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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}](img7.gif) |
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
.
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}](img9.gif) |
(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
mode:
![\begin{displaymath}p_n \; = \; {n h \over 2 \ell}
\end{displaymath}](img11.gif) |
(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}](img12.gif) |
(24.4) |
then we have the energy of the electron in its
mode:
![\begin{displaymath}E_n \; = \; {n^2 h^2 \over 8 m \ell^2} .
\end{displaymath}](img13.gif) |
(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}](img14.gif) |
(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: Fudging The Bohr Atom
Up: Particle in a Box
Previous: Particle in a Box
Jess H. Brewer -
Last modified: Wed Nov 18 16:47:45 PST 2015