Post on 05-Jun-2020
Making Movies - Time Development
Recall the rectangular initial wave packet in the infinite square well shownbelow. How does it evolve in time?
V (x) = 0 0 < x < a
= ∞ x ≤ 0 and x ≥ a
|Ψ(x , 0)〉 = 1√d
x0 ≤ x ≤ x1 and d = x0 − x1
= 0 otherwise
x0 x1
V(x)
Ψ(x,0)
0 a x
a = 1.0 Åx0 = 0.3 Åx1 = 0.5 Å
Jerry Gilfoyle Time Development 1 / 13
Probabilities of Different States
a = 1.0 Å
x0 = 0.3 Å
x1 = 0.5 Å
0 500 1000 1500 2000 2500 30000.0
0.1
0.2
0.3
0.4
Energy (eV)
Pro
babili
ty
Rectangular Wave in a Square Well
Jerry Gilfoyle Time Development 2 / 13
Time Development of a Square Wave
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
x
Pro
babili
tyD
ensity
time=0.0000⨯10-16s.
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
x
Pro
babili
tyD
ensity
time=0.0040⨯10-16s.
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
x
Pro
babili
tyD
ensity
time=0.0080⨯10-16s.
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
x
Pro
babili
tyD
ensity
time=0.0120⨯10-16s.
Jerry Gilfoyle Time Development 3 / 13
Comparison of Bound and Free Particles
Particle in a Box
The potential
V =0 0 < x < a
=∞ otherwise
Eigenfunctions and eigenvalues
|φn〉 =√
2
asin(nπx
a
)En = n
2 ~2π2
2ma2
Superposition
|ψ〉 =∞∑n=1
bn|φn〉 〈φm|φn〉 = δm,n
Getting the coefficients
bn = 〈φn|ψ〉 Pn = |bn|2
Free Particle
The potential
V = 0
Eigenfunctions and eigenvalues
|φ(k)〉 = 1√2π
e±ikx E =~2k2
2m
Superposition
|ψ〉 =∫ ∞−∞
b(k)φ(k)dk
〈φ(k ′)|φ(k)〉 =δ(k − k ′)
Getting the coefficients
b(k) = 〈φ(k)|ψ〉 Pn = |b(k)|2
Jerry Gilfoyle Time Development 4 / 13
Time Development of the Initial Gaussian
Recall the Gaussian initial wave packet for the free particle shown below.How does it evolve in time?
V (x) = 0 |Ψ(x , 0)〉 = 1(2πσ2)1/4
e−x2/4σ2
-4 -2 0 2 40.0
0.1
0.2
0.3
0.4
0.5
x
ψ
Jerry Gilfoyle Time Development 5 / 13
Time Development of the Initial Gaussian
0 1 2 3 4 5 60
1
2
3
4
5
x
|ψ2
t=0
0 1 2 3 4 5 60
1
2
3
4
5
x
|ψ2
t=1.5
0 1 2 3 4 5 60
1
2
3
4
5
x
|ψ2
t=3.
0 1 2 3 4 5 60
1
2
3
4
5
x
|ψ2
t=4.5
Jerry Gilfoyle Time Development 6 / 13
Time Development of Nuclear Fusion
Consider a case of one dimensional nuclear ‘fusion’. A neutron is in thepotential well of a nucleus that we will approximate with an infinite squarewell with walls at x = 0 and x = L. The eigenfunctions and eigenvalues are
En =n2~2π2
2ma2φn =
√2
asin(nπx
a
)0 ≤ x ≤ a
= 0 x < 0 and x > a .
The neutron is in the n = 4 state when it fuses with another nucleus thatis the same size, instantly putting the neutron in a new infinite square wellwith walls at x = 0 and x = 2a.
1 What are the new eigenfunctions and eigenvalues of the fused system?
2 How will the initial wave packet evolve in time?
Jerry Gilfoyle Time Development 7 / 13
Time Development of Nuclear Fusion
0 5 10 15 200.00
0.05
0.10
0.15
0.20
x(fm)
|ψ2
t=0×10-21 s
0 5 10 15 200.00
0.05
0.10
0.15
0.20
x(fm)
|ψ2
t=0.3×10-21 s
0 5 10 15 200.00
0.05
0.10
0.15
0.20
x(fm)
|ψ2
t=0.6×10-21 s
0 5 10 15 200.00
0.05
0.10
0.15
0.20
x(fm)
|ψ2
t=0.9×10-21 s
Jerry Gilfoyle Time Development 8 / 13
Comparison of Bound and Free Particles
Particle in a Box
The potential
V =0 0 < x < a
=∞ otherwise
Eigenfunctions and eigenvalues
|φn〉 =√
2
asin(nπx
a
)En = n
2 ~2π2
2ma2
Superposition
|ψ〉 =∞∑n=1
bn|φn〉 〈φm|φn〉 = δm,n
Getting the coefficients
bn = 〈φn|ψ〉 Pn = |bn|2
Time Dependence
Ψ(x , t) =∞∑n=1
bn|φn(x)〉e−iωnt
Free Particle
The potential
V = 0
Eigenfunctions and eigenvalues
|φ(k)〉 =1√
2πe±ikx E =
~2k2
2m
Superposition
|ψ〉 =∫ ∞−∞
b(k)φ(k)dk
〈φ(k ′)|φ(k)〉 =δ(k − k ′)
Getting the coefficients
b(k) = 〈φ(k)|ψ〉 Pn = |b(k)|2
Time Dependence
Ψ(x , t) =
∫ ∞−∞
b(k)φk (x)e−iω(k)tdk
Jerry Gilfoyle Time Development 9 / 13
Liboff 6.4 - 1
Jerry Gilfoyle Time Development 10 / 13
Liboff 6.4 - 2
Jerry Gilfoyle Time Development 11 / 13
Liboff 6.4 - 3
Mathematica result
Jerry Gilfoyle Time Development 12 / 13
Liboff 6.4 - 4
Mathematica result
Jerry Gilfoyle Time Development 13 / 13
Liboff 6.4 - 5
Black Curve - Initial
Probability
Density.
Red Curve - Probability
Density at
t = 10 s.
Chance of being in
the original box: 0.993
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.070
2
4
6
8
10
12
x (m)
|ψ2(m
-1)
Square Wave Time Development
Jerry Gilfoyle Time Development 14 / 13