Download Localized particle - Physics | Oregon State University

FREE PARTICLE GAUSSIAN
WAVEPACKET
Reading:
QM Course packet
Wavepacket
1
GAUSSIAN WAVE PACKET - REVIEW
•
•
•
•
•
•
Time dependent Schrödinger equation
Energy eigenvalue equation (time independent SE)
Eigenstates
Time dependence
(Connection to separation of variables)
Mathematical representations of the above
Wavepacket
2
Build a "wavepacket" from free particle eigenstates
•Ask 2 important questions:
•Given a particular superposition, what can we learn
about the particle's location and momentum?
HEISENBERG UNCERTAINTY PRINCIPLE
•How does the wavepacket evolve in time?
GROUP VELOCITY
Wavepacket
3
We have already discussed the principle of
superposition & the time evolution of that superposition
in the context of the discrete quantum mechanical states
of the infinite potential energy well.
We have also already discussed how build a Gaussian
wave packet from harmonic waveforms (with a
continuous frequency distribution) in the context of the
classical rope problem.
We now look at the case of superposition of quantum
mechanical states of the free particle, which are no
longer discrete.
Wavepacket
4
Free particle eigenstates
y k (x,t) = eikx e
-i
Ek
•Oscillating function
•Definite momentum p = hk/2π
•Subscript k on w reminds us that
w depends on k
• What are E and w in terms of
given quantities?
t
k
Ek =
2m
2
wk º
Ek
2
k2
=
2m
1
0.5
0
j k (x) º yk (x,0)
-0.5
y
-1
-2
j k (x) = e
x
0
2
ikx
Wavepacket
5
1
Superposition of eigenstates
(Fourier series)
0.5
0
-0.5
j ( x ) = å c kj k ( x )
k
k
Ek =
2m
2
-2
+
2
-2
+
2
+
2
0
1
0.5
j k (x) = e
2
-1
v phase,k
ikx
0
-0.5
wk
Ek
k
=
=
=
k
k 2m
-1
0
1
0.5
0
-0.5
y
1
-1

0.5
-2
1
x
0
0
0.5
-0.5
0
y
-0.5
-1
-2
x
0
2
Wavepacket
6
-1
-2
0
2
1
Superposition of eigenstates
(Fourier integral)
¥
0.5
0
-0.5
j (x) = ò dk A(k)e
ikx
-1
+
-2
-¥
0
1
2
0.5
0
-0.5
-1
Definite momentum
Extended+ position
-2
1
0
2
0.5
0
-0.5
y
1
-1

0.5
-2
1
x
+
0
2
0
0.5
-0.5
y
-1
Localized particle
x
Indefinite
momentum
-2
0
0
-0.5
2
Wavepacket
7
-1
-2
0
2
1
Superposition of eigenstates
How does it develop in time?
0.5
0
-0.5
¥
j (x,0) = ò dk A(k)e
ikx
-1
+
-2
-¥
0
1
2
0.5
¥
j (x,t) = ò dk A(k)e e
ikx -iEk t /
0
-¥
-0.5
-1
Definite momentum
Extended+ position
-2
k
Ek =
2m
2
2
1
0
2
0.5
0
-0.5
y
1
-1

