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A 1-Dimensional Box


 
Figure: - First three allowed modes of a standing wave confined to a 1-dimensional box.
\begin{figure}
\begin{center}\epsfysize 1.4in
\epsfbox{PS/st_waves.ps}\end{center}
\end{figure}

Suppose an electron is confined somehow to a ``1-dimensional box'' (like a bead on a wire). Actually there are many examples of such systems; a DNA molecule is an interesting example. The ``box'' (or string, or however you want to think of it) has a length $\ell$. If the electron is truly confined to the box, then its ``wave'' must have nodes (zeroes) at the ends of the box -- and be zero everywhere outside the box. This is the familiar condition defining the allowed ``modes'' of vibrations in a string or in a closed organ pipe:

 \begin{displaymath}\lambda_n \; = \; {2 \ell \over n}
\end{displaymath} (24.2)

where n is any nonzero integer.

If we put this together with de Broglie's formula (1), we get an equation for the momentum of the electron in it's $n^{\rm th}$ mode:

 \begin{displaymath}p_n \; = \; {n h \over 2 \ell}
\end{displaymath} (24.3)

and if we recall that the kinetic energy associated with a particle of mass m having momentum p is given by

 \begin{displaymath}E \; = \; {p^2 \over 2m}
\end{displaymath} (24.4)

then we have the energy of the electron in its $n^{\rm th}$ mode:

 \begin{displaymath}E_n \; = \; {n^2 h^2 \over 8 m \ell^2} .
\end{displaymath} (24.5)

The electron not only has discrete ``energy levels'' but it has an irreducible minimum energy for the lowest possible state (the `` GROUND STATE''):

 \begin{displaymath}E_1 \; = \; {h^2 \over 8 m \ell^2} .
\end{displaymath} (24.6)

The smaller the box, the bigger the ground state energy. Particles don't ``like'' to be confined! This has a number of profound consequences which we will revisit shortly. But first let's do a little trick and turn our string into a circle . . . .


next up previous
Next: Fudging The Bohr Atom Up: Particle in a Box Previous: Particle in a Box
Jess H. Brewer
2000-01-17