The Quantum Mechanical Particle in a Box

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

Y(x, t) = S c_n y_n(x)exp(-i E_n t / h)

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(nπx/a) and E_n = h²/(2m) (n π/a)²

Since all of the energies E_n are multiples of E₁, any state has period

T = 2πh/E₁ = 4 m a²/(hπ)

For a 1 gram particle in a 1 cm box, this period is about 1.2*10²⁷sec, or about 4*10¹⁹ 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 y₁(x)exp(-i E₁ t / h) + b y₂(x)exp(-i E₂ t / h) + c y₃(x)exp(-i E₃ t / h) ]/sqrt(a² + b² + c²)

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.