This preview shows pages 1–3. Sign up to view the full content.
1
EEE 352: LECTURE 7
Quantum Mechanics and the Schrödinger Equation
* The Schrödinger equation
⇒
Time dependent
&
independent
forms
⇒
Free electron propagation
⇒
Potential energy
Ψ
=
Ψ
+
Ψ
∇
−
E
V
m
)
(
2
2
2
r
h
Erwin Schrödiner
Nobel Prize in Physics, 1933
Where are we going for the next few lectures?
Schrödinger
equation
Simple
examples
Band
structure
Electron waves
in the
semiconductor
Bragg reflection in the crystal
due to the periodic structure.
Fourier transform space for
E
vs
k
Band structure describes the
dispersion relation between energy and
wave number,
e.g. between energy and
momentum
, for ALLOWED wave states
in the semiconductor.
Solving The Schrödinger Equation
We wish to solve the timeindependent form of the Schrödinger equation:
If we assume a solution of the form
then
2
2
2
2
2
2
2
2
2
,
0
)
(
)
(
)
(
)
(
2
h
h
mE
k
x
k
dx
x
d
x
E
dx
x
d
m
=
=
+
=
−
ψ
x
i
e
α
(
)
ikx
ikx
x
i
Be
Ae
x
k
e
k
−
+
=
±
=
=
+
−
)
(
0
2
2
The Schrödinger Equation
•
These
GENERAL
solutions contain contributions from
TWO
spatially varying waves
* That propagate in
OPPOSITE
directions to each other!
* The constants
A
&
B
are determined by suitable
BOUNDARY CONDITIONS
⇒
Such as the
INITIAL
direction of the particle
( ) ( )
[ ]
t
i
e
ikx
B
ikx
A
t
x
ω
−
−
+
=
Ψ
exp
exp
)
,
(
[]
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
Ψ
⎥
⎦
⎤
⎢
⎣
⎡
−
=
Ψ
h
h
iEt
ikx
B
t
x
iEt
ikx
A
t
x
exp
exp
)
,
(
exp
exp
)
,
(
FREELY MOVING PARTICLE
THAT MOVES ALONG THE xAXIS
IN THE
+x
DIRECTION
FREELY MOVING PARTICLE
THAT MOVES ALONG THE xAXIS
IN THE
x
DIRECTION
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
The Schrödinger Equation
FREELY MOVING
PARTICLE
THAT MOVES ALONG
THE xAXIS
IN THE +x DIRECTION
FREELY MOVING
PARTICLE
THAT MOVES ALONG
THE xAXIS
IN THE x DIRECTION
[]
0
0
exp
exp
)
,
(
)
(
>
=
=
−
=
=
=
⎥
⎦
⎤
⎢
⎣
⎡
−
=
Ψ
−
k
t
x
v
t
kx
Ae
iEt
ikx
A
t
x
phase
t
kx
i
ω
φ
h
0
0
exp
exp
)
,
(
)
(
<
−
=
=
+
=
=
=
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=
Ψ
+
−
k
t
x
v
t
kx
Ae
iEt
ikx
A
t
x
phase
t
kx
i
h
The Schrödinger Equation
•
Classically a freely moving particle should propagate with CONSTANT velocity
* What do the
QUANTUM
solutions tell us about the motion of the particle?
* We have already found the connection between energy and wavenumber
•
Given this last result we may compute the MOMENTUM of the moving particle
* And we find this to be CONSTANT as expected classically
⇒
We also obtain the de Broglie relation!
m
k
E
2
/
2
2
h
=
⎥
⎦
⎤
⎢
⎣
⎡
=
=
=
=
=
⇒
=
∴
λ
π
h
h
k
m
k
m
mE
p
m
p
E
2
2
2
2
2
2
2
2
2
h
h
SINCE E IS
CONSTANT
p IS
CONSTANT
The de Broglie relation is
recovered!
The Free Particle
For the free particle, we have now found that
m
k
m
k
E
2
/
2
/
2
2
2
h
h
=
⇒
=
k
m
k
m
mE
p
h
h
=
=
=
2
2
2
2
2
The two velocities that result from this PARTICLE WAVE are
particle
group
particle
phase
v
m
p
m
k
k
v
v
m
p
m
k
k
v
=
=
=
∂
∂
=
=
=
=
=
h
h
2
2
2
As indicated earlier, we connect the free particle velocity with
the GROUP VELOCITY of the wave.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/19/2009 for the course EEE 352 taught by Professor Ferry during the Spring '08 term at ASU.
 Spring '08
 Ferry

Click to edit the document details