When this differential equation is solved without the previously used boundary conditions
(2.29)
the new solutions represent a particle–wave that travels along the positive or negative x‐direction. The most general solution of the differential Eq. (2.23) is
where b is a constant.
The second derivative of Eq. (2.48) is given by
(2.49)
with
or
Equation (2.51) was obtained by substituting
(2.52)
into Eq. (2.50). Thus, the unbound particle can be described by a traveling wave (as opposed to a standing wave)
(2.53)
carrying a momentum
(2.54)
into the positive or negative x‐direction. k is the wave vector defined in Eq. (1.6).
2.4.3 The Particle in a Box with Finite Energy Barriers
Finally, the particle in a box with a finite energy barrier, V0, will be discussed qualitatively. This is a situation where the particle is no longer strictly forbidden outside the confinement box and leads to the concept of tunneling, that is, the probability of the electron found outside the box. The shape of the potential function is shown in Figure 2.5b.
The potential energy for this case is written as
(2.55)
and
(2.56)
(Notice that the boundaries of the box were shifted from 0 to L to −L/2 to +L/2 for symmetry reasons that will be taken up again in Section 3.2.) The Schrödinger equation is written in two parts: Inside the box, where the potential energy is zero, the same equation holds that was used earlier:
(2.23)
Outside the box, the Schrödinger equation is
(2.57)
Figure 2.5 (a) Particle in a box with infinite potential energy barrier. (b) Particle in a box with infinite potential energy barrier.
The solutions of this equation will be of the form
(2.58)
(2.59)
where
(2.60)
For