where .

Now, if then the particle is unbounded. Thus, when the particle encounters the well
it is either reflected or transmitted. As is easily demonstrated, the reflection and transmission
probabilities are given by Eqs. (327) and (328), respectively,
where

(373) | |||

(374) |

Suppose, however, that . In this case, the particle
is bounded (*i.e.*,
as
).
Is is possible to find bounded solutions of Schrödinger's equation
in the finite square potential well (372)?

Now, it is easily seen that independent solutions of Schrödinger's equation (301)
in the symmetric [*i.e.*, ] potential (372)
must be either totally symmetric [*i.e.*,
], or
totally anti-symmetric [*i.e.*,
]. Moreover,
the solutions must satisfy the boundary condition

(375) |

Let us, first of all, search for a totally symmetric solution.
In the region to the left of the well (*i.e.* ), the
solution of Schrödinger's equation which satisfies the
boundary condition
and
is

(376) |

(377) |

(378) |

(379) |

(380) |

Let . It follows that

(382) |

(383) |

with

(385) |

Now, the solutions to Eq. (384) correspond to the
intersection of the curve
with the curve
. Figure 16 shows these two curves plotted for
a particular value of . In this case, the curves intersect
twice, indicating the existence of two totally symmetric bound states in the well.
Moreover, it is evident, from the figure, that as increases (*i.e.*, as the well becomes
deeper) there are more and more bound states. However, it is also evident that there is
always at least one totally symmetric bound state, no matter how small
becomes (*i.e.*, no matter how shallow the well becomes). In the limit
(*i.e.*, the limit in which the well becomes very deep), the
solutions to Eq. (384) asymptote to the roots of
.
This gives
, where is a positive integer, or

(386) |

For the case of a totally anti-symmetric bound state, similar analysis to the
above yields

(388) |