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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