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A 1-Dimensional Box

 

Figure: - First three allowed modes of a standing wave confined to a 1-dimensional box.

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:

$$\lambda_n \; = \; {2 \ell \over n}$$ (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:

$$p_n \; = \; {n h \over 2 \ell}$$ (24.3)

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

$$E \; = \; {p^2 \over 2m}$$ (24.4)

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

$$E_n \; = \; {n^2 h^2 \over 8 m \ell^2} .$$ (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"):

$$E_1 \; = \; {h^2 \over 8 m \ell^2} .$$ (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 . . . .