量子物理课堂习题

量子物理课堂习题

Lecture 1: 旧量子论

1.求氘原子Hα线n=2到n=3的波数

2.Ce的逸出功是1.9eV, 求阈值频率和波长

3.对于氢原子、一次电离的氦离子He+和两次电离的锂粒子Li++,分别计算它们的：

a)第一、第二波尔轨道半径及电子在这些轨道上的速度

b)电子在基态的结合能

c)第一激发态退激到基态所放光子的波长

Lecture 2：波粒二象性不确定性原理

=a（n＝1，2，…, a 为弦长）。按照“定态即驻波”的说

1.已知琴弦振动的驻波条件为nλ

2

法，束缚在长宽高分别为a，b，c 的三维势箱中的粒子(质量为m)的定态能量取值是

多少？

2.一原子的激发态发射波长为600nm 的光谱线，测得波长的精度为?λ/λ=10?7,试问该

原子态的寿命为多长？

3.1，3—丁二烯分子长度a≈7?，试用测不准关系估计其基电子态能级的大小（量级）Lecture 3: 波函数薛定谔方程

1.下列哪些函数不是品优函数，说明理由：f(x)=x2,e?|x|,sin(x),e?x2

2.试写出下列体系的定态薛定谔方程：（a）He 原子（b）H2 分子

3.写出一个被束缚在半径为a的圆周上运动的粒子的Schr?dinger 方程，并求其解Lecture 4: 势箱模型

1.（

2.7）Consider a particle with quantum number n moving in a one-dimensional box of length

l.

(a) Determine the probability of finding the particle in the left quarter of the box.

(b)For what value of n is this probability a maximum?

(c) What is the limit of this probability for n→∞?

(d) What principle is illustrated in (c)?

2.（2.17）A crude treatment of the pi electrons of a conjugated molecule regards these

electrons as moving in the particle-in-a-box potential of Fig. 2.1, where the box length is

somewhat more than the length of the conjugated chain. The Pauli exclusion principle

(Chapter 10) allows no more than two electrons to occupy each box level. (These two have

opposite spins.) For 1,3-butadiene, CH2 =CHCH=CH2, take the box length as 7.0 ? and use

this model to estimate the wavelength of light absorbed when a p electron is excited from the

highest-occupied to the lowest-vacant box level of the molecular electronic ground state. The experimental value is 217 nm.

3.(2.18) For the particle in a one-dimensional box of length 1, we could have put the coordinate

origin at the center of the box. Find the wave functions and energy levels for this choice of origin.

4.试用一维势箱模型(6个电子)计算如下分子的电子光谱最大吸收波长（第一吸收峰）。

5.边长等于L 的三维立方无限深势箱中，写出质量为m粒子对应的波函数、及其能量，

并给出能量小于等于12 h2 / 8mL2的可能的状态数。

Lecture 5: 谐振子

1.（4.11）Use the recursion relation (4.46) to find the ν=3normalized harmonic-oscillator

wave function.

2.（4.20）(a)The three-dimensional harmonic oscillator has the potential-energy function V=

1 2k x x2+1

2

k y y2+1

2

k z z2 , where the k's are three force constants. Find the energy

eigenvalues by solving the Schrodinger equation.

(b) If k x=k y=k z,find the degree of degeneracy of each of the four lowest energy levels.

3.（

4.19）Find the eigenvalues and eigenfunctions of H?for a one-dimensional system

with V(x)=∞ for x<0,V(x)=1

2

kx2for x≥0.

Lecture 6: 算符与量子力学

1.(3.48)For the particle confined to a box with dimensions a, b, and c, find the following values

for the state with quantum numbers n x,n y,n z.

(a) ?x?;

(b) ?y?,?z?. Use symmetry considerations and the answer to part a.

(c) ?p x?;

(d) ?x2?. Is ?x2?=?x?2?Is ?xy?=?x??y??

2.(

3.25) Find the eigenfunctions of ??2

2m

d2/dx2. If the eigenfunctions are to remain finite for x→±∞, what are the allowed eigenvalues?

3.(3.30) Evaluate the commutators

(a)[x?,p?x],

(b) [x?,p?x2],

(c) [x?,p?y],

(d) [x?,V?(x,y,z)],

(e) [x?,H?], where the Hamiltonian operator is given by Eq. (3.45);

(f) [x?y?z?,p?x2]

Lecture 7: 量子力学基本定理

1.(7.6) Let A and B be Hermitian operators and let c be a real constant.

