Elements of Modern Physics
78
∆p x ≈
w p
d
(3.67)
where w is the width of the central fringe. This uncertainty results from the fact
that the particle may come to any point within this fringe. Since w =
d
a
λ , and
p = h/λ by the de Broglie relation, we get
(∆x) (∆p x ) ≈ h
(3.68)
which is the same as Eq. (3.63) in order of magnitude. This demonstration
brings out the fact that the Heisenberg uncertainty principle is essentially a
consequence of associating wave properties with the particles.
3.7 STEP POTENTIAL
As the first example of nonzero potentials, the one-dimensional problem of a
article which comes across a sudden change in the potential is considered. The
potential may be approximated by
V(x) = 0
for x < 0
= V
for x ≥ 0
(3.69)
as shown in Fig. 3.2(a).
Fig. 3.2 (a) A step potential, (b) A potential barrier.
This Schrödinger equation in one dimension, is
( , )
x t
i
t
∂ψ
∂
=
2
2
2
( )
( , )
2
V x
x t
m x


∂
−
+
ψ


∂


(3.70)
As before, for states with energy E,
y (x, t) = exp (
/ ) ( )
iEt
x
−
φ
(3.71)
where φ (x) satisfies the equation [see Eq. (3.24)]