Elements of Modern Physics
82
This wave function could have been obtained from Eqs. (3.73) and (3.86)
with b – = 0, by just requiring that the wave function vanishes at x = 0,
φ (0) = φ r (0) = 0, but no further conditions on dφ/dx. Indeed this prescription
considerably simplifies that calculations whenever the potentials jump to
infinity.
5. The quantum mechanical phenomenon of a particle penetrating classically
forbidden barriers gives rise to an interesting observation of trapped
particles escaping through classically forbidden barriers. Consider a situation
[Fig. 3.2(b)] where the barrier exists only for 0 ≤ x ≤ d, i.e. V (x) = V for
0 ≤ x ≤ d and V (x) = 0 elsewhere. In this case, if a beam of particles, is
incident from the left with energy E < V, the wave function is given by
φ(x) = a + e
ipx
+ a – e
–ipx
for x < 0,
= b + e
–
α
x
+ b – eα
x
for 0 ≤ x ≤ d,
(3.94)
= c + e
ipx
for x > d
where p and α are given in Eqs. (3.74) and (3.85) respectively. Continuity
of the wave function and its derivative at x = 0 and x = d, allows us to
determine a – , b + , b – and c + in terms of a + . Since the particles can penetrate
the forbidden barrier, in general b + , b – and c + are nonzero. Thus, the particles
can cross a barrier even if classically, the energy is insufficient to pass
over the barrier, and the probability of transmission is given by the ratio
|c + |
2
/|a + |
2
. This effect is termed as tunnelling and provides a satisfactory
explanation for the decay of unstable particles (e.g. U 235 , etc.) as a
tunnelling of trapped particles through a potential barrier.
A scanning tunneling microscope (STM) is an instrument for imaging
surfaces at the atomic level. It is based on the concept of quantum
tunneling. When a conducting tip is brought very near to the surface to
be examined, a bias (voltage difference) applied between the two can
allow electrons to tunnel through the vacuum between them. The resulting
tunneling current is a function of tip position, applied voltage, and the
local density of states (LDOS) of the sample. Information is acquired by
monitoring the current as the tip’s position scans across the surface. For
an STM, good resolution is considered to be 0.1 nm lateral resolution and
0.01 nm depth resolution. With this resolution, individual atoms within
materials are routinely imaged and manipulated. (Source : Wikipedia)
82
This wave function could have been obtained from Eqs. (3.73) and (3.86)
with b – = 0, by just requiring that the wave function vanishes at x = 0,
φ (0) = φ r (0) = 0, but no further conditions on dφ/dx. Indeed this prescription
considerably simplifies that calculations whenever the potentials jump to
infinity.
5. The quantum mechanical phenomenon of a particle penetrating classically
forbidden barriers gives rise to an interesting observation of trapped
particles escaping through classically forbidden barriers. Consider a situation
[Fig. 3.2(b)] where the barrier exists only for 0 ≤ x ≤ d, i.e. V (x) = V for
0 ≤ x ≤ d and V (x) = 0 elsewhere. In this case, if a beam of particles, is
incident from the left with energy E < V, the wave function is given by
φ(x) = a + e
ipx
+ a – e
–ipx
for x < 0,
= b + e
–
α
x
+ b – eα
x
for 0 ≤ x ≤ d,
(3.94)
= c + e
ipx
for x > d
where p and α are given in Eqs. (3.74) and (3.85) respectively. Continuity
of the wave function and its derivative at x = 0 and x = d, allows us to
determine a – , b + , b – and c + in terms of a + . Since the particles can penetrate
the forbidden barrier, in general b + , b – and c + are nonzero. Thus, the particles
can cross a barrier even if classically, the energy is insufficient to pass
over the barrier, and the probability of transmission is given by the ratio
|c + |
2
/|a + |
2
. This effect is termed as tunnelling and provides a satisfactory
explanation for the decay of unstable particles (e.g. U 235 , etc.) as a
tunnelling of trapped particles through a potential barrier.
A scanning tunneling microscope (STM) is an instrument for imaging
surfaces at the atomic level. It is based on the concept of quantum
tunneling. When a conducting tip is brought very near to the surface to
be examined, a bias (voltage difference) applied between the two can
allow electrons to tunnel through the vacuum between them. The resulting
tunneling current is a function of tip position, applied voltage, and the
local density of states (LDOS) of the sample. Information is acquired by
monitoring the current as the tip’s position scans across the surface. For
an STM, good resolution is considered to be 0.1 nm lateral resolution and
0.01 nm depth resolution. With this resolution, individual atoms within
materials are routinely imaged and manipulated. (Source : Wikipedia)