(a) Show that cA is Hermitian.

(b) Show that A+B is Hermitian.

2.(7.7) (a) Show that d2/dx2and T x are Hermitian, where T x≡?(?2/2m)d2/dx2

(b)show that ?T x?=(?2

2m )∫|eΨ

ex

|

2

dτ

H3C

N H3C H

C+

CH3

CH3 l =8A

(c) For a one-particle system, does ?T?equal ?T x?+?T y?+?T z??

(d) Show that ?T?≥0for a one-particle system.

Lecture 8: 角动量

1.Show the three commutation relations [L?x,L?y]=i?L?z, [L?y,L?z]=i?L?x and [L?z,L?x]=i?L?y

are equivalent to the single relation L?×L?=i?L?.

2.（5.18）find [L?x2,L?y].

3.（5.23）Calculate the possible angles between L and the z axis for l=2.

4.（

5.30）Show that the spherical harmonics are eigenfunctions of the operator L?x2+L?y2.(The

proof is short.) What are the eigenvalues?

5.（5.35）Apply the lowering operator L??three times in succession to Y11θ,(θ,φ)and verify

that we obtain functions that are proportional to Y10,Y1?1, and 0.

6.（

7.11）Let A?be a Hermitian operator. Show that ?A2?=∫|A?ψ|2dτand therefore ?A2?≥

0.

Lecture 9：H原子

1.（6.2）The particle in a spherical box has V=0 for r≤b and V=∞for r＞b. For this system:

(a) Explain why ψ=R(r)f(θ,Φ), where R(r)satisfies (6.17). What is the function f(θ,Φ)?

(b) Solve (6.17) for R(r)for the l = 0 states.

Hints: The substitution R(r)=g(r)/r reduces (6.17) to an easily solved equation. Use the boundary condition that ψis finite at r = 0 [see the discussion after Eq. (6.83)] and use a second boundary condition. Show that for the l = 0 states, ψ=N[sin (nπr/b)]/r and E=

n2h2/8mb2with n = 1, 2, 3……(For l ≠0, the energy-level formula is more complicated.)

2.（6.3）If the three force constants in Prob.4.20 are all equal, we have a three-dimensional

isotropic harmonic oscillator.

(a) State why the wave functions for this case can be written as ψ=f(r)G(θ,Φ).

(b) What is the function G?

(c) Write a differential equation satisfied by f(r).

(d) Use the results found in Prob. 4.20 to show that the ground-state wave function does have

the form f(r)G(θ,Φ), and verify that the ground-state f(r) satisfies the differential equation in

(c).

3.（6.6）For a system of two noninteracting particles of mass 9.0×10?26g and 5.0×10?26g

in a one-dimensional box of length 1.00×10?8cm, calculate the energies of the six lowest

stationary states.

4.（6.23）For the ground state of the hydrogenlike atom, show that ?r?=3a/2Z.

5.（

6.24）Find ?r?for the 2p0state of the hydrogenlike atom.

6.（6.33）For the H-atom ground state, find the probability of finding the electron in the

classically forbidden region.

Lecture 10：变分法

1.（8.10）Apply the variation function Φ=e?cr to the hydrogen atom. Choose the parameter

c (which is real) to minimize the variational integral, an

d calculat

e the percent error in the

ground-state energy.

2.（8.3）Verify the result for ?Φ|L?|Φ?in the last example of Section 8.1.

3.（8.4）If the normalized variation function Φ=(3/l2)1/2x for 0≤x ≤l is applied to the

particle-in-a-one-dimensional-box problem, one finds that the variational integral equals zero, which is less than the true ground-state energy. What is wrong?

4.（8.2）(a) Consider a one-particle, one-dimensional system with potential energy

V=V0 for 1

4

l≤x≤

3

4

l,

V=0 for 0≤x≤1

4

l and

3

4

l≤x≤ l

and V=∞elsewhere (where V0 is a constant). Plot V versus x. Use the trial variation

function Φ1=(2/l)1/2sin (πx/l)for 0≤x≤ l to estimate the ground-state energy for

V0=?2/ml2and compare with the true ground-state energy E=5.750345?2/ml2. To save time in evaluating integrals, note that ?Φ1|H?|Φ1?=?Φ1|T?|Φ1?+?Φ1|V?|Φ1?and explain

why ?Φ1|T?|Φ1? equals the particle-in-a-box ground-state energy ?2/8ml2.

