j/An electron in a gentle
electric field gradually shortens
its wavelength as it gains energy.
(As discussed on p. 894, it is
actually not quite correct to graph
the wavefunction of an electron
as a real number unless it is a
standing wave, which isn’t the
case here.)k/The wavefunction’s tails
go where classical physics says
they shouldn’t.its charge is negative, it loses electrical energy by moving to a higher
voltage, so its kinetic energy increases. As its electrical energy goes
down, its kinetic energy goes up by an equal amount, keeping the
total energy constant. Increasing kinetic energy implies a growing
momentum, and therefore a shortening wavelength, j.
The wavefunction as a whole does not have a single well-defined
wavelength, but the wave changes so gradually that if you only look
at a small part of it you can still pick out a wavelength and relate
it to the momentum and energy. (The picture actually exagger-
ates by many orders of magnitude the rate at which the wavelength
changes.)
But what if the electric field was stronger? The electric field in
an old-fashioned vacuum tube TV screen is only∼ 105 N/C, but the
electric field within an atom is more like 10^12 N/C. In figure l, the
wavelength changes so rapidly that there is nothing that looks like
a sine wave at all. We could get a rough idea of the wavelength in a
given region by measuring the distance between two peaks, but that
would only be a rough approximation. Suppose we want to know
the wavelength at pointP. The trick is to construct a sine wave, like
the one shown with the dashed line, which matches the curvature of
the actual wavefunction as closely as possible nearP. The sine wave
that matches as well as possible is called the “osculating” curve, from
a Latin word meaning “to kiss.” The wavelength of the osculating
curve is the wavelength that will relate correctly to conservation of
energy.
l/A typical wavefunction of an electron in an atom (heavy curve)
and the osculating sine wave (dashed curve) that matches its curvature
at point P.Tunneling
We implicitly assumed that the particle-in-a-box wavefunction
would cut off abruptly at the sides of the box, k/1, but that would be
unphysical. A kink has infinite curvature, and curvature is related
to energy, so it can’t be infinite. A physically realistic wavefunction
must always “tail off” gradually, k/2. In classical physics, a particle
can never enter a region in which its interaction energyUwould be
greater than the amount of energy it has available. But in quantum
Section 13.3 Matter as a wave 903