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THE BARRIER
POTENTIALChapter 6.5
ROBERT EISBERG
Quantum Mechanics:
&
Chapter 8.3
Ajoy Ghatak
and S. Lokanathan
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PYL100 Electromagnetic Waves and Quantum Mechanics
THE BARRIER POTENTIAL
Classical Mechanics
A particle of total energy E in the region x < 0,
• Particle will have probability of reflected, If total energy E < V0• Particle will have probability of being transmitted to region x>a, If total
energy E > V0
Even if E is smaller than V 0, quantum mechanics predicts that there
is a certain probability that the particle will be transmitted through
the barrier into the region x > a.
Quantum Mechanics
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PYL100 Electromagnetic Waves and Quantum Mechanics
Case-I E
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PYL100 Electromagnetic Waves and Quantum Mechanics
Boundary conditions
Boundary Condition: matching and
at the points = 0 and
=
At = 0
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PYL100 Electromagnetic Waves and Quantum Mechanics
Boundary conditions
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Reflection and Transmission coefficient
Where
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Reflection and Transmission coefficient
Where
=>Very remarkable from the point of view of classical mechanics.
=>T is vanishingly small in the classical limit because quantity α, which
is a measure of the opacity of the barrier, is extremely large.
For ≫ 1
Very small
but non-zero
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Case-II E > V0
= + − where
Thus we have to replace by , everywhere
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PYL100 Electromagnetic Waves and Quantum Mechanics
Reflection and transmission coefficients = 10 = 3
Thus whenever the barrier width is a multiple of /2, perfect
transmission occurs.
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PYL100 Electromagnetic Waves and Quantum Mechanics
Application of Quantun Tunneling: Alpha
Decay
Alpha decay is spontaneous emission of 4 from anunstable nucleus
Two forces acting in nucleus:
Coulomb force: Protons really close together, so very big
repulsion from coulomb force
Nuclear force: (attraction between nuclear particles is
very strong if very close together) - called the STRONGForce.
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PYL100 Electromagnetic Waves and Quantum Mechanics
Alpha decay: Quantum tunneling1928 Gamow, Condon, and Gurney
The probability persecond that a radioactive
nucleus will emit an α
particle of
energy E
barrier penetration theory
