function is considered to have zero concavity over all.
A classically allowed region with constantU example 21
In a classically allowed region with constantU, we expect the
solutions to the Schrodinger equation to be sine waves. A sine ̈
wave in three dimensions has the formΨ= sin(
kxx+kyy+kzz)
.
When we compute∂^2 Ψ/∂x^2 , double differentiation of sin gives
−sin, and the chain rule brings out a factor ofkx^2. Applying all
three second derivative operators, we get∇^2 Ψ=(
−kx^2 −ky^2 −kz^2)
sin(
kxx+kyy+kzz)
=−
(
kx^2 +ky^2 +kz^2)
Ψ.
The Schrodinger equation gives ̈E·Ψ=−
~^2
2 m∇^2 Ψ+U·Ψ
=−
~^2
2 m·−
(
kx^2 +ky^2 +kz^2)
Ψ+U·Ψ
E−U=
~^2
2 m(
kx^2 +ky^2 +kz^2)
,
which can be satisfied since we’re in a classically allowed region
withE−U>0, and the right-hand side is manifestly positive.Use of complex numbers
In a classically forbidden region, a particle’s total energy,U+K,
is less than itsU, so itsKmust be negative. If we want to keep be-
lieving in the equationK=p^2 / 2 m, then apparently the momentum
of the particle is the square root of a negative number. This is a
symptom of the fact that the Schr ̈odinger equation fails to describe
all of nature unless the wavefunction and various other quantities
are allowed to be complex numbers. In particular it is not possible
to describe traveling waves correctly without using complex wave-
functions. Complex numbers were reviewed in subsection 10.5.5,
p. 625.
This may seem like nonsense, since real numbers are the only
ones that are, well, real! Quantum mechanics can always be re-
lated to the real world, however, because its structure is such that
the results of measurements always come out to be real numbers.
For example, we may describe an electron as having non-real mo-
mentum in classically forbidden regions, but its average momentum
will always come out to be real (the imaginary parts average out to
zero), and it can never transfer a non-real quantity of momentum
to another particle.
Section 13.3 Matter as a wave 911