Download Lecture 17 Example: Quantum mechanics

EE263 Autumn 2008-09
Stephen Boyd
Lecture 17
Example: Quantum mechanics
• wave function and Schrodinger equation
• discretization
• preservation of probability
• eigenvalues & eigenstates
• example
17–1
Quantum mechanics
• single particle in interval [0, 1], mass m
• potential V : [0, 1] → R
Ψ : [0, 1] × R+ → C is (complex-valued) wave function
interpretation: |Ψ(x, t)|2 is probability density of particle at position x,
time t
Z 1
(so
|Ψ(x, t)|2 dx = 1 for all t)
0
evolution of Ψ governed by Schrodinger equation:
!
2
h̄
∇2x Ψ = HΨ
ih̄Ψ̇ = V −
2m
where H is Hamiltonian operator, i =
Example: Quantum mechanics
√
−1
17–2
Discretization
let’s discretize position x into N discrete points, k/N , k = 1, . . . , N
wave function is approximated as vector Ψ(t) ∈ CN
∇2x operator is approximated as matrix

−2
1
 1 −2
1

1 −2
∇2 = N 2 

...

1
...

...
1 −2





so w = ∇2v means
(vk+1 − vk )/(1/N ) − (vk − vk−1)(1/N )
wk =
1/N
(which approximates w = ∂ 2v/∂x2)
Example: Quantum mechanics
17–3
discretized Schrodinger equation is (complex) linear dynamical system
Ψ̇ = (−i/h̄)(V − (h̄/2m)∇2)Ψ = (−i/h̄)HΨ
where V is a diagonal matrix with Vkk = V (k/N )
hence we analyze using linear dynamical system theory (with complex
vectors & matrices):
Ψ̇ = (−i/h̄)HΨ
solution of Shrodinger equation: Ψ(t) = e(−i/h̄)tH Ψ(0)
matrix e(−i/h̄)tH propogates wave function forward in time t seconds
(backward if t < 0)
Example: Quantum mechanics
17–4
Preservation of probability
d
d ∗
2
kΨk =
Ψ Ψ
dt
dt
= Ψ̇∗Ψ + Ψ∗Ψ̇
= ((−i/h̄)HΨ)∗Ψ + Ψ∗((−i/h̄)HΨ)
= (i/h̄)Ψ∗HΨ + (−i/h̄)Ψ∗HΨ
= 0
(using H = H T ∈ RN ×N )
hence, kΨ(t)k2 is constant; our discretization preserves probability exactly
Example: Quantum mechanics
17–5
U = e−(i/h̄)tH is unitary, meaning U ∗U = I
unitary is extension of orthogonal for complex matrix: if U ∈ CN ×N is
unitary and z ∈ CN , then
kU zk2 = (U z)∗(U z) = z ∗U ∗U z = z ∗z = kzk2
Example: Quantum mechanics
17–6
Eigenvalues & eigenstates
H is symmetric, so
• its eigenvalues λ1, . . . , λN are real (λ1 ≤ · · · ≤ λN )
• its eigenvectors v1, . . . , vN can be chosen to be orthogonal (and real)
from Hv = λv ⇔ (−i/h̄)Hv = (−i/h̄)λv we see:
• eigenvectors of (−i/h̄)H are same as eigenvectors of H, i.e., v1, . . . , vN
• eigenvalues of (−i/h̄)H are (−i/h̄)λ1, . . . , (−i/h̄)λN (which are pure
imaginary)
Example: Quantum mechanics
17–7
• eigenvectors vk are called eigenstates of system
• eigenvalue λk is energy of eigenstate vk
• for mode Ψ(t) = e(−i/h̄)λk tvk , probability density
2
|Ψm(t)|2 = e(−i/h̄)λk tvk = |vmk |2
doesn’t change with time (vmk is mth entry of vk )
Example: Quantum mechanics
17–8
Example
Potential Function V (x)
1000
900
800
700
V
600
500
400
300
200
100
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
• potential bump in middle of infinite potential well
• (for this example, we set h̄ = 1, m = 1 . . . )
Example: Quantum mechanics
17–9
lowest energy eigenfunctions
0.2
0
−0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2
0
−0.2
0
0.2
0
−0.2
0
0.2
0
−0.2
0
x
• potential V shown as dotted line (scaled to fit plot)
• four eigenstates with lowest energy shown (i.e., v1, v2, v3, v4)
Example: Quantum mechanics
17–10
now let’s look at a trajectory of Ψ, with initial wave function Ψ(0)
• particle near x = 0.2
• with momentum to right (can’t see in plot of |Ψ|2)
• (expected) kinetic energy half potential bump height
Example: Quantum mechanics
17–11
0.08
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
10
20
30
40
50
60
70
80
90
100
0.15
0.1
0.05
0
0
eigenstate
• top plot shows initial probability density |Ψ(0)|2
• bottom plot shows |vk∗Ψ(0)|2, i.e., resolution of Ψ(0) into eigenstates
Example: Quantum mechanics
17–12
time evolution, for t = 0, 40, 80, . . . , 320:
|Ψ(t)|2
Example: Quantum mechanics
17–13
cf. classical solution:
• particle rolls half way up potential bump, stops, then rolls back down
• reverses velocity when it hits the wall at left
(perfectly elastic collision)
• then repeats
Example: Quantum mechanics
17–14
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
t
1.2
1.4
1.6
1.8
2
4
x 10
N/2
plot shows probability that particle is in left half of well, i.e.,
versus time t
Example: Quantum mechanics
X
k=1
|Ψk (t)|2,
17–15