1
Introduction to Quantum and Statistical Mechanics
Fall 2009
Handout 2
The Schrödinger Equation from Intuition and as a Postulate
Complementary Reading:
Chap. 4 (pp. 43 – 64), Hagelstein, Senturia and Orlando
2.1
Wave packet for intuitive construction of the Schrödinger equation
After learning the central concept of deBroglie wavelength, we can intuitively understand how
classical particle mechanics and wave mechanics can be unified in the eigenfunction of the wave
equation as a “wave packet” (or similarly as superposition of plane waves) by
a(x)e
i(kx
ω
t)
, where
(
)
ω
ω
ψ
h
h
h
h
=
=
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
−
=
−
=
E
k
p
t
E
x
p
i
t
kx
i
t
x
exp
)
(
exp
)
,
(
(2.1)
The question in hand is: what is the overarching equation that can capture this waveparticle
duality and satisfy the experimental evidence from photons and electrons. Namely,
k
p
h
=
and
ω
h
=
E
.
In the wave equation, if we substitute the plane wave into Eq. (1.5), we get a
dispersion relation of
k
c
ω
±
=
, where the wave velocity
c
is also the group velocity
k
ω
v
g
∂
∂
=
,
and the phase velocity
v
p
.
We will now see what dispersion relation and governing equation can
be constructed for particlelike behavior.
The classical kinetic energy of a free particle is:
m
k
k
ω
ω
m
p
E
2
)
(
2
2
2
h
h
=
→
=
=
(2.2)
We now ask the question: what relation of the derivatives in time and space will give this
quadratic dispersion relation? For the plane wave, we know:
)
(
)
(
)
(
)
(
;
t
ω
kx
i
t
ω
kx
i
t
ω
kx
i
t
ω
kx
i
ike
e
x
e
ω
i
e
t
−
−
−
−
=
∂
∂
−
=
∂
∂
, and the dispersion relation of
m
k
k
ω
2
)
(
2
h
=
can
be obtained from:
)
,
(
2
)
,
(
2
2
2
t
x
ψ
x
m
t
x
ψ
t
i
∂
∂
−
=
∂
∂
h
h
or in 3D:
)
,
(
2
)
,
(
2
2
t
x
m
t
x
t
i
v
h
v
h
ψ
ψ
∇
−
=
∂
∂
(2.3)
Remember that the dispersion relation is obtained from equating the wave energy
ω
E
h
=
with
the freeparticle kinetic energy
E = p
2
/2m
.
This suggests that if we map the two “operators” on
the plane wave function:
x
i
p
t
i
E
∂
∂
−
=
∂
∂
=
h
h
;
or in 3D:
i
i
x
i
p
∂
∂
−
=
h
or
∇
−
=
h
v
i
p
(2.4)
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We can then obtain the dispersion relation of Eq. (2.2) and the wave behavior of a particle!
(
Exercise 2.1
) What is the group velocity (
v
g
) behind the dispersion relation of
m
k
k
ω
2
/
)
(
2
h
=
?
What does it mean when
v
g
depends on the wave vector
k
or the deBroglie wavelength?
Equation (2.4) is however rather strange when the imaginary number and the operator were
introduced.
This is counterintuitive as the real world is described by imaginary number and
operators!
However, this will eventually become the most fundamental description of the world
we understand today.
We will now proceed to obtain a consistent mathematical formulation to
unify the classical and wave mechanics in the spirit of the deBroglie particlewave duality.
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 '05
 TANG
 wave packet, wave function, uncertainty principle.

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