------- 11/9/2016 PHYS7014 Quantum Mechanics I Midterm Exam INSTRUCTION 1. This is an open-book, 120-minute exam. You are only allowed to use the main textbook (by Sakurai and Napolitano), your own notes and homework assighments. You are allowed to refer to derived results in the textbook only. Please indicate both equation and page numbers when you do so. For example, Eq. (3.1) on p.90 of Sakurai and Napolitano. 2. Each sub-problem has 10 points. The total points is 100. Please arrange your time and do not waste too much time on a single problem. 3. To avoid any misunderstanding, ask if you have any question about the problems or notations. PROBLEMS 1. Remember the operator identity for operators A and B and a real parameter λ: λ^2 λ^3 exp(λA) B exp(－λA) ＝ B＋λ[A, B]＋──[A,[A,B]]＋──[A,[A,[A,B]]]＋… 2! 3! (a) If [A,[A,B]] ＝ βB for constant β, show that [A,B] exp(λA) B exp(－λA) ＝ B cosh(λ√B) ＋ ─── sinh(λ√B). √B (b) If A(t) is an operator that depends on t, then show that d 1 1 ─ ＝ exp{A(t)} {A'(t)－─[A(t),A'(t)]＋─[A(t),[A(t),A'(t)]]－＋…}, dt 2! 3! where A'(t) ＝ dA/dt. 2. The wave function ψ(x,t) expanded in terms of plane waves with definite momenta is 1 1 ψ(x,t) ＝ ──────── ∫dp φ(p,t) exp(─── p‧x). (2πhbar)^(3/2) hbar (a) Work out the Schrodiner equation for φ(p,t). [State clearly what assumption, if any, you have used in the derivation.] (b) Simplify the above result by further assuming that the potential V(x) is an analytic function of x. 3. Suppose the Hamiltonian H for a particular quantum system is a well-behaved (differentiable) function of some parameter λ. Denote E_n and ∣ψ_n〉 as the eigenvalues and the corresponding orthonormal eigenkets of H(λ). The subscript n may be a disrete or continuous set of index used to count states. One may assume either that E_n is nondegenerate, or -if degenerate- that the eigenkets are the "good" combinations. (a) Prove the so-called Feynman-Hellmann theorem: dE dH ── ＝ 〈ψ_n∣──∣ψ_n〉. (1) dλ dλ (b) Apply the theorem to the one-dimensional simple harmonic oscillator, using λ ＝ m, and explain what physics you find. Here you can directly make use the energy spectrum 1 E_n ＝ (n ＋ ─) hbar ω. 2 [Feymann derived this theorem as part of his undergraduate Senior Thesis at MIT.] 4. Consider a one-dimensional simple harmonic oscillator of mass m and charge q. Suppose the system is placed in a static electric field of strength E. Therefore, the Hamiltonian of this oscillator is given by p^2 1 H ＝ ── ＋ ─ m ω^2 x^2 － qEx. (2) 2m 2 (a) If the electric field is a constant, i.e., E ＝ E_0. Derive the energe level for all states. (b) Write down the wave function ψ_0(x) for the ground state in Part (a). Determine the most likely position of the oscillator in this ground state and give the physical interpretation for your result. 5. Work in Heisenberg picture. (a) Derive the quantum mechanic version of the Lorentz force: d^2 x 1 dx dx m ─── ＝ Q e [E ＋ ─ (─ ×B － B ×─)]. dt^2 2c dt dt (b) Explain whether the terms in the round parenthness are symmetric in the two operators? ------------------------------------------------------------------------------- ★★★Note that quantities in color purple are operators while those in color light white are vector.★★★ --

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