5量子力学

量子力学 1. The wave function } 2 exp{)( 22xbAx −=ψ , where A and b are real constants, is a normalized eigenfunction of the Schrödinger equation for a particle of mass M and energy E in a one dimensional potential V(x) such that V(x) = 0 at x = 0. Which of the following is correct? (A) M bV 2 42h= (B) M xbV 2 242h= (C) M xbV 2 462h= (D) ( )2222 1 xbbE −= h (E) M bE 2 42h= 解：Schrödinger方程 ψψ )( 2 2 22 VE dx d M −=− h ， ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛−−=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛−−− 2 exp)( 2 exp)1( 2 2222 222 2 xbVExbxbb M h 。 在上式中令 x=0，得 M bE 2 22h= 。 将 E的达式代回 Schrödinger方程，得 2 42 2 x M bV h= 。 选（B）。 ε x V O x1 - V0 x2 2. An attractive one-dimensional square well has depth V0 as shown above. If there is a bound state at an energy ε<0, which of the following best shows a possible wave function for this state? x ψ O x1 x2 x ψ O x1 x2 x ψ O x1 x2 x ψ O x1 x2 x ψ O x1 x2 ( A ) ( B ) ( C ) ( D ) ( E ) 1 量子力学 解：（A）不对，当x→∞ 时，波函数ψ没有趋于 0。 （C）不对，波函数ψ在x1和x2处不连续。(D)不对， 波函数ψ在x1和x2处导数不连续（以上两条对波函数 的要求可参阅曾谨言《 量子力学导论》第二版， pp. 53，北京大学出版社）。在 x > x2 和x < x1 处， V(x) = 0 。由Schrödinger方程 ( ) ( )[ ] ( ) ( )xmxxVEmx ψεψψ 22 22 hh =−−=′′ 所以 ( )xψ 大于零处， )(xψ ′′ 大于零； ( )xψ 小于零 处， )(xψ ′′ 小于零。其行为正如(B)中所描述，而不 应像(E)那样振荡。 3. If an electron were confined to nuclear dimensions, the uncertainty in its momentum would be most nearly (A) 0.2 eV/c (B) 200 eV/c (C) 200 KeV/c (D) 200 MeV/c (E) 200 GeV/c 解：测不准关系 2 h≥Δ⋅Δ px 。 取 ，原子核的尺度为 m，动 量不确定度为 h=Δ⋅Δ px 1510~ −Δx 34 19 8 15 19 1.054 10 1.054 10 3 10 eV/c 10 1.6 10 219 MeV/c p x - - - - 创 创D = = = D � = h 。 选(D)。 4. The total energy in quantum mechanics corresponds to the differential operator given by (A) t i ∂ ∂h (B) 2 2 2 t i ∂ ∂h (C) ∇− hi (D) ∇hi (E) ∇⋅∇h 解：Schrödinger方程 ψψψ EH t i ==∂ ∂ )h 。 选（A）。 5. The third lowest energy level of a one-dimensional quantum mechanical harmonic oscillator of frequency f has an energy of (A) 0 (B) hf 2 3 (C) 2hf (D) hf 2 5 (E) 3hf 解：由一维谐振子能级公式 hfnnEn ⎟⎠ ⎞⎜⎝ ⎛ +=⎟⎠ ⎞⎜⎝ ⎛ += 2 1 2 1 ωh 而第三个最低能级对应 n = 2，能量 hfE 2 5 2 = 。 选（D）。 ( ) ⎩⎨ ⎧ > <= 0for 0for 0 0 xV x xV V = 0 V = V0 0 x 6. The figure above represents a step function in potential energy for electrons moving along the x-direction in a one-dimensional problem. A monochromatic beam of electrons of energy E is incident on the barrier from the left. If E>V0, then which of the following is correct? (A) At x=0, the sum of the amplitudes of the incident wave and the reflected wave is equal to the amplitude of the transmitted wave. (B) At x=0, the amplitude of the transmitted wave is zero. (C) The wave number of the reflected wave is less than that of the incident wave. (D) The wave number of the reflected wave is equal to that of the transmitted wave. 2 量子力学 (E) Electrons pass over the potential barrier without reflection but with reduced speed. 解：设入射平面波为 ，其中ikxe h/2mEk = 。 在 x < 0 处有入射和反射波，在 x > 0 处仅有透射 波。波函数为 ⎪⎩ ⎪⎨⎧ > <+= ′ − 0 x ,Te 0; x,Re )( xki ikxikxe xψ 其中 h/)(2 0VEmk −=′ 。 在 x = 0 处有波函数连续条件 TR =+1 。 选（A）。 7. If f is frequency and h is Planck’s constant, the zero-point energy of a one-dimensional quantum mechanical harmonic oscillator is (A) 0 (B) hf 3 1 (C) hf 2 1 (D) hf (E) hf 2 3 解：由一维谐振子能级公式 hfnnEn ⎟⎠ ⎞⎜⎝ ⎛ +=⎟⎠ ⎞⎜⎝ ⎛ += 2 1 2 1 ωh ， 而零点能对应于 n=0，所以 hfE 2 1 0 = 。 