|
The quasiclassical qualitative condition imply that de Broglie wavelength of the particle  is less than characteristic length 
determining the conditions of the problem. This condition means that
the particle wavelength should not change considerably within the
length of the wavelength order
 |
, |
(1) |
where  ,  – de Broglie wavelength of the particle expressed by way of the particle classical momentum p(z) [1].
Condition (1) can be expressed in another form taking into account that
 |
, |
(2) |
where  means classical force acting upon the particle in the external field.
Introducing this force, we get
 |
(3) |
From (3) it is
clear that the quasiclassical approximation is not valid at too small
momentum of the particle. In particular, it is deliberately invalid
near positions in which the particle, according to classical mechanics,
should stop, then start moving in the opposite direction. These points
are the so called "turning points". Their coordinates  and  are determined from the condition  .
It should be
emphasized that condition (3) itself can be insufficient for the
permissibility of the quasiclassical approach. One more condition
should be met: the barrier height should not change much over the
length .
Let us consider the particles move in the field shown in Fig. 1 which is characterized by the presence of the potential barrier with potential energy  exceeding particle total energy  and meeting all the quasiclassics conditions. In this case points and are the turning points.
 |
| Fig. 1. Potential barrier of arbitrary shape. |
The approximation
technique of the Schrodinger equation solution when quasiclassical
conditions are met was first used by Wentzel, Kramers and Brillouin.
This technique is known as WKB approximation or quasiclassical
quantization method. In this textbook we do not present the Schrodinger
equations solution for the given case. However, it can be found in [1,2] and the barrier transparency in this case is
 |
(4) |
Comparing expressions (3) in chapter 1.1.1
for transmission coefficients of rectangular barrier (precise solution
of Shrodinger equation) and (4) for quasiclassical approximation, we
can notice that there is no qualitative difference between them. In
both cases the transparency decreases exponentially with the barrier
width.
Summary.
- If the problem parameters satisfy
quasiclassical conditions, then transmission coefficient can be
calculated in a general form using (4).
- In case of the square barrier
there is no qualitative difference between transmission coefficients
calculated using quantum mechanics and quasiclassical approximation. In
both cases the transparency decreases exponentially with the barrier
width.
References.
- Landau L.D., Lifshitz E. M. Quantum mechanics. Nauka, 1989 (In Russian)
- Mott N., Sneddon I. Wave mechanics and its application. Nauka, 1966 (In Russian)
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