Square Potential Barrier

(313) |

where is given by Eq. (305).

Let us adopt the following solution
of the above equation to the left of the barrier (*i.e.*, ):

(315) |

(316) |

Let us adopt the following solution to Eq. (314) to the right
of the barrier (*i.e.* ):

(317) |

(318) |

In other words, the probabilities of reflection and transmission sum to unity, as must be the case, since reflection and transmission are the only possible outcomes for a particle incident on the barrier.

Inside the barrier (*i.e.*,
), satisfies

(321) |

Let us, first of all, consider the case where . In this case, the general
solution to Eq. (320) inside the barrier takes the
form

(322) |

Now, the boundary conditions at the edges of the barrier (*i.e.*, at
and ) are that and are both
continuous. These boundary conditions ensure that the probability current
(155) remains finite and continuous across the edges of the boundary, as must be the
case if it is to be a spatial constant.

Continuity of and at the left edge of
the barrier (*i.e.*, ) yields

(323) | |||

(324) |

Likewise, continuity of and at the right edge of the barrier (

(325) | |||

(326) |

After considerable algebra, the above four equations yield

and

Note that the above two expression satisfy the constraint (319).

It is instructive to compare the quantum mechanical probabilities
of reflection and transmission--(327) and (328), respectively--with those derived from classical physics. Now, according
to classical physics, if a particle of energy is incident on
a potential barrier of height then the particle slows down
as it passes through the barrier, but is otherwise unaffected.
In other words, the classical probability of reflection is
*zero*, and the classical probability of transmission is *unity*.

The reflection and transmission probabilities obtained from Eqs. (327) and (328), respectively, are plotted in Figs. 10 and
11. It can be seen, from Fig. 10, that the classical
result, and , is obtained in the limit where the height of the barrier
is relatively small (*i.e.*, ). However, when is
of order , there is a substantial probability that the incident particle
will be *reflected* by the barrier. According to classical physics, reflection is impossible when .

It can also be seen, from Fig. 11,
that at certain barrier widths the probability of reflection goes to *zero*. It turns out that this is true irrespective of the energy of the incident particle.
It is evident, from Eq. (327), that these special barrier widths
correspond to

(329) |

Let us, now, consider the case . In this case, the general
solution to Eq. (320) inside the barrier takes the
form

(330) |

(331) | |||

(332) |

Likewise, continuity of and at the right edge of the barrier (

(333) | |||

(334) |

After considerable algebra, the above four equations yield

and

These expressions can also be obtained from Eqs. (327) and (328) by making the substitution . Note that Eqs. (335) and (336) satisfy the constraint (319).

It is again instructive to compare the quantum mechanical probabilities
of reflection and transmission--(335) and (336), respectively--with those derived from classical physics. Now, according
to classical physics, if a particle of energy is incident on
a potential barrier of height then the particle is reflected.
In other words, the classical probability of reflection is
*unity*, and the classical probability of transmission is *zero*.

The reflection and transmission probabilities obtained from Eqs. (335) and (336), respectively, are plotted in Figs. 12 and
13. It can be seen, from Fig. 12, that the classical
result, and , is obtained for relatively
thin barriers (*i.e.*, ) in the limit where the height of the barrier
is relatively large (*i.e.*, ). However, when is
of order , there is a substantial probability that the incident particle
will be *transmitted* by the barrier. According to classical physics, transmission is impossible when .

It can also be seen, from
Fig. 13, that the transmission probability decays *exponentially*
as the width of the barrier increases. Nevertheless, even for very
wide barriers (*i.e.*, ), there is a small but *finite*
probability that a particle incident on the barrier will be
*transmitted*. This phenomenon, which is inexplicable within
the context of classical physics, is called *tunneling*.