For a box of length *a*, the wave functions which satisfies the Schrödinger equation is:

Y(*x, t*) = S *c _{n}* y

where the sum goes from 1 to infinity, *h* is ``h bar,'' for which I can't find the markup code, and

y_{n}(*x*) = sqrt(2/a) sin(*n*π*x/a*) and E_{n} = *h*^{2}/(2*m*) (*n* π/*a*)^{2}

Since all of the energies E_{n} are multiples of E_{1}, any state has period

T = 2πh/E_{1} = 4 *m a*^{2}/(*h*π)

For a 1 gram particle in a 1 cm box, this period is about 1.2*10^{27}sec, or about 4*10^{19} years. On the other hand, for an electron in a box 10^{-10} meters across, the period is about 10^{-16} seconds.

The figures show the probability density, |Y(*x, t*)|^{2}, as a function of time for a particle in a superposition of the first three modes:

Y(*x, t*) = [
a y_{1}(*x*)exp(-i E_{1} *t / h*) +
b y_{2}(*x*)exp(-i E_{2} *t / h*) +
c y_{3}(*x*)exp(-i E_{3} *t / h*) ]/sqrt(a^{2} + b^{2} + c^{2})

The wave function is normalized for any choice of a, b, and c.
The dotted line shows the average probability density, 1/*a*.

First mode (stationary state).

Second mode (stationary state).

Initial state in left.

Symmetric initial state.

In all of these, you can click on ``scrollbar'' to change a, b, c, and d. The parameters a, b, and c are the amplitudes of the first three modes. The parameter d determines how fast time advances. (The period T is scaled to 2π/d seconds, with the default value of d=1/2.) The ``clock'' in the upper right goes around once for each period.

These graphs were produced by
DPGraph,
which is a ** fast** program for
viewing time-dependent graphs.

e-mail: Jim.Swift@nau.edu