Elements of Quantum Theory
81
This means that |a – | = |a + | and
φ (x) = 2a + e
–i
δ cos (px + δ), x < 0
(3.89)
where
δ = tan
–1
(α/p)
(3.90)
i.e. φ (x) represents a standing wave.
There are several significant points about the results, that should be noted:
1. Since
2
–
| |
∞
∞
ψ
∫ dx = ∞, the wave function is not normalizable.
However, it could be used to describe a beam of particles with a density of
|a + |
2
per unit length, moving to the right, which get either transmitted or
reflected at x = 0.
2. For E > V, i.e. case (i), the rate of flow of the incoming particles must be
equal to the sum of the rates of flow of transmitted and reflected particles.
Since the momenta of these particles are p, q and – p respectively, one
has the condition
p|a + |
2
= q |b + |
2
+ p|a – |
2
(3.91)
which is equivalent to the requirement that T + R = 1 and is easily seen to
be satisfied by the relations is Eq. (3.83).
3. For E < V, i.e. case (ii), a standing wave is obtained for x < 0, which
means that all the incoming particles are reflected. However, the wave
function is nonzero for x > 0 though it vanishes exponentially as x → ∞.
Thus, there is a finite probability of finding the particles in the region x > 0
which is a forbidden region in classical mechanics. This is called barrier
penetration and has no classical analogue in the mechanics of particles
(such a phenomenon was observed in optics by Newton). It does not
however lead to a paradox since localization or observation of the particle
in the classically forbidden region involves a change in the momentum and
the energy of the particle which may then have sufficient energy to make
this a classically allowed region.
Electrons in a metal are a case with E < V but when metal is heated it
satisfies condition E > V and hence we see thermionic emission. This is
used in cathode ray tubes.
4. A special case of interest is the one of V → ∞ in case (ii) for which
α → ∞ and the wave function is given by
φ(x) = 2ia + sin (px),
for x < 0
(3.92)
φ r (x) = 0,
for x ≥ 0
(3.93)
81
This means that |a – | = |a + | and
φ (x) = 2a + e
–i
δ cos (px + δ), x < 0
(3.89)
where
δ = tan
–1
(α/p)
(3.90)
i.e. φ (x) represents a standing wave.
There are several significant points about the results, that should be noted:
1. Since
2
–
| |
∞
∞
ψ
∫ dx = ∞, the wave function is not normalizable.
However, it could be used to describe a beam of particles with a density of
|a + |
2
per unit length, moving to the right, which get either transmitted or
reflected at x = 0.
2. For E > V, i.e. case (i), the rate of flow of the incoming particles must be
equal to the sum of the rates of flow of transmitted and reflected particles.
Since the momenta of these particles are p, q and – p respectively, one
has the condition
p|a + |
2
= q |b + |
2
+ p|a – |
2
(3.91)
which is equivalent to the requirement that T + R = 1 and is easily seen to
be satisfied by the relations is Eq. (3.83).
3. For E < V, i.e. case (ii), a standing wave is obtained for x < 0, which
means that all the incoming particles are reflected. However, the wave
function is nonzero for x > 0 though it vanishes exponentially as x → ∞.
Thus, there is a finite probability of finding the particles in the region x > 0
which is a forbidden region in classical mechanics. This is called barrier
penetration and has no classical analogue in the mechanics of particles
(such a phenomenon was observed in optics by Newton). It does not
however lead to a paradox since localization or observation of the particle
in the classically forbidden region involves a change in the momentum and
the energy of the particle which may then have sufficient energy to make
this a classically allowed region.
Electrons in a metal are a case with E < V but when metal is heated it
satisfies condition E > V and hence we see thermionic emission. This is
used in cathode ray tubes.
4. A special case of interest is the one of V → ∞ in case (ii) for which
α → ∞ and the wave function is given by
φ(x) = 2ia + sin (px),
for x < 0
(3.92)
φ r (x) = 0,
for x ≥ 0
(3.93)