Waves and the Schroedinger Equation
5 april 2010
1
The Wave Equation
We have seen from previous discussions that the waveparticle duality of
matter requires we describe entities through some waveform based represe
nation. The most natural consideration are classical waves, and arriving at
a way to describe their
spatial
and
temporal
evolution. In the following
discussion, pursuing this description for quantum entities will lead us to the
Schroedinger equation, our starting point for treating atomic and molecular
systems.
The motion of classical, nondispersive waves requires some definitions:
frequency
=
1
T
=
ν
wavelength
=
λ
velocity
=
v
=
λν
A general expression for a wave moving in the +x direction:
ψ
(
x, t
) =
A sin
2
π
x
λ

t
T
ψ
(
x, t
) =
A sin
2
πx
λ

2
πt
T
ψ
(
x, t
) =
A sin
(
kx

ωt
)
•
wavevector,
k
k
=

k

=
2
π
λ
wave vector units of inverse length (
1
length
)
Recall:
λ
=
h
p
(1)
1
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•
Angular frequency (radians/second)
ω
= 2
πν
Note that a phaseshift can be introduced in order to change the origin of
the waveform:
ψ
(
x, t
) =
A sin
(
kx

ωt
+
φ
)
Now, let’s consider
stationary, or standing, waves
; for such entities,
the nodes remain fixed in time and space (though the wave is moving with
velocity, v).
Standing waves are generated from interference of waves of
equal frequency and amplitude traveling in
opposite directions
.
ψ
(
x, t
) =
A sin
(
kx

ωt
) +
A sin
(
kx
+
ωt
)
ψ
(
x, t
) = 2
Asin
(
kx
)
cos
(
ωt
)
ψ
(
x, t
) =
ψ
(
x
)
cos
(
ωt
)
(2)
ψ
(
x, t
) =
ψ
(
x
)

{z
}
time

independent
cos
(
ωt
)

{z
}
time

dependent
Thus, from the last expression, we see that stationary waves have fixed nodal
points; zero amplitude versus time at fixed points). Now, we have gone about
things in a reverse manner, but we can consider the following.
We have
written a representation of a waveparticle entity as a sinusoidal function.
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 Spring '08
 Dybowski,C
 Physical chemistry, pH, Eigenfunctions, Schroedinger equation

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