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Consider a quantum-superposition of two plane-waves: The interference pattern can move! – How does this `wavepacket' then move? ,σ› evolve in time?
Typology: Schemes and Mind Maps
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Wavefunction in K-space
( )
( , ) ( )
2
0
2
0
e k k
k t k t e k k
= −
= =
δ
ψ ψ
k
k
ψ( k,t )
Let ψ( 0 ) = k
2
0
( t ) e k
Then: ψ =
`Motion’ in QM
interference effect
plane-waves:
= +
− −
0
2
0
2
2
2
1
( )
2
0
2
0
t e k e k
t
M
k
t i
M
k
i
h h
ψ
π π
ψ
2 2
( , )
0
0
0
0
2
2
−
−
= +
t
M
k
t i k x
M
k
ik x
e e
x t
h h
= + − t
M
k
k x
2
3
1 cos
2
1 h
π
This is a moving
`standing wave’
= − t
M
k
k x
2
3
2
1
cos
1
h
π
= + +
2 2
1
2
1
( , )
0
0
0
0
t
k
t i k x
k
i k x
e e
x t
h h
π
ψ
Wavepacket formation
`localized’ state
Physical versus non-physical states:
〉 and |k
〉 have 〈 ψ| ψ〉= ∞
〉 and |k
〉
are non-physical, and can therefore only be
used as intermediate states in calculations
produces the nonphysical state |x
〉, such a
measurement must be impossible
onto this subspace, as
= ± σ
0
x x
ψ ψ
σ
σ
dx x x
x
x
∫
−
=
′
0
0
Phase and Group Velocities
moves at the velocity
with the velocity of a classical particle having
the same momentum
m
p
v
2
=
( )
( )
( )
2
2
2
0
( )
1
( , )
t
x x t
e
t
x t
σ
π σ
ψ
−
−
=
t
m
p
x t x
( )= +
( ) 1
= +
σ
σ σ
m
t
t
h
m
p
v
=
€
ψ( x , t ) = π σ +
i h t
m σ
e
x −
p
0
t
m
− x
0
2
2 σ
2
i h t
m σ
2
p
0
x −
p
0
t
2 m
h
Wavepacket Spreading
s
we can ignore spreading
s
the size of the wavepacket grows
linearly in time:
spread!
( ) 1
= +
σ
σ σ
m
t
t
h
€
σ( t ) ≈ σ
h
m σ
t
s
=
t
m
t
σ
σ
h
( ) ≈
m σ
v
s
h
=
t
Δx
t=t
s
Δx = v
s
t
Δx = σ
( )
( ) 1 /
s
σ t = σ + t t
Wavepacket for a Baseball
to be localized by neutron diffraction to .1 nanometer
10
14
s = 30 million years
m/s.
size
defined/deterministic position and momentum)
should do pretty well
s
kgm s
m kg m
t
s
10
10
1 10
=
⋅
= =
h
σ
s
m
kg m
kgm s
m
v
s
10
1 10
10
=
⋅
= =
σ
h 1 nm
300 million years
