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University Physics Notes: Quantum Mechanics – Quantum Tunnelling (1)
The particle has total energy_html_mfcd8365.gif){kind=small}
and
is incident on the barrier_html_3c9cf313.gif){kind=small}
from
the left. The Schrodinger equation in each of regions I, II and III
takes the form_html_d06c824.gif){kind=small}
In region I_html_m571cef2c.gif){kind=small}
so_html_197959db.gif){kind=small}
The solution is_html_m7d672a78.gif){kind=small}
where
_html_m13cdc950.gif){kind=small}
The_html_m294b8089.gif){kind=small}
term
represents motion to the right and the_html_m32ff94fa.gif){kind=small}
term
represents motion to the left – indicating reflection of the
wavefunction from the barrier.
In region II the particle has total energy_html_3e949a10.gif){kind=small}
and
The solution is_html_27d3822f.gif){kind=small}
where_html_555866e0.gif){kind=small}
ere
Since_html_m108d7cd5.gif){kind=small}
this
represents exponential decay of the wavefunction in this region.
In region IIIso
The solution is_html_m5af6f9f6.gif){kind=small}
where
_html_779ea015.gif){kind=small}
is
a measure of finding the partiacle at a point and since this value is
greater that 0 in region III there is a non zero probability of
finding the particle in region III, which is forbidden by classical
mechanics since it has insufficient energy to surmount the barrier.