Download Momentum vs. Wavevector

Momentum vs. Wavevector
• Instead of momentum, it is often convenient
to use wavevector states:
Wavevector definition:
Important commutators:
K=
P
h
[X , K ]= i
Eigenvalue equation:
K k =k k
k =c p
Relation to momentum eigenstates:
k k ′ = δ (k − k ′)
[P, K ]= 0
p = hk
Normalization
+∞
∫ dk
k k =1
−∞
ei k x
xk =
2π
Closure
Wavefunction
1
1
′
′
′
′
p p = δ ( p − p ) = δ (hk − hk ) = δ (k − k ) = k k ′
h
h
∴k = h p
€
p= hk
Wavefunction in K-space
• What is the wavefunction in K-basis?
Let
ψ (0) = k0
Then:
ψ (t ) = e
−i
ψ (k , t ) = k ψ (t ) = e
=e
hk 02t
−i
2m
hk 02
t
2M
hk 02t
−i
2m
k0
k k0
δ (k − k0 )
ψ (k , t )
k0
k
Wavefunction in X-space
• Quantum mechanical kinetic-energy
eigenstate:
p02
ψ (t ) = e
−i
2 Mh
t
p0
– Only Global phase is changing in time
– Global phase can’t be observed
• Is anything actually moving?
ψ ( x, t ) =
e
p 
p
i  x− 0 t 
h  2M 
2πh
1
ψ ( x, t ) =
2πh
2
– The spatial phase-pattern is moving, but at:
v = p0 2 M
– This is called the `phase velocity’
• Not related to particle velocity in classical limit
• Classical free-particle motion:
x(t ) = x0 + v t
p (t ) = p0
• How can we have a `classical limit’ of QM if
nothing moves at v=p /M?
`Motion’ in QM
• The answer is that in QM motion is in
interference effect
• Consider a quantum-superposition of two
plane-waves:
2
2
hk
2 hk 0
−i
t
1  −i 2 M0 t
M
ψ (t ) =
k0 + e
2k 0
e
2 
ψ ( x, t ) =
e
 hk 
ik 0  x − 0 t 
 2M 
2 π
+
e



 hk 
i 2 k0  x − 0 t 
M 

2 π
3 hk 0 
3 hk 0 



i  k0 x −
t
−i  k 0 x −
t 
2 M 
2 M 


1  e
e
2

ψ ( x, t ) =
1
+
+

2π 
2
2




1 
3 hk0 

=
1
+
cos
k
x
−
t 

0

2π 
2 M 

=
1 
1
3 hk0 
cos 2   k0 x −
t 
π
2 M 
2 
This is a moving
`standing wave’
• The interference pattern can move!
Wavepacket formation
• Measurement of position must produce a
`localized’ state
– How does this `wavepacket’ then move?
Physical versus non-physical states:
• States like |x0〉 and |k0〉 have 〈ψ|ψ〉=∞
• All physical states must have 〈ψ|ψ〉=1
• CONCLUSION 1: states such as |x0〉 and |k0〉
are non-physical, and can therefore only be
used as intermediate states in calculations
• CONCLUSION 2: Since a measurement of X
produces the nonphysical state |x0〉, such a
x = x0 ± σ
measurement must be impossible
• However, real detectors have finite resolution
– call the resolution σ
– Result of measurement is therefore:
– After measurement, state vector is projected
onto this subspace, as
x0 +σ
ψ′ =
∫ dx x
xψ
x0 −σ
– This state will be a `wavepacket’ with width σ
Gaussian Wavepackets
• A proto-typical wave packet is the Gaussian state
|x0,σ〉, defined via:
x x 0 ,σ =
€
1
1/ 4 e
−
( x−x 0 ) 2
2σ
2
[πσ ]
2
– Has width σ, centered at x=x0
– The width σ can be arbitrarily small, but we will
always have 〈ψ|ψ〉=1
• Such wavepackets are the physical `position’
states
• A wavepacket centered at x0 and moving with
velocity v=p0/m, has the wavefunction:
x x 0 , p0 ,σ =
1
2 1/ 4
[πσ ]
e
−
( x−x 0 ) 2
2σ
2
+i
p0x
h
• How does the intial state |x0,p0,σ〉 evolve in time?
2
€
Analytic
solution
exists:
2
 
