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PHY3063
R. D. Field
Department of Physics
Chapter5_1.doc
University of Florida
De Broglie’s Pilot Waves
Bohr’s Model of the Hydrogen Atom:
One way to arrive at Bohr’s hypothesis is to think of the
electron
not as a particle
but as
a standing wave
at
radius
r
around the proton.
Thus,
r
n
π
λ
2
=
and
2
n
r
=
with n = 1, 2, 3, …
The orbital angular momentum is
h
n
h
n
p
n
rp
L
=
=
=
=
2
2
,
which is Bohr’s hypothesis provided that
p = h/
λ
!
Pilot Waves:
In his doctoral dissertation (1924) Louis De Broglie suggested that if waves
can act like particles (
i.e.
the photon), why not particles acting like waves?
He called these waves “pilot waves” (wave of what?) and assigned them the
following wavelength and frequency:
h
E
f
p
h
=
=
or
k
p
E
h
h
=
=
ω
where
k = 2
π
/
λ
and
ω
= 2
π
f
.
Phase Velocity of a Plane Wave:
For a traveling plane (
i.e.
monochromatic) wave
given by
)
sin(
)
,
(
t
kx
A
t
x
−
=
Ψ
the position of the n
th
node (
i.e.
zeros) is given by
n
t
kx
n
=
−
and the speed of the n
th
node along the xaxis is
f
k
dt
dx
v
n
phase
=
=
=
Phase Velocity of a Plane Pilot Wave:
For a plane De Broglie pilot wave we get
c
cp
c
m
cp
c
cp
E
p
E
k
v
phase
2
2
0
2
)
(
)
(
+
=
=
=
=
,
which is
greater than c for particles with nonzero rest mass!
Plane Wave
1
0
1
024681
0
x
v
phase
r
λ
= 2
π
r/4
Bohr’s Postulate!
Particles cannot be
plane pilot waves!
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View Full Document PHY3063
R. D. Field
Department of Physics
Chapter5_2.doc
University of Florida
Wave Superposition and Wave Packets
Suppose we add two plane waves
together one with wave number
k
and
frequency
ω
and the other with wave
number
k+
∆
k
and frequency
ω
+
∆ω
as
follows
)
)
(
)
sin((
)
,
(
)
sin(
)
,
(
)
,
(
)
,
(
)
,
(
2
1
2
1
t
x
k
k
t
x
t
kx
t
x
t
x
t
x
t
x
ω
∆
+
−
∆
+
=
Ψ
−
=
Ψ
Ψ
+
Ψ
=
Ψ
Then
∆
+
−
∆
+
∆
−
∆
=
Ψ
t
x
k
k
t
x
k
t
x
2
2
2
2
sin
2
2
cos
2
)
,
(
where I used
))
(
sin(
))
(
cos(
2
sin
sin
2
1
2
1
B
A
B
A
B
A
+
−
=
+
Now suppose that
∆
k << 2k
and
<< 2
ω
so that
()
t
kx
t
x
P
t
kx
t
x
k
t
x
−
=
−
∆
−
∆
≈
Ψ
sin
)
,
(
sin
2
2
cos
2
)
,
(
.
The term P(x,t) describes the “wavepacket”.
The position of the n
th
node
(
i.e.
zeros) of the wavepacket is given by
π
)
1
2
(
+
=
∆
−
∆
n
t
kx
n
and the
speed of the wavepacket along the xaxis is
dk
d
k
dt
dx
v
n
group
→
∆
∆
=
=
Group Velocity of a Pilot WavePacket:
For a De Broglie pilot wavepacket we get
c
E
cp
dp
dE
dk
d
v
group
=
=
=
,
where I used
h
/
E
=
,
h
/
p
k
=
, and
( )
E
p
c
c
m
cp
dp
d
2
2
2
0
2
)
(
)
(
=
+
.
Note that for pilot waves
v
phase
·
v
group
= c
2
.
The pilot wavepacket travels
at the speed of the particle!
Wave Packet
3
2
1
0
1
2
3
x
v
group
Wave Packet!
Wave Superposition
2
1
0
1
2
x
PHY3063
R. D. Field
Department of Physics
Chapter5_3.doc
University of Florida
Re(
Ψ
)
A
Ψ
= Ae
i(kx
ω
t)
Im(
Ψ
)
φ
= kx
ω
t
t
Im(
Ψ
) = Asin(kx
ω
t)
A
A
Crest
Trough
A
Ψ
(0,t) = Ae
i
ω
t
Distance r
φ
= kr
ω
t
x = 0
φ
=
ω
t
A
x = r
Ψ
(r,t) = Ae
i(
kr

ω
t)
Representing Waves as Complex Numbers
We can use complex numbers to represent traveling waves.
If we let
)
(
t
kx
i
Ae
ω
−
=
Ψ
then
)
sin(
)
Im(
t
kx
A
−
=
Ψ
is a traveling plane wave with wave number
k = 2
π
/
λ
, “angular” frequency
ω
= 2
π
f
, and amplitude
A
.
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This note was uploaded on 04/17/2008 for the course PHY 3063 taught by Professor Field during the Spring '07 term at University of Florida.
 Spring '07
 Field
 Physics

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