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PHY4604
R. D. Field
Department of Physics
Chapter1_1.doc
University of Florida
Classical Mechanics vs Quantum Mechanics
Classical Mechanics:
The goal of
classical
mechanics is to determine the
position of a particle at any given time,
x(t)
.
Once we know
x(t)
then we
can compute the velocity
v
x
= dx/dt
, the momentum
p
x
= mv
x
, the kinetic
energy
T = p
x
2
/2m
, or any other dynamical variable.
Classical Equations of Motion:
Newton’s Laws for a particle under the
influence of the potential
V(x)
are as follows:
dx
x
dV
dt
dp
ma
F
x
x
x
)
(
−
=
=
=
with
dt
dx
m
p
x
=
.
To determine
x(t)
one must solve the Newton’s equation
dx
x
dV
dt
t
x
d
m
)
(
)
(
2
2
−
=
with the appropriate initial conditions (typically the position and velocity at t
= 0).
Quantum Mechanics:
In Quantum Mechanics the situation is much
different.
In this case we are looking for the particles wave function
Ψ
(x,t)
which is the solution of Schrödinger’s equation:
t
t
x
i
t
x
x
V
x
t
x
m
∂
Ψ
∂
=
Ψ
+
∂
Ψ
∂
−
)
,
(
)
,
(
)
(
)
,
(
2
2
2
2
h
h
.
where
1
−
=
i
and
s
J
h
⋅
×
=
=
−
34
10
054572
.
1
2
π
h
.
The wave function is a complex function and Schrödinger’s equation is
analogous to Newton’s equation. Given suitable initial conditions (typically
Ψ
(x,0)
), one can solve Schrödinger’s equation for
Ψ
(x,t)
for all future times,
just as in classical physics, Newton’s equation determines
x(t)
for all future
times.
Schrödinger Equation:
1.
Ignores the creation and annihilation of particles, but photons may be
emitted and absorbed.
2.
Assumes that all relevant velocities are much less than the speed of
light (
i.e.
nonrelativistic).
Newton’s Equation!
Schrödinger’s Equation!
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View Full DocumentPHY4604
R. D. Field
Department of Physics
Chapter1_2.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!
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 Spring '07
 FIELDS
 mechanics

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