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
, *v*_{0}, and *x*_{0}.
From these one can form two scales for both time and length:
time scales:
1/ & *x*_{0}/*v*_{0}
and length scales:
*x*_{0} & *v*_{0}/.
Thus the classical problem is actually a generally more difficult
"two-scale" problem that in this case has an easy solution because
one time scale enters in only in the dimensionless ratio:
*x*_{0}/*v*_{0} = tan .
Thus we end up with simple one-scale dependencies like:
*t*+.
Similarly there is really only one length scale in the solution:
*A* which is just the square root of the sum of the squares of the
above two length scales.

In the quantum mechanical problem we loose the "givens"
*v*_{0} and *x*_{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 *½k*=*B*. *B* has the units of force/length.
*A* has the units of
energy·length^{2}=force·length^{3}.
Thus *l*=(*A/B*)^{1/4} has units of length
and *e*=*B·l ^{2}*=(

particle | frequency | length scale | energy scale |
---|---|---|---|

H_{2} vibration | 1.3×10^{14} Hz | .01 nm | .23 eV |

O_{2} vibration | 5×10^{13} Hz | .005 nm | .10 eV |

CCl_{4} symmetric stretch | 1.4×10^{13} Hz | .002 nm | .03 eV |

simple nuclear model | 2×10^{20} Hz | 7 fm | .4 MeV |

simple meson (bb) model | 1×10^{23} Hz | .2 fm | .2 GeV |

(Note f=femto=10^{-15}.) Molecules typically vibrate a small
fraction of their interatomic separation. Nuclear vibrations
are often modeled with a liquid drop model---a lot of the nuclear mass
is not moving, instead we have something more like a surface wave.
Room temperature thermal energy (~ .03 eV) is comparable to (often a bit smaller than)
molecular vibration
energy.