reveal many of the qualitative characteristics of
quantum mechanical (*QM*) systems.

* THINK FIRST*!

The first step in any problem is to gather together all your
* qualitative* knowledge about the situation
before you start working out any quantitative details.
This is especially true in * QM*,
where the "blind calculation" approach is often not only
a waste of effort but actually intractable!

* SYMMETRY*:

In this case we save ourselves a mind-bogglingly difficult
mathematical nightmare by making a few simple observations
about ** symmetry**: the potential
is symmetric
about

* BOUND or UNBOUND*?

A * potential well* generally has bound states
of well-defined energy * E * < 0 unless the mass is too small
(see below).
There is also a continuum of * unbound* states with * E * > 0,
whose behaviour we may also want to examine.
[For instance, the classical particle will "pass over"
the well and continue on the other side every time;
this will not be the case for the * QM * result!]
We will start with the * bound* state
.
If
is localized around the potential well,
then to be * normalizable* it must obey
.
[Moreover, we certainly expect
as we get further into regions where the
* classical* particle cannot penetrate * at all*
due to its inadequate energy!]
Our first guess for such functions
is always the * decaying exponential* function, here
(2)
for region * 1 * .
[Remember, * x * = -|*x*| in this region.]
Since we are always free to choose the * overall phase*
of
arbitrarily [multiplying by a constant factor
of
has no effect on the physics],
we may do so immediately by choosing that
.
In this case the
symmetry requirement
gives two possible solutions for region * 3 *
:
where the **+** sign is for the symmetric (** even**) solution
and the

On region * 2 *
we expect some sort of oscillatory function,
for which the obvious choices are
(3)
where the **-** sign is chosen for the (** odd**)
function
for the following reason:

* BOUNDARY CONDITIONS*:

We must always satisfy the matching condition
for the wavefunction [
must be * continuous*]
and the matching condition for its spatial derivative
[
must also be continuous (except where
the potential is infinite)]
at the boundaries
and .
The first matching condition immediately implies that
must be * positive* at ,
since we chose positive.
For the odd function this mandates the negative sign
in Eq. (3) above.
If we now explicitly apply the matching condition
for at the boundaries, we get
(4)
where
is the magnitude of
* at the boundary*.
The matching condition for the derivative gives
(5).
Dividing Eq. (5) by Eq. (4) gives
(6).
Adding the squares of Eqs. (4) and (5) gives
(7).

* APPLYING the SCHRÖDINGER EQUATION*:

How do these guesses fare with our original equation?
On regions * 1* and * 3* we get
,
which ensures
(8)
since * E * < 0 is the same throughout.
On region * 2* we have
or
(9).
Substituting these values for
and * k *
back into Eqs. (7) and (6) gives
(10)
and (11),
respectively.

Equation (11) implies restrictions on the allowed values of * E*.
This is anticipated [remember, * E * is * quantized*]
but you may be surprised to find a * transcendental*
equation governing * E*. * There is no algebraic solution*
to Eq. (11)! [Actually, Eq. (11) is * two* transcendental
equations for
.
As * n * and increase,
we alternate between ** even** and

* ARE THERE ANY BOUND STATES*?

Under what circumstances does Eq. (11) * have* a solution?
On the one hand, the general rule that * confinement costs
energy* would seem to dictate that a narrower well must be
deeper in order to "hang on to" a particle:
smaller * L * (or * m*)
should require larger .
The definition of
reflects this aspect of the problem:
is smaller
for smaller
and/or smaller .
However, while
, so that
the two must intersect somewhere, no matter how small
is.
Thus * there is always at least one
(even) bound state* for this potential!
The apparent paradox is resolved when we realize that
the exponentially decaying "tails" of
penetrate deeper and deeper into the classically forbidden region
as
gets smaller and smaller, until the region
where
is sinusoidal (inside the well) becomes a
negligible point at the centre of a wavefunction
that decays away exponentially from a centre cusp!
The stability of this solution
is extremely sensitive to the "flatness" of * V * = 0 in regions
far from the well, for obvious reasons.

An interesting limit is obtained by allowing the well
to shrink
while holding constant the ratio
(so that in compensation).
This is called the ** delta function potential**.

* IS THAT THE WHOLE STORY*??

After all this work, what have we learned?
We know
everywhere to within a normalization constant * A *
which can be found by applying
if we need it
( if we want to calculate expectation values)
and we have what we need to find
(and therefore
and ) for the
stationary states allowed in this potential well.
It would be nice to have a tidy algebraic solution,
but even simple problems are not necessarily nice!
If you want exact solutions, you will have to solve the
transcendental equations each time you specify the
parameters ,
* L * and * m * which govern the
physics of this problem. However, it is possible to
make many qualitative obervations (see class notes)
based on simple right-hemisphere graphical arguments.
We may also examine some ** limiting cases**:

* DEEPLY BOUND STATES*:

If ,
the lowest few eigenstates will have energies
that are only a little above the bottom of the well
[,
where ].
Then Eq. (11) reads approximately
or, even more approximately,
.
You may want to play with this approximation to see
the spectrum of deeply-bound states.
Note that ,
so in the limit of the ** infinite square well**
the solutions are simply
,
which is satisfied for .
Check that this agrees with the formula you know already,
bearing in mind that here we have defined the

* UNBOUND STATES*:

In all the equations above we have assumed * E * < 0 (bound states).
** What happens when E > 0**?
Taking the equations at face value,
we would conclude that
is

Jess H. Brewer - Last modified: Mon Nov 23 17:39:04 PST 2015