iht  
ψ ( x, t ) = π  σ +
 
m
σ
 
 
−1 / 4
ψ ( x,0) = x x0 , p0 , σ
e
pt
p t



p0  x − 0 
 x − 0 − x0 
m
 +i  2 m 
−
iht 
h

2σ 2  1+
2 
 mσ 
Phase and Group Velocities
 p0t
2
 p t
−x 0 
p 0  x− 0 
 x−
m


 2m 
−
+i
−1/
4

iht 
h
2
2σ 2 1+

2
 mσ 
 
iht  
ψ (x,t) = π σ +

mσ  
 
e
• We can see that the phase velocity is
vp =
€
p0
2m
• What does the probability density look like?
1
ψ ( x, t ) =
e
π σ (t )
2
p
x0 (t ) = x0 + 0 t
m
(
x − x0 ( t ) )2
−
σ 2 (t )
 ht 
σ (t ) = σ 1 + 
2 
 mσ 
2
• We see that the center of the wavepacket
moves at the velocity
p0
vg =
m
– We call this the `group velocity’
• We can see that the group velocity correlates
with the velocity of a classical particle having
the same momentum
Wavepacket Spreading
• The width of the Gaussian wavepacket is:
 ht 
σ (t ) = σ 1 + 
2 
 mσ 
2
• Thus the timescale for wavepacket spreading is:
mσ 2
ts =
h
2
σ (t ) = σ 1 + (t / t s )
– For t << ts we can ignore spreading σ (t) ≈ σ
– For t >>ts the size of the wavepacket grows
linearly in time:
h
σ (t ) ≈
t
€mσ
– Thus `spreading’ at the velocity:
vs =
h
mσ
– The smaller the wavepacket, the faster it will
spread!
Δx
Δx = vst
Δx = σ
t=ts
t
Electron Wavepacket
• Lets consider an electron:
– m = 10-30 kg
– If position was measured using light, then the
electron will at most be localized to the size of
the wavelength
– Visible light λ = 10-7 m
mσ 2 10 −30 kg ⋅10 −14 m 2
−10
ts =
=
=
10
s
−34
2 −1
h
10 kg m s
• Electron wavepacket will start spreading
ns after being localized
0.1
h
10 −34 kg m 2 s −1
km
3 m
vs =
= −30
=
10
=
1
mσ 10 kg ⋅10 −7 m
s
s
•
Spreading velocity, 1 km/s, is fairly large
• Hard to keep an electron pinned down to a
deterministic position
 Classical Mechanics will not describe physics
correctly
– Repeated measurement of the electrons
position could maintain localization, but
random nature of measurements would
introduce `quantum Brownian motion’ so that
CM will still not be correct
Wavepacket for a Baseball
•
Here we have
m = 1 kg
– Let us still consider the center-of-mass of the electron
to be localized by neutron diffraction to .1 nanometer
mσ 2 1kg ⋅10 −20 m 2
14
ts =
= −34
=
10
s
2 −1
h
10 kg m s
•
So the wavepacket will only start spreading after
1014 s = 30 million years
h
10 −34 kg m 2 s −1
− 24 m
vs =
=
=
10
mσ
1kg ⋅10 −10 m
s
•
1 nm
300 million years
After which it will start to spread at a rate of 10-24
m/s.
– In another 30 million years, it will have doubled in
size
•
•
Note: the age of the universe is 13 billion years
In that time it will have reached 10 nm.
•
So classical mechanics (i.e. welldefined/deterministic position and momentum)
should do pretty well