弘毅量子力学2013A

武汉大学物理科学与技术学院2012－2013(一)

《量子力学》课程期末考试试题A 卷

学号： 姓名： 专业： 得分：

1. Only one of the four selections is correct for every problem, please choose it. (3 points every one, total 36 points)

(1). A particle, is in a combination of stationary states: ()∑=n

n n t x c t x ),(,ψψ,what

will we get if we measure its energy?( ) A. H B.

∑n n n E c C. one of the values of {}n E D. ∑n n E

(2). A particle, is in a combination of stationary states: ()∑=n

n n t x c t x ),(,ψψ, What

is the probability of measuring the energy n E ?( )

A. n c

B. ∑n

n

n c c C. 2n c D. ∑m m n c

c 2

2

(3). How many terms are in the Schr?dinger equation?( )

A. 1

B. 2

C. 3

D. 4

(4). A particle emits a certain radiation of energy E with a band width E ?. What can we say about its characteristic emission time?( )

A. Emission time is a least 2ηE E ?

B. Emission time is a most 2

ηE E ? C. Emission time is a least E ?2η D. Emission time is a most E

?2η (5). Which one of the following quantities could not physically correspond to a spherical harmonic?( )

A. ()?θ,ll Y

B. ()?θ,1,+-l l Y

C. ()?θ,1,+l l Y

D. ()?θ,1,2Y

(6). When a particle is subject to a potential that depends on the radius only, which quantum numbers apply to quantize the energy?( )

A. Only the principal quantum number n

B. Only the azimuth quantum number l

C. Possibly both numbers (n,l)

D. Possibly all three numbers (n,l,m)

(7). What is the degeneracy of the 5th energy band of the hydrogen atom?( )

A.11

B. 5

C. 25

D. 50

(8). When measuring the vertical component of the angular momentum (Lz ) on ()?θ,?1

,53Y L -, what will we get?( ) A. η B. η-

C. η2

D. η2-

(9). For a given n value of free atomic hydrogen’s wave function , how many

eigenstates can we find for the operator 2?L

?( ) A. n B. 2(n+1)

C. n(n+1)

D. n 2

(10). When measuring Sx of an electron, what are the possible results?( )

A. η±

B. η2

1± C. η21 D. η3

1± (11). If two electrons would occupy a triplet state (S=1) what can we say about their spatial wave function for exchange?( )

A. It is antisymmetric

B. It is symmetric

C. It could be both

D. It could not be both

(12). Which is wrong of the following four formulae about operator?( )

A. ()++++=+B A B A

???? B. ()++=A A ?*?λλ C. ()+++=B A B A ???? D. ()A

A ??=++

2. A particle in the infinite square well has the initial wave function:

()a x a x A x ≤≤??

? ??=0,sin 0,3πψ. Please determine A (4 points), and find ()t x ,ψ (8 points).

3. The Hamiltonian for a certain two-level system is:

()12212211?++-=εH

, where 1, 2 is an orthogonal basis and ε is a number with dimension of the

energy, please find the eigenvalues(6 points) and eigenstates as linear combinations of 1 and 2(8 points).

4. An electron is in the spin state ???

? ??=i A 125χ. Please determine the normalization constant A(4 points), and find the expectation values of x S , y S and z S .(8 points)

5. Consider the isotropic three-dimensional harmonic oscillator, please find the first-order correction of energy in the perturbation: yz x H 2'λ= (λ is constant) on

(1). The ground state(6 points),

(2). The first excited state(8 points).

The Hermit polynomials are as: x x H x H 2)(,1)(10==

6. In the quantum theory of scattering, we imagine an incident plane wave traveling in the z direction, and then producing an outgoing spherical wave caused by scattering potential. Please give the general form of solution of the Schrodinger equation (6 points), and the formula of differential scattering cross-section )(θD (6 points).