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Chapter
4
Schroedinger
Equation
Einstein’s
relation
between
particle
energy
and
frequency
Eq.(3.83)
and
de
Broglie’s
relation
between
particle
momentum
and
wave
number
of
a
corre
sponding
matter
Eq.(3.84)
suggest
a
equation
for
matter
waves.
This
search
for
an
equation
describing
matter
waves
was
carried
out
by
Erwin
Schroedinger.
He
successful
in
the
year
1926.
The
energy
of
a
classical,
nonrelativistic
particle
with
momentum
p±
that
is
subject
to
a
conservative
force
derived
from
a
potential
V
(
±
r
)
is
2
E
=
+
V
(
±
r
)
.
(4.1)
2
m
For
simplicity
lets
begin
ﬁ
rst
with
a
constant
potential
V
(
±
r
)=
V
0
=
const.
This
is
the
force
free
case.
According
to
Einstein
and
de
Broglie,
the
dis
persion
relation
between
ω
and
±
k
for
describing
the
particle
motion
should
be
~
2
±
k
2
~
ω
=
+
V
0
.
(4.2)
2
m
Note,
so
far
we
had
a
dispersion
relation
for
in
one
dimension,
where
the
wavenumber
k
(
ω
)
,
a
function
of
frequency.
in
three
dimen
sions
the
frequency
of
the
is
rather
a
function
of
the
three
components
o
f
thewavevector
.
Eachwavew
ithag
ivenwavevector
±
k
has
the
following
time
dependence
e
j
(
±
k
·
±
r
−
ωt
)
,
with
ω
=
~
±
k
2
+
V
0
(4.3)
2
m
~
Note,
thisisawave
with
phasefrontstravelingto
the
right.
In
contrast
to
our
notation
used
in
chapter
2
for
rightward
traveling
electromagnetic
waves,
199
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CHAPTER
4.
SCHROEDINGER
EQUATION
sw
itched
thes
igninthe
exponent
.
Th
isnotat
ioncon
formsw
iththe
phys
ics
oriented
literature.
A
superposition
of
such
waves
in
±
k
−
space
enables
us
to
construct
wave
packets
in
real
space
Z
³
´
Ψ
(
±
r,
t
)=
φ
ω
±
k,ω
e
j
(
±
k
·
±
r
−
ωt
)
d
3
kdω
(4.4)
The
inverse
transform
of
the
above
expression
is
³
´
Z
±
φ
ω
±
k,
ω
=
1
4
Ψ
(
±
t
)
e
−
j
(
±
k
·
r
−
ωt
)
d
3
rd
t
,
(4.5)
(2
π
)
with
Ã
!
³
´
~
±
k
2
V
0
φ
ω
±
ω
=
φ
(
k
)
δ
ω
−
2
m
−
~
.
(4.6)
Or
we
can
rewrite
the
function
in
Eq.(4.4)
by
carrying
out
the
trivial
frequency
integration
over
ω
Ã "
Ã
! #!
Z
Ψ
(
±
t
φ
(
k
)exp
j
±
k
·
±
r
−
~
2
±
k
m
2
+
V
~
0
t
d
3
k.
(4.7)
Due
to
the
Fourier
relationship
between
the
function
in
space
and
time
coordinates
and
the
function
in
vector
and
frequency
coordinates
³
´
φ
ω
±
ω
Ψ
(
±
t
)
(4.8)
↔
have
ωφ
ω
(
ω
)
j
∂
Ψ
(
±
t
)
,
(4.9)
↔
∂t
±
k φ
ω
(
ω
)
↔−
j
∇
Ψ
(
±
t
)
,
(4.10)
±
k
2
φ
ω
(
ω
)
∆Ψ
(
±
t
)
.
(4.11)
where
∂
∂
∂
∇
=
±e
x
∂x
+
y
∂y
+
z
∂z
,
(4.12)
∂
2
∂
2
∂
2
∆
=
∇
·
∇≡∇
2
=
2
+
2
+
2
.
(4.13)
201
From
the
dispersion
relation
follows
by
multiplication
with
the
wave
function
in
thewavevectorand
frequencydoma
~
2
k
2
~
ωφ
ω
(
k,
ω
)=
φ
ω
(
ω
)+
V
0
φ
ω
(
ω
)
.
(4.14)
2
m
With
the
inverse
transformation
the
corresponding
equation
in
the
space
and
tieme
domain
is
j
~
∂
Ψ
(
±
r,
t
)
=
−
~
2
∆Ψ
(
±
t
V
0
Ψ
(
±
t
)
.
(4.15)
∂t
2
m
Generalization
of
the
above
equation
for
a
constant
potential
to
the
instance
of
an
arbitrary
potential
in
space
leads
ﬁ
nally
to
the
Schroedinger
equation
j
~
∂
Ψ
(
±
t
)
=
−
~
2
(
±
t
V
(
±
r
)
Ψ
(
±
t
)
.
(4.16)
2
m
Note,
the
last
few
pages
ar
not
a
derivation
of
the
Schroedinger
Equation
but
rather
a
motivation
for
it
based
on
the
ﬁ
ndings
of
Einstein
and
deBroglie.
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This note was uploaded on 12/02/2010 for the course ECE 6.641 taught by Professor Zahn during the Spring '09 term at MIT.
 Spring '09
 Zahn
 Electromagnet, Frequency

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