Page 239 - Instant notes

The wave nature of matter 225

A solution to this equation is (see section on Schrödinger equation):

In fact, the general solution is:

where any value of E and a forms a suitable wavefunction. However, because the particle
is confined to a box of finite length, the walls impose boundary conditions on which
wavefunctions are physically allowable. Since the potential energy rises to infinity at the
walls the probability of finding the particle outside the box is zero, so the wavefunction at
the walls of the box, and everywhere outside the box, must be zero. Therefore, all
acceptable wavefunctions for the particle must fit exactly inside the box, like the
vibrations of a string fixed at both ends. To satisfy this condition requires that the
wavelength, λ, of allowed wavefunctions must be one of the values:

or, more concisely, that:

The relationship between λ and the mathematical description of a sine wave is

sin(2πx/λ) so the wavelength of the wavefunction is:

Therefore, allowed wavefunctions of the particle in a box must satisfy:

which, on rearranging gives: