Quantum - Mechanics - examanation(03)解答

华南理工大学2005—2006学年上学期期末考试答卷

物理科学与技术学院 应物专业 03级 课程 量子力学

1. Calculation the following commutation relations:

(a) , for any function f(x).

(b) ,

(c)

Solution: (a) For any test function g(x)

(b)

(c) Using

2. An operator, representing observable A, has two normalized eigenstates 1and 2, with eigenvalues a1 and a2, respectively. Operator, representing observable B, has two normalized eigenstates 1and 2, with eigenvalues b1 and b2. The eigenstates are related by

1=(31+42)/5, 2=(41- 32)/5

(a) Observable A is measured and the value a1 is obtained. What is the state of the system (immediately) after this measurement?

(b) If B is now measured, what are the possible results, and what are their probabilities?

(c) Right after the measurement of B, A is measured again. What is the probability of getting a1?

Solution: (a) 1

(b) in state 1=(31+42)/5, measure B, get b1 (with probability 9/25) or b2 (with probability 16/25)

(c) Right after the measurement of B,

·with probability 9/25 the particle is in state 1=(31+42)/5; in that case, the probability of getting a1 is 9/25,

·with probability 16/25 the particle is in state 2=(41-32)/5; in that case, the probability of getting a1 is 16/25,

So the total probability of getting a1 is

(9/25)(9/25)+ (16/25)(16/25)=337/625=0.5392

3 Suppose a spin particle in the state,

(a) What are the probabilities of gettingand, if you measure and?

(b) Find the expectation values of and.

Solution:

(a) The eigenspinors of are,(eigenvalue) ;,(eigenvalue)， then

So for,the probability of getting is;

The probability of getting is.

The eigenspinors of are

,(eigenvalue);,(eigenvalue)

Then

So for,the probability of getting is; The probability of getting is.

(b) The expectation value of is

or

The expectation value of is

or

4. For two noninteraction identical particles, the possible single particle states are 1, 2, 3, find the number of possible state of the system (a) particles are bosons, (b) particles are fermions, (c) particles are classical ones.

Solution: (a) particles are bosons, there are six possible states of the system: 11, 22, 33,

(b) particles are fermions, there are 3 possible states of the system:

(c) classical particles are distinguishable, so 12 (particle 1 in state 1,particle 2 are in state 2 )and 21 (particle 1 in state 2,particle 2 are in state 1 )are different. There are 9 possible states of the system:

11, 12, 13, 21, 22, 23, 31, 32, 33

5. For one infinite square well, find the effect (in first order) of the perturbation

on the ground state.

Solution: the un-perturbed ground energy and wavefunction are

so the first correction to ground state is:

6. A particle of mass m moves in a one dimension potential V(x)=k x 4 (k>0), use the variation method to obtain an upper limit for the energy of the ground state of the particle.

Solution: Chose the trail wavefunction as

, A is determined by normalization:

7. Consider a particle scattered off the three-dimensional potential V(r)=V0(r-a), calculate the differential scattering cross-section by Born approximation.

Solution: The first approximation for this potential is

As a0,

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