(b) For this system, use the variation function Φ2=x(l?x). To save time, note that

?Φ2|T?|Φ2?is given by Eq. (8.12). (Why?)

Lecture 11 微扰论

1.（9.3）For the anharmonic oscillator with Hamiltonian (9.3), evaluate E(1)for the first

excited state, taking the unperturbed system as the harmonic oscillator. What is E(0)?

H?=??2

2m

d2

dx2

+1

2

kx2+cx3+dx4(9.3)

2.（9.4）Consider the one-particle, one-dimensional system with potential-energy

V=V0 for 1

l＜x＜

3

l,

V=0 for 0≤x≤1

4

l and

3

4

l≤x≤ l

and V=∞elsewhere, where V0=?2/ml2. Treat the system as a perturbed particle in a box.

(a) Find the first-order energy correction for the general stationary state with quantum

number n.

(b) For the ground state and for the first excited state, compare E(0)+E(1)with the true

energies 5.750345?2/ml2and 20.23604?2/ml2. Explain why E(0)+E(1)for each of these two states is the same as obtained by the variational treatment of Probs. 8.2a and 8.18.

3.（9.22）(a) For a particle in a square box of length l with origin at x = 0, y = 0, write down

the wave functions and energy levels.

(b) If the system of (a) is perturbed by

H?′=b for 1

4

l ≤x≤

3

4

l and

1

4

l ≤y≤

3

4

l

where b is a constant and H?′=0elsewhere, find E(1)for the ground state. For the first

excited energy level, find the E(1)values and the correct zeroth-order wave functions.

Lecture 12：He原子基态

1.（9.13）There is more than one way to divide a Hamiltonian H?into an unperturbed part

H?0and a perturbation H?′. Instead of the division (9.40) and (9.41), consider the following

way of dividing up the helium-atom Hamiltonian:

What are the unperturbed wave functions? Calculate E(0)and E(1)for the ground state.

(See Section 9.4.)

Lecture 13：自旋与Pauli原理

1.（10.2）Calculate the angle that the spin vector S makes with the z axis for an electron with

spin function a

2.（10.6）(a) Show that P?12commutes with the Hamiltonian for the lithium atom.

(b) Show that P?12and P?23do not commute with each other.

(c) Show that P?12and P?23commute when they are applied to antisymmetric functions.

3.（10.26）(a) If the spin component S x of an electron is measured, what possible values can

result?

(b) The functions a and b form a complete set, so any one-electron spin function can be

written as a linear combination of them. Use Eqs. (10.72) and (10.73) to construct the two

normalized eigenfunctions of S?x with eigenvalues +1

2?and ?1

2

?.

(c) Suppose a measurement of S?z for an electron gives the value +1

2

?. If a measurement of

S?x is then carried out, give the probabilities for each possible outcome.

(d) Do the same as in (b) for S?y instead of S?x. In the Stern–Gerlach experiment, a beam of particles is sent through an inhomogeneous magnetic field, which splits the beam into several beams each having particles with a different component of magnetic dipole moment in the field direction. For example, a beam of ground-state sodium atoms is split into two beams, corresponding to the two possible orientations of the valence electron’s spin. (This neglects the effect of the nuclear spin; for a complete discussion, see H. Kopfermann, Nuclear Moments, Academic Press, 1958, pp. 42–51.) Problem 10.26c corresponds to setting up a

Stern–Gerlach apparatus with the field in the z direction and then allowing the +1

2

?beam

from this apparatus to enter a Stern–Gerlach apparatus that has the field in the x direction.

4.（10.25）Show that α and β are each eigenfunctions of S?x2(but not of S?x). Give a physical

explanation of why these results make sense (see Prob. 10.26a).

Lecture 14 HF方法简介

1.（11.3）If R(r1)is the radial factor in the function t1in the Hartree differential equation

(11.9), write the differential equation satisfied by R.

Lecture 15 原子光谱项

1.（11.15）Find the terms that arise from each of the following electron configurations:

(a) 1s22 s22p63s23p5g;

(b) 1s22s22p3p3d;

(c) 1s22s22p44d. You may use Table 11.2a for part (c)

2.（11.35）Draw a diagram similar to Fig. 11.6 for the carbon 1s22s22p2 configuration. (The 1S

term is the highest.)