0.5
-2
1
x
+
0
2
0
0.5
-0.5
y
-1
Localized particle
x
Indefinite
momentum
-2
0
0
-0.5
2
Wavepacket
8
-1
-2
0
2
Gaussian wave packet
1
4
2
æ 2a ö -a 2 x2
j (x, 0) = ç
e
÷
è p ø
2
2a
Localized particle
x
What is A(k)?
We'll ignore overall constants that are not of primary importance
(there are conventions about factors of 2π that are important to take
care of to get numerical results, but we're after the physics!)
A(k) = j k j
Projection of general
function on eigenstate
t =0
Wavepacket
9
Gaussian wave packet
1
4
2
æ 2a 2 ö -a 2 x2
j (x, 0) = ç
e
÷
è p ø
2a
Localized particle
x
A(k) = j k j t = 0 =
¥
*
dx
j
ò k (x)j (x,0)
-¥
¥
A(k) = j k j t = 0 @
ò dx e
-ikx - a 2 x 2
e
-¥
We did this integral (by hand).
¥
using:
ò dy e
-¥
Wavepacket
vy -uy 2
e
=
p
e
u
v2
4u
10
Gaussian wave packet
1
4
æ 2a 2 ö -a 2 x2
j (x, 0) = ç
e
÷
è p ø
Localized particle
x
A(k) = j k j t = 0 @
¥
ò dx e
-ikx - a 2 x 2
e
-¥
A(k) @ e
Wavepacket
- k2 4a 2
11
Gaussian wave packet
j (x) @ e
2
-a2 x2
2a
Localized particle
x
A(k) @ e- k
2
4a
2
2 2a
k
If j(x) is wide, A(k) is narrow and vice versa.
Wavepacket
12
j (x) @ e
A(k) @ e
-a2 x2
- k2 4a 2
A(k) is the projection of j(x) on the momentum eigenstates eikx, and
thus represents the amplitude of each momentum eigenstate in the
superposition.
We need the contribution of a wide spread of momentum states to
localize a particle. If we have the contribution of just a few, the
Wavepacket
13
location of the particle is uncertain
j (x) @ e
-2a 2 x 2
2
x
-2k 2 4 a 2
A(k) @ e
2
k
To define "uncertainty" in position or momentum, we must
consider probability, not wave function.
Wavepacket
14
j (x) @ e
-2a 2 x 2
2
x
-2k 2 4 a 2
A(k) @ e
2
Wavepacket
15
k
j (x) @ e
2
-2a 2 x 2
x
HEISENBERG UNCERTAINTY PRINCIPLE
A(k) @ e
2
- 2 k 2 4a 2
Wavepacket
16
k
Next, we’ll ask how a general wave packet propagates,
And deal with the particular example of the Gaussian
wavepacket.
In short, we simply attach the exp(-iE(k)t/hbar) factor to
each eigenstate and let time run.
Difference to non-dispersive equation: not all waves
propagate with same velocity. “Packet” does not stay
intact! Need to invoke “group velocity” to follow the
progress of the “bump”.
Wavepacket
17
1
Superposition of eigenstates
How does it develop in time?
0.5
0
¥
j (x,t) = ò dk A(k)e e
ikx -iw k t
-0.5
Localized particle
y
-1
-¥
-2
0
x
2
If w depends on k, the different eigenstates (waves) making up this
"packet" travel at different speeds, so the feature at x=0 that exists
at t=0 may not stay intact at all time.
It may stay identifiably intact for some reasonable time, and if it
does, how fast does it travel?
The answer is "it travels at the group velocity dw/dk"
wk º
vgroup
dw
k
=
=
dk m
Ek
k
=
2m
Wavepacket
2
w
k
=
k 2m
18
1
Superposition of eigenstates
How does it develop in time?
0.5
0
¥
j (x,t) = ò dk A(k)e e
ikx -iw k t
-¥
-0.5
Localized particle
y
-1
-2
0
x
2
