Our aim is to solve the time independent Schrödinger's equation to find the probability amplitude (i.e., the wavefunction) as a function of position.

where *U*(*x*) consists of linear combinations
of Dirac delta functions with singular points at various locations.
((*x-a*) has its singular point
at *x*=*a*.)

As usual our first step is to look at units and seek a dimensionless version of Schrödinger's equation. We immediately find an anomaly: is a function which has units (if its argument has units). Look at the fundamental properties of :

*f*, of course, has the same units on both sides of the
equation, so the units of must cancel the
units of *dx*...i.e., the units of
are the reciprocal of the units of its argument.

is unitless (0 or 1) and so the units
of ' = *d/dx*
come just from the *dx* in the denominator. Again
the units of
are the reciprocal of the units of its argument.

Thus if we make a potential energy:
*U*(*x*)=*w*(*x-a*),
has the units of 1/length (since *x*
has units of length) so *w* must have units of energy·length.

^{2}/*m*=*A* has the
units of energy·length^{2}.

Thus we can make our length scale *L*=*A/w*, and our
energy scale *e*=*w*^{2}/*A*.

The algebra of this conversion can be a little tricky we note:

(*x*)= (*x'·L*)=
(1/*L*) (*x'*)

So our dimensionless version of Schrödinger's equation becomes:

In the following pages we will drop all primes, so beware that on the following
pages *x'* is called simply *x*.

Note that our strength of potential *w* has completely disappeared.
It is, of course, still present in our length and energy scale factors. Thus
shrinking the dimensionless location of all delta function singularities corresponds to
fixed real-space positions with a reduced potential.

Note that we will mostly be considering *attractive* delta function
potentials so our Schrödinger's equation will have -
rather than the + shown above.