答案选（C）。 Questions 8-9 3 Two spin- 2 1 particles, 1 and 2, have spins in a singlet state with spin wave function )]1()2()2()1([ 2 1)2,1( βαβα −=Ψ , where α and β refer to up and down spins, respectively, along any chosen axis. The spin of particle 1 is measured along the z-axis and found to be up. 8. A simultaneous measurement of the spin of particle 2 along the z-axis would yield which of the following results? (A) Up with 100% probability (B) Down with 100% probability (C) Up with 25% probability, down with 75% probability (D) Up with 50% probability, down with 50% probability (E) Up with 75% probability, down with 25% probability 解：选（B）。 9. A simultaneous measurement of the spin of particle 2 along the x-axis would yield which of the following results (A) Up with 100% probability (B) Down with 100% probability (C) Up with 25% probability, down with 75% probability (D) Up with 50% probability, down with 50% probability (E) Up with 75% probability, down with 25% probability 解：由上一问讨论，总角动量为零，则要求角动量 的各分量均为零。对于本题，要求 021 =+= xxx ssS ， 所以s2x平均值也应为零，所以自旋x分量向上向下 的几率应该相同。选（D）。 10. The wave function of a particle is , where x is position, t is time, and k and ω are positive real numbers. The wave function represents a simultaneous eigenstate of )( tkxie ω− (A) position and momentum (B) energy and time (C) energy and momentum (D) position and time (E) energy, momentum, position, and time 解：此为一维平面波，代表一维自由运动粒子。所 以能量和动量为守恒量。选(C)。 11. The three operators (Lx, Ly, Lz) for the components 量子力学 of angular momentum commute with the Hamiltonian. Therefore the angular momentum is 4 (A) equal to zero (B) equal to energy in magnitude (C) a unit vector (D) proportional to sinθ (E) a constant of motion 解：因为 ，其中 α = x, y, z，所以 [ ] 0, =Hlα [ ] [ ] [ ] 0,,,2 =+= ααααα lHlHllHl ， [ ] [ ] 0,, 22 == ∑ α α HlHl ， 角动量为守恒量，或称运动常数。选（E）。 12. Under an exchange of both coordinates and spins, the complete wave function for a system of two electrons must be (A) antisymmetric (B) symmetric (C) additive (D) incoherent (E) orthogonal to either independent wave function 解：电子为 Fermion，而 Fermion 要求波函数交换 反对称。选(A)。 13. A spinless particle is confined in a cubical box of side L for which the potential is V = 0, for 0 ≤ x, y, z ≤ L V = ∞, otherwise. What is the degeneracy of the third quantum level in the box? (A) 1 (B) 3 (C) 6 (D) 9 (E) 12 解：为使波函数在盒壁上为零，各方向波矢必须满 足 ...3,2,1 , == nnLk πα ， 其中α=x, y, z。因此粒子的能级为 ( ) ( ) 2 2 2 2 2 22 2 2 2 2 2 2 lmn x y z k 2E k k k m m l m n m L p = = + + 骣 ÷ç= + +÷ç ÷ç桫 h h h 。 第一能级对应（1, m, n）取值为（1, 1, 1）, 第二能 级对应（1, m, n）取值为（1, 1, 2）、（1, 2, 1）、（2, 1, 1），第三能级对应（1, m, n）取值为（1, 2, 2）、（2, 1, 2）、（2, 2, 1）。所以简并度为 3。选（B）。注意这 与用周期性边界条件所的情况不同，在那种情况下 n可以取 0。 14. The hypothesis that an electron possesses spin is qualitatively significant for the explanation of all of the following topics EXCEPT the (A) structure of the periodic table (B) specific heat of metals (C) anomalous Zeeman effect (D) deflection of a moving electron by a uniform magnetic field (E) fine structure of atomic spectra 解：（D）涉及的为普物甚至是高中课程，显然 不需要考虑自旋就可解释。选（D）。（B）比较容易 混，关键是自旋的引入使态的数目加倍。 