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

Note again the huge shift between classical mechanics and
quantum mechanics: what was the
dependent variable in classical mechanics (*x*) has become the independent variable;
what was the independent variable (*t*) has disappeared.
Later we'll see how motion is possible if time plays no role.
It turns out energy (*E*) is more measurable than motion,
hence energy has replaced time as the most important variable.

Almost any "real" problem in classical mechanics (e.g., a real cannon ball flying through the Earth's atmosphere) is so complex that it cannot be solved using paper-and-pencil methods: the differential equation must be solved "numerically" using a computer. Quantum mechanics only makes things more difficult. Only the simplest examples have "analytical" solutions (i.e., the answer can be written down using "well known" functions). The purpose of these textbook problems is to provide examples where all the dependencies are explicit so that they are available for play/understanding.

If a problem has an analytical solution, say,

=*f(x)*

where *f(x)* is some "well-known" function, consider the
consequences. Just about "every-known" function requires
that its argument be dimensionless. (I can think of only two
counterexamples.) Thus =sin(*x*),
or =exp(-*x ^{2}*) cannot be
proper answers if

=*f(x/L)*

where *L* "sets the scale" of the problem, i.e.,
has units that cancel those of *x*. Thus the first
point-of-attack on any physics problem is to determine
how you could form quantities like *L* that allow
you to form dimensionless quantities like *x/L*.

In the classical mechanics problem, we have "givens" of
*g*, *v*_{0}, and *z*_{0}.
From these one can form two scales for both time and length:
time scales:
*v*_{0}/*g* & *z*_{0}/*v*_{0}
and length scales:
*z*_{0} & *v*_{0}^{2}/*g*.
Thus the classical problem is actually a generally more difficult
"two-scale" problem that in this case has an easy solution: a polynomial.
(If a multi-scale problem can't be solved with a polynomial, then
its probably time to give up on a nice solution. Such problems
at least have a dimensionless variable to classify types
of solutions. In this case:
*D*=(*z*_{0}*g*)/(*v*_{0}^{2}) divides solutions
that are initially mostly potential energy
from solutions that are initially mostly kinetic energy
[*D*=½ is the dividing line].)

In the quantum mechanical problem we lose the "givens"
*v*_{0} and *z*_{0} (there is
no *single* position or velocity), and we gain a quantity
with . In particular, Schrödinger's
equation has constants: ^{2}/2*m*=*A*
and *mg*=*B*. *B* has the units of force.
*A* has the units of
energy·length^{2}=force·length^{3}.
Thus *l*=(*A/B*)^{1/3} has units of length
and *e*=*B·l*=(*AB ^{2}*)

particle | force | length scale | energy scale |
---|---|---|---|

He atom | mg | 2.3 µm | .95 peV |

BuckyBall (C60) | mg | 73 nm | 5.4 peV |

Planck Mass | mg | ~1 am | ~1 µeV |

electron | 10^{6} V/m electric field | 3.4 nm | 3.4 meV |

(Note p=pico=10^{-12}, a=atto=10^{-18}.) A am is about
the smallest distance measured, about 1/1000 the size of a proton.
A peV is a very tiny amount of energy. The thermal energy at 0.01°K above
absolute zero (about as cold as we can make bulk matter) is about a million times
larger than a peV. A peV photon would have a frequency of a couple of hundred
Hz. In short, its hard to imagine how we could measure quantum mechanical
"falling" -- however in 2001 it was done by a group in Grenoble, France
using falling neutrons. An easier approach is to use a constant electric field
which produces a constant
force on an electron much as gravity (near the Earth's surface)
produces a constant force on an object with mass. Thus an electron
in a large uniform electric field has the same physics as a falling
object. The "electrically falling" electron has measurable length
and energy scales.