This preview shows pages 1–3. Sign up to view the full content.
[email protected]
116
04/04/2005
Projection amplitudes and ket vectors for free states
∑
∑

=
Ψ
=
Ψ
x
x
x
e
x
x
x
all
1
all
2
2
2
σ
π
∑



=
Ψ
k
k
e
k
all
1
2
2
2
1
2
2
all
all
x
ik
x
x
ik
k
e
x
k
x
x
k
k
e
k
x
k
k
x
x

=
→
Ψ
=
Ψ
=
→
Ψ
=
Ψ
∑
∑
Gaussian probability distribution at t=0:
2
2
2
1
)
0
,
(
x
e
x

=
Ψ
Satisfies the requirements for projection amplitudes:
And is a
stationary state of the free Schrödinger equation:
*
x
k
k
x
=
k
x
x
k
=
Φ
)
(
m
k
k
k
x
m
E
x
E
x
2
2
2
2
2
2
2
),
(
)
(
=
Φ
=
Φ

∂
∂
While these stationary basis states cannot be normalized to 1,
any wave function is normalized to 1.
Ψ
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document [email protected]
117
04/04/2005
The uncertainty principle and groundstate energies
x
∆
Area in which the ground state wave function has to be confined
Momentum uncertainty in all three dimensions
x
x
p
∆
≥
∆
1
2
2
2
2
3
0
)
(
x
m
E
∆
≈
Example: Hydrogen atom
2
2
2
4
1
2
0
4
0
1
8
0
)
(
0
0
0
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 09/28/2008 for the course PHYS 316 taught by Professor Hoffstaetter during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 HOFFSTAETTER
 Physics

Click to edit the document details