15. Eigenfunctions for a rigid dumbbell rotating about its center have a dependence of the form ( ) φφψ imAe= , where m is a constant. Which of the following values of A will properly normalize the eigenfunction? (A) m2 (B) 2π (C) (2π)2 (D) π2 1 (E) π2 1 解：最简单的量子力学题。令 122 2 0 2 0 === ∫∫ −∗ πφφψψ π φφπ AdAeAed imim ， 量子力学 π2 1A = 。 选(D)。 16. The Hamiltonian operator in the Schrödinger equation can be formed from the classical Hamiltonian by substituting (A) wavelength and frequency for momentum and energy (B) a differential operator for momentum (C) transition probability for potential energy (D) sums over discrete eigenvalues for integrals over continuous variables (E) Gaussian distributions of observables for exact values 解：只需做如下变换： ∇−→ hip ， 22 2 2 ∇−→ h m p 。 选（B）。 17. If ψ is a normalized solution of the Schrödinger equation and Q is the operator corresponding to a physical observerble x, the quantity ψ*Qψ may be integrated to obtain the (A) normalization constant for ψ (B) special overlap of Q with ψ (C) mean value of x (D) uncertainty in x (E) time derivative of x 解：量子力学中，可观测力学量 A 的平均值 τψψ dAA ∫ ∗= ˆ ，其中 Aˆ为 A对应的算符。选（C）。 18. Which of the following is an eigenfunction of the linear momentum operator x i ∂ ∂− h with a positive eigenvalue ; i.e., an eigenfunction that describes a particle that is moving in free space in the direction of positive x with a precise linear momentum? kh (A) kxcos (B) kxsin (C) ikxe− (D) ikxe (E) kxe− 解：自由粒子本征态为量子力学基础知识，选（D）。 若不敢肯定，可以现场求导： ikxikxikx keikeie x i hhh =⋅−=∂ ∂− 。 19. A system containing two identical particles is described by a wave function of the form ( ) ( ) ( ) ( )[ ]212121 xxxx αββα ψψψψψ += where x1 and x2 represent the spatial coordinates of the particles and α and β represent all the quantum numbers, including spin, of the states that they occupy. The particles might be (A) electrons (B) positrons (C) protons (D) neutrons (E) deuterons 解：由交换对称性可知，应为 Boson。(A)、(B)、(C)、 (D)分别为电子、正电子、质子、中子，均为典型的 Fermion，交换反对称。(E)为氘核，为 Boson。选 （E）。 O-a +a x Ψ (x) 20. The figure above shows one of the possible energy eigenfunctions ψ(x) for a particle bouncing freely back and forth along the x-axis between impenetrable walls located at x=–a and x=+a. The potential energy equals zero for |x|=′=< ∫− xdxxeEHE aa ψψψ 。 一级微扰为 0。选（C）。 25. A particle of mass M is infinitely deep quare well potential V where V = ∞ for x<−a, a E) and width L. If 20 2 /8 hyEmq −= π , A is constant, and aL >> 1, the transmission coefficient is best approximated by (A) qLAe 2− (B) qLAe 2 (C) A sinh qL (D) A sin qL (E) A/[1+(qL)2] 解：详细的推导可参阅曾谨言《量子力学导论》， 第二版，63页，北京大学出版社。考试现场显然没 时间去推公式，可定性考虑。显然势垒宽度越大， L越大，透射系数越小。而（B）、（C）随 L增大， （D）随 L成振荡关系，均被排除。对于（A）、（E）， 联想到 α衰变中也发生势垒贯穿，而半衰期随能量 减小而迅速变长，透射系数与粒子能量（由 q代表） 的关系应为指数形式。选（A）。 27. A one-dimensional beam of particles each of kinetic energy E travels along the x-axis from left to right. It encounters a potential energy step of height E0, 7 量子力学 with E > E0.What is the reflection probability? (A) 2 0 )( E E (B) )( )( 00 00 EEEE EEEE −+ −− (C) )(2 )(2 00 00 EEEEE EEEEE −+− −−− (D) )(2 )(2 00 00 EEEEE EEEEE −++ −−+ (E) )(22 )(22 00 00 EEEEE EEEEE −+− −−− 解：详细的推导可参阅《量子力学导论》，第二版， 63页，北京大学出版社。考试现场可用极限法挑出 正确答案。