Chapter VII Tunneling Phenomena

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Chapter VII Tunneling Phenomena
Forbidden
Quantum mechanically ?
? particles decay
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y(x)
U(x)
n 1, 2, 3,
n 0, 1, 2, 3,
The wavefunctions for the finite square wells and
the quantum oscillators look roughly similar The
particle has a finite probability of leaking
out of the well !
No classical analogy !!
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The Step Potential
time-independent potential
I
II
Separation of variables
time-independent Schrödinger equations
Case (I) total energy E lt Uo
general solutions
Condition 1. x gt 0 and x ? ?, ?(x) ? 0
C 0
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Condition 2. continuity condition on boundary x
0
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Transmission and Reflection coefficients
The incident wave
The reflected wave
The transmitted wave
The reflection coefficient
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A particle incident on the step potential with
E lt Uo has a unity (100 !) probability of being
reflected
R 1
For x gt 0
Probability density
k2 gt 0. P(x,t) decreases rapidly with increasing x
Penetration depth
The closer is E to Uo, the greater is ? and
the slower is the decay of P (x,t )
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Example. Consider a thought experiment capable
of proving that the particle is located somewhere
in the region x gt 0
The experiment amounts to locate the particle in
the range
The uncertainty in momentum
The energy of the particle is uncertain by an
amount
? It is no longer possible to say that the total
energy E of the particle is definitely less
than the potential energy U0
The particle could be found inside the barrier
for a finite amount of time
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Example. Penetration depth for a tunneling
electron
Assume U0 E a few eV 10-19 J
Notice that, for a macroscopic particle, the
penetration depth is undetectably small
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(Eisberg)
The probability density ?? for a group
wavefunction. As time evolves, the group moves up
to the step, penetrates slightly into the
classically excluded region, and then is
completely reflected from the step
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Analogy with Wave Optics
If light passing through a glass prism reflects
from an internal surface with an angle greater
than the critical angle, total internal
reflection occurs. However, the electromagnetic
field is not exactly zero just outside the
prism If we bring another prism very close to
the first one, experiments show that the
electromagnetic wave (light) appears in the
second prism The situation is analogous to the
tunneling described here. This effect was
observed by Newton and can be demonstrated with
two prisms and a laser. The intensity of the
second light beam decreases exponentially as the
distance between the two prisms increases
frustrated total internal reflection
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(a) Total internal reflection of light waves at a
glassair boundary. An evanescent wave penetrates
into the space beyond the reflecting surface. (b)
Frustrated total internal reflection. The
evanescent wave is picked up by a neighboring
surface, resulting in transmission across the
gap. Notice that the light beam does not appear
in the gap
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Case (II) total energy E gt Uo
general solutions
Let the particle be initially at x lt 0 and move
toward x 0 ? D 0, there is only a transmitted
wave in the region x gt 0
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The reflection coefficient
The transmitted coefficient
The particle has different velocities in regions
I and II
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Ex. E (4/3) Uo
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The Square Barrier Potential
time-independent
Separation of variables
time-independent Schrödinger equations
(I)
(II)
(III)
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I
III
general solutions
(G 0)
General solution for region II
Case I total energy E lt Uo
Case II total energy E gt Uo
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Consider case (II) E gt Uo
(G 0)
Find coefficients B/A, C/A, D/A, and F/A
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transmission coefficient
reflection coefficient
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Resonant transmission
A novel quantum phenomenon
Reflection coefficient
n 0, 1, 2, 3, integer
R 0 (i.e. T 1), implying that no reflection
occurs at certain energies
Resonances arise from the superposition of de
Broglie waves reflected from the leading and
trailing edges of the barrier
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Example. Resonant transmission and barrier width
multiple transmission and reflection in the
barrier
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Evidence for the wave nature of matter!!
resonance (T 1, R 0)
(12/7/2009)
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Consider case (I) E lt Uo
(G 0)
To calculate coefficients B/A, C/A, D/A, F/A
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A stationary wave for a particle in the presence
of a square barrier (for the case E lt U0). Since
the wave amplitude is nonzero in the barrier,
there is some probability of finding the particle
there
To have finite transmission, the width of the
barrier must be smaller than a few nm Barrier
width ? ?
Transmission coefficient for a square barrier