The quantity dw/dk may (does!) vary depending on the k value at
which you choose to evaluate it. So it must be evaluated at a
particular value k0 that represents the center of the packet. The next
few pages spend time deriving the basic result. The derivation is
not so important. The result is important:
vgroup
dw
=
dk
k0
k0
=
m
w
k
=
k 2m
Wavepacket
19
For those of you who want more:
1
0.5
¥
j (x,t) = ò dk A(k)e e
ikx -iw k t
-¥
0
-0.5
Localized particle
y
-1
-2
dw
w (k) = w 0 +
dk
( k - k0 )
k = k0
j (x,t) =
¥
x
0
2
w is a smooth function of k
and w0= w(k0)
æ
dw
-i ç w 0 +
dk
è
ikx
ò dk A(k)e
e
ö
( k - k0 )÷ t
ø
k=k0
-¥
k - k0 = s
Wavepacket
20
For those of you who want more:
1
0.5
0
-0.5
Localized particle
y
-1
-2
j (x,t) =
¥
ò dk A(k)e
æ
dw
-i ç w 0 +
dk
è
ikx
e
x
ö
( k - k0 )÷ t
ø
k0
j (x,t) =
ò d(k
0
æ
dw
-i ç w 0 +
dk
i ( k0 + s ) x
è
+ s ) A(k0 + s)e
2
k - k0 = s
-¥
¥
0
e
ö
( k0 + s - k0 )÷ t
ø
k0
-¥
j (x,t) = e
æ
dw
-i ç w 0 - k0
dk
è
ö
÷t ¥
k0 ø
ò d(k
0
-¥
æ dw
i ( k0 + s ) ç x dk
è
+ s ) A(k0 + s)e
Wavepacket
ö
t÷
k0 ø
21
For those of you who want more:
1
0.5
0
k = k0 + s
-0.5
Localized particle
y
-1
-2
j (x,t) = e
æ
dw ö
-i ç w0 -k0
÷t ¥
dk k0 ø
è
j (x,t) = e
ò d ( k0
-¥
æ
dw
-iççw 0 -k0
dk
è
x
0
+ s) A(k0 + s)e
ö
÷÷ t ¥
k0 ø
ò dk A(k)e
2
æ dw
ö
i ( k0 +s ) ç xt÷
è dk k0 ø
æ dw
ik çç xè dk
ö
t ÷÷
k0 ø
-¥
Phase factor - goes to 1 in probability
Wavepacket
22
For those of you who want more:
1
0.5
j (x,0) =
¥
ò dk A(k)e
0
ikx
-0.5
-¥
Localized particle
y
-1
-2
¥
j (x,t) = phase ò dk A(k)e
0
x
æ dw
ik çç xè dk
2
ö
t ÷÷
k0 ø
-¥
¥
j (x,t) = phase ò dk A(k)e
-¥
Wavepacket
ik ( x-vgt )
= j (x - vgt,0)
dw
vg =
dk
23
k = k0
Particular example of the Gaussian wavepacket.
Wavepacket
24
j (x,t) @
¥
ò dk e
-k 2 4 a 2 ikx -iwt
e e
-¥
j (x,t) @
¥
ò dk e
é 1
ù
- ê 2 +i
t ú k2
2m û
ë 4a
eikx
-¥
1
- a 2 x 2 [1+it t ]
j (x,t) @
e
1 + it t
P(x,t) @
1
1+ t t
2
2
e
k2
w (k) =
2m
Use same integral as before
t=
m
2 a2
- 2a 2 x 2 éë1+t 2 t 2 ùû
Wavepacket
25
P(x,t) @
1
1+ t t
2
2
e
- 2a 2 x 2 éë1+t 2 t 2 ùû
Wavepacket
t=
m
2 a2
26
P(x,t) @
t = 0t
1
1+ t t
2
2
e
- 2a 2 x 2 éë1+t 2 t 2 ùû
1
- a 2 x 2 [1+it t ]
j (x,t) @
e
1 + it t
t = 2t Zero-momentum wavepacket:
t = 5t
Wavepacket
Spreads but doesn’t travel!
It has many positive k
components as it has negative
k components.
27
P(x,t) @
1
1+ t2 t 2
e
- 2a 2 x 2 éë1+t 2 t 2 ùû
t is characteristic time for
wavepacket to spread
t=
t for macroscopic things:
m ≈ 10-3kg; 1/a ≈ 10-2 m
t ≈ 1027 s ≈ 1020 yr !! long
m
2 a
2

t for atomic scale:
m ≈ 10-?kg; 1/a ≈ 10-? m
t ≈ 10-? s

t for nuclear scale:
m ≈ 10-?kg; 1/a ≈ 10-? m
t ≈ 10-? s

Wavepacket
28
FREE PARTICLE QUANTUM
WAVEPACKET - REVIEW
•
•
•
Gaussian superposition of free-particle eigenstates (of energy
and momentum!)
Localized in space means dispersed in momentum and vice
versa.
Look at time-dependent probability distribution: packet
broadens and moves
Wavepacket
29