当E0 → 0时，相当于没势垒，反射几率 为 0。几个选项中只有（E）符合此极限。选（E）。 28. A free particle moving in one dimension line has following wave function at time = 0: dxedX /2 1 )( − −=ψ , where d is a constant. What is the probability that a measurement of the position of the particle at time = 0 will yield a result between x1 and x2 (x2 > x1 > 0)? (A) 0 (B) (x2 – x1) / d (C) (x2 – x1)2 / d2 (D) )( 2 1 // 21 dxdx ee −− − (E) )( 2 1 /2/2 21 dxdx ee −− − 解：几率积分为 ( ) ( ) ( ) 2 1 2 1 /22 12 2 1 x x dxx x edxxxxxp −−=Ψ=>> ∫ 。 选（E）。 29. Two particles have angular-momentum quantum numbers l1 = l2 = 4, m1 = 3, and m2 = 2. Which of the following is an allowed value for l corresponding to L = L1 + L2? (A) 0 (B) 1 (C) 2 (D) 4 (E) 6 解：根据矢量合成的原理，合角动量的投影 521 =+= mmm ， 而合角动量的大小显然大于它的投影值，l > m = 5。 选（E）。 30. A particle of energy E moving in one dimension is scattered by a potential barriers of height V0(V0 > E) and width L. If 20 2 /8 hVEmq −= π . A is a constant and qL >> 1, the transmission coefficient is best approximated by (A) Ae−2qL (B) Ae2qL (C) A sinh qL (D) A sin qL (E) A / [1+(qL)2] 解：一维势垒穿透的隧道效应。在 qL >> 1的条件 下，shqL >> 1，透射系数 ( ) qLqL ee V EVET 22 0 016 −− ∝−≈ 。 选（A）。 31. The wave function for the lowest energy level of hydrogen is 2/3 / 0 o ar a e π − =Ψ , where a0 is the Bohr radius. The expectation value r 1 of reciprocal distance of the electron from the nucleus is (A) 1 / a0 (B) 1.5 / a0 (C) 2 / a0 (D) a0 (E) ∞ 解：直接积分求平均值 0 0 22 1411 a drr rr =⋅Ψ= ∫∞ π 。 选（A）。 8 量子力学 b V0 E 32. A wave packet of electrons having a mean energy E is incident from the left on the one-dimensional square well of depth V0 and width b shown above. Unusually high transmission of the electrons (analogous to the Ramsauer-Townsend effect) will take place when the energy of the electrons most nearly satisfies which of the following conditions? (A) b VEM h 2 )(2 0 =+ (B) b VEM h 4 )(2 0 =+ (C) b VEM h 8 )(2 0 =+ (D) b ME h 4 2 = (E) b ME h 2 2 = 解：形成共振透射（T = 1）的条件是 πnbk =' ， ( ) 2 02' h VEm k += 。 上式中当 n = 1时即是选项（A）。选（A）。 33. The Pauli exclusion principle results from the quantum mechanical fact that no two electrons in an atom can (A) have the same set of quantum numbers (B) have the same spatial wave function (C) have the same spin (D) interact with each other (E) be in an excited state simultaneously 解：Pauli不相容原理：不可能由两个全同的 Fermi 子处于同一个单粒子态。而电子属于 Fermion。选 （A）。 x ∞ ∞ O V (x) 34. A one-dimensional square well potential with infinitely high sides is shown above. In the lowest energy state, the wave function is proportional to sin kx. If the potential is altered slightly by introducing a small bulge in the middle as shown, which of the following is true of the ground state? (A) The energy of the ground state remains unchanged. (B) The energy of the ground state is increased. (C) The energy of the ground state is decreased. (D) The original ground state splits into two states of lower energy. (E) The original ground state splits into two states of higher energy. 解：微扰的Hamilton量始终为正，在以及近似下的 修正能量<Ψ0’|H’|Ψ>显然为正，因此基态能级上升。 选（B）。 