Download 2. ANALYSIS OF THE PROPERTIES OF A MICROPARTICLE 2.1

2. ANALYSIS OF THE PROPERTIES OF A
MICROPARTICLE
2.1. Objective of the test
Simulation and investigation of behaviour of microparticles.
2.2. Theory and the main formulae
Microparticles have wave-like properties. The length  and
frequency  of de Broglie’s wave that corresponds to a microparticle
with energy W and momentum p is given by
(2.1)
h p
and
(2.2)
 W h.
Here h is Planck’s constant.
The energy spectrum of a free microparticle is continuous. Its
kinetic energy is given by
2 2
k ,
2m
where k  2  k is the wave number,   h / 2 .
Wk 
(2.3)
A microparticle can be reflected by a low potential barrier. The
reflection coefficient from a low and infinitely wide barrier (Fig
2.1(a)) is given by
2
 k  k2 
R 1
 ,
 k1  k 2 
(2.4)
where k1  2mW1  , k2  2mW2  , W1 is the initial kinetic energy
of the microparticle, W2  W1  Wb is the kinetic energy of the
microparticle in the barrier region, Wb is the height of the barrier.
A microparticle as de Broglie’s wave is reflected by a low barrier
or passes the barrier. So the transmission coefficient D  1  R .
If the low barrier has finite width d (Fig 2.1(b)), the transmission
coefficient is given by
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W
W
M
0
x
1
2
W2
2
W1
1
W2
W1
M
0
(a)
d
x
(b)
Fig 2.1. Microparticle M meets (a) low infinitely wide potential barrier
and (b) low potential barrier having finite width d

W 2 sin 2 k 2 d 

D  1  b
4W1 W1  Wb  

1
.
(2.5)
A microparticle can penetrate through a high and thin potential
barrier. This is the essence of the tunnel effect.
The transmission coefficient through the high rectangular barrier
(Fig 2.2(a)) is given by
(2.6)
D  D0 exp  2d 2mW  ,


where D0 is the proportionality coefficient having value close to 1;
W  Wb  W1 .
Wb
W
W
W
W
M
M
d
0
x
d
W1
0
(a)
(b)
Fig 2.2. (a) Rectangular and (b) triangular high barriers
8
x
If the barrier has triangular form (Fig 2.2(b)),
D  exp  4d 2mW 3 .


(2.7)
Electrons can meet the high triangular barrier in a reverse biased
pn junction. Then the thickness d of the barrier is given by
(2.8)
d  W qE .
Here q  e is magnitude of the electronic charge, E – strength of
the electric field in the depletion region of the junction.
A microparticle in a potential well cannot have arbitrary energy.
The energy spectrum of a microparticle in a potential well is discrete.
When a microparticle exists in a one-dimensional infinitely deep
potential well, its energy has particular allowed value dependent upon
quantum number n :
h2 2
(2.9)
n .
8ma 2
Here a is the width of the well, m is microparticle mass.
Wn 


1.
2.
3.
4.
5.
6.
7.
8.
2.3. Preparing for the test:
Using lecture-notes and referenced literature [1, p. 3–21] examine
the properties of a microparticle and electrons in atoms.
Prepare to answer the questions:
What determines the length of de Broglie’s wave?
What determines the frequency of de Broglie’s wave?
What is the idea of uncertainty principle?
Write and explain Heisenberg’s inequalities.
What information about a microparticle provides amplitude of the
wavefunction corresponding to the microparticle?
What happens when a microparticle meets a low potential barrier?
What happens when a microparticle meets the potential barrier
that is high for it?
Explain the essence of the tunnel effect. What is the relationship
between the probability of tunneling and parameters of the
barrier?
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9. Describe the essential features of the energy spectrum of a free
microparticle and a microparticle in a potential well.
10. How can we describe the state of the electron in an atom?
2.4. In laboratory:
1. Answer the test question.
2. According to specified data simulate behaviour and parameters of
a microparticle:
a) passing the low infinitely wide potential barrier (find
reflection coefficient R versus W1 Wb );
b) passing the low potential barrier having finite width (find
reflection coefficient R versus W1 Wb at given values of
barrier thickness d );
c) penetrating the high rectangular barrier (find how
transmission coefficient D depends on barrier thickness d
and particle mass m );
d) penetrating the high triangular barrier (find how transmission
coefficient D depends on barrier thickness d and mass m );
e) existing in a one-dimensional infinitely deep potential well
(find how allowed energy values depend on the width of the
well).
3. After necessary calculations, plot graphs and examine the results.
4. Prepare the report.
1.
2.
3.
4.
5.
2.5. Contents of the report
Objectives.
Initial data.
Results of calculations.
Graphs.
Conclusions.
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