35. X and Y are two stationary states of a particle in a spherically symmetric potential. In which of the following situations will the wave functions of the two states be orthogonal? I. X and Y correspond to different energies. II. X and Y correspond to different total orbital angular momenta L. III. X and Y correspond to the same L but different Lz. (A) Not neccesarily in any of these situations. (B) In situation I, but not neccesarily in II or III. (C) In situation I and II, but not neccesarily in III. 9 量子力学 (D) In situation II and III, but not neccesarily in I. (E) In all three situations. 解：在球对称势场的问题中，不考虑自旋的情况下， 完备的量子数为（n, l , lz)，解为三者的共同本征函 数。对于三者中任何不完全相同的两个量子态，都 是正交的。选（E）。 36. A particle of mass m moves in a thress-dimensional potential V(r) = 2 1 kr2. If k is halved, what is the ratio of the new ground-state energy to that of the old ground-state energy? (A) 6/1 (B) 1/2 (C) 2/1 (D) 2 (E) 6 解：一维谐振子的零点能 μω kE 22 1 0 hh == 。 因此前后零点能之比 2 1'' 0 00 == k k E E o 。 选（C）。 - l la- a V0 ∞ ∞ x 37. A spinless nonrelativistic particle of mass m is placed in the divided suqare well shoqn above. The potential rises to infinity at x = ± l. Which of the following is NOT true for states ψ(x) of definite energy E? (A) If V0 →∞, there are two states for each allowed energy. (B) If V0 →0, the allowed energies satisfy 2 22 h mEl π = integer. (C) For a general V0 , solutions of definite E have a definite reflection symmetry. ψ(x) = ψ(−x) or ψ(x) = − ψ(−x). (D) ψ(l) = 0 = ψ(−l). (E) ψ(a) = 0 = ψ(−a). 解：对一般的V0，在x = ±a处波函数只需满足连续 性及导数连续的边条件。当V0 → 0时，相当于一个 无穷高势垒中的束缚态。而当V0 → ∞时，相当于彼 此隔断的两个无穷高势垒，分别对应各自的束缚 态。选（E）。 38. If S is the total spin quantum number, which of the following lists all possible spin states for three electrons? (A) Two S = 1/2 doublets and one S = 1 triplet (B) Two S = 0 singlets, one S = 1/2 doublet, and one S = 3/2 quartet (C) One S = 0 singlet, one S = 1 triplet, and one S = 3/2 quartet (D) Two S = 1/2 doublets and one S = 3/2 quartet (E) Four S = 1/2 doublets 解：三个电子总共可能的自旋取法为： 1、两个同向，一个反向，S = 1/2，为双重态； 2、三个都同向，S = 3/2，为四重态。 选（D）。 39. A quantum system has two eigenstates : )(1 xΨ with energy E1 and )(2 xΨ with energy E2. These are normalized and orthogonal: that is, ∫∫ ∞ ∞− ∗ ∞ ∞− ∗ ΨΨ==ΨΨ dxdx 2211 1 and 10 量子力学 ∫∞ ∞− ∗ΨΨ= dx210 If at t = 0 the system was in the state )]()([ 2 1)( 21 xxx Ψ+Ψ=Ψ , the probability of finding it in this same state )(xΨ at a later time t is (A) zero (B) 1 (C) htEE /)sin( 21 − (D) ][ 2 1 // 21 htiEhtiE ee −− + (E) ]/)cos(1[ 2 1 21 htEE −+ 解：体系的随时间演化的波函数为 ( ) ( ) ( )( hh /2/1 2121, tiEtiE exextx −− Ψ+Ψ=Ψ 11 )。 t时刻粒子处于ψ(x, 0)态的几率是 ( ) ( ) 2//2* ][ 2 1,0, 21 htiEhtiE eedxtxxp −− ∞ ∞− +=ΨΨ= ∫ 。 选（E）。 Questions 40-41 The sketch below shows a one-dimensional potential for an electron. The potential is symmetric about the V-axis. x V (x) 40. Which of the following statements correctly describes the ground state of the system with on electron present? (A) A single electron must be localized in one well. (B) The ground state will accommodate up to four electrons. (C) The kinetic energy of the ground state will be one-half its potential energy. (D) The wave function of the ground state will be antisymmetric with respect to the V-axis. (E) The wave function of the ground state will be symmetric with respect to the V-axis. 解：因为势垒关于 y轴对称， )()( xVxV −= ， 所以波函数有确定的宇称。由于基态波函数