Download lectures10-11.ppt - Projects at Harvard

Numerical Solution of Quantum
Wave Equations
Non-Relativistic (Schroedinger)
Relativistic (Klein-Gordon, Dirac)
Sauro Succi
The Schroedinger equation
The state of a given non-relativistic particle is decribed by a COMPLEX wavefunction:
y (x;t)
The dynamics is described by the Schroedinger equation (SCE):
Kinetic+Potential Energy:
Initial + Boundary Conditions
y (x;0) = y0 (x) y (±L;t) = 0
+L
Normalization:
2
|
y
(x)
|
dx =1
ò
-L
2
How to derive the SCE?
Classical Hamiltonian:
p2
E = H(p, q) =
+V (q)
2m
Plane waves:
Quanta of momentum/energy
Correspondence Rules:
q«x
Whence:
(q.e.d)
3
The wavefunction
Complex wavefunction
Where S is the action:
S=
ò p dq
Probability Distribution:
r(x) = y *y =| y |2 = R2 = A2 + B2
The (gradient) of the phase gives the velocity/momentum of the wavepacket:
4
Time-Independent SCE
The Schroedinger equation is linear, hence suitable to spectral
decomposition and superposition of basic EIGENfunctions
Hyn = Enyn
This a standard matrix eigenvalue problem of size n=1,N
Note that eigenvalues are real because the Hamiltonian is hermitian
Very often only the first few eigenvalues are required:
Ground state + low-lying excited modes.
The quantization emerges from the Boundary Conditions:
y (L) = 0 Þ kn L = 2p n
Discrete Spectrum
5
Spectral Decomposition
¥
Eigenvalues
y (x, t) = å cn e
-iwnt
yn (x)
Eigenfunctions
n=0
Slowest frequency:
Ground state.
Higher frequencies:
Excitations
Oscillations do not
damp out:
Reversible Dynamics
6
Quantum weirdness: tunnel effect
E < V(x) «ik
Spatial Decay
7
Hydrogen atom
8
Physical observables
Classical observables:
L
A(t) = ò y * (x;t) Ây (x;t)dx º< y * | Ay >
-L
Examples:
Position:
 º x̂ = x
L
x(t) = ò y * (x;t)xy (x;t)dx º< x >
+L
-L
The classical position is the average over the quantum density rho(x)
Potential Energy:
 º V̂ = V(x)
2
|
y
(x)
|
dx =1
ò
-L
L
V(t) = ò y * (x;t)V (x)y (x;t)dx º< V(x) >¹ V(< x > (t))
0
Kinetic Energy:
9
Classical Limit
Generically:
De Broglie wavelength:
Singles out minimum S = Classical physics
Classical size ~ Interaction range
For a proton:
Vth = kBT / m » 2000 m / s
Human: the De Broglie length is below the Planck scale!
10
Heisenberg Principle
Generically:
Where deltax is the spread in the position and deltap the spread in momentum of
the Fourier-transformed wavefunction Psi(p) = \int exp(-ipx/hbar) * psi(x) dx
This is an exquisite consequence of the wavelike nature:
Fully Delocalized: plane wave
f (p) = d (p - p0 )
Fully Localized: Dirac delta
y (x) = d (x - x0 )
11
Numerical solution of the SCE
12
Diffusion-Reaction in imaginary time
¶ty = i[DDy -W(x)]y
¶ty = iLy
L= Liouvillean operator
Self-adjoint: nice properties
The solution is a superposition of non-decaying oscillations
13
(that’s why it is called a wave equation…)
Transfer Operator
y (x;t + h) = e y (x;t) º Th × y
iLh
y (x;t + h) = e
*
y (x;t)
-iLh
*
In QM the propagator is often called U for unitary:
Uh º e
iLh
Remarkable properties:
U-h = U
-1
h
Hence
Uh ×U-h =U-h ×Uh = I
This reflects the reversibility of QM
14
Transfer Operator
y (x;t + h) = e y (x;t) º Th × y
iLh
y (x;t + h) = e
*
y (x;t)
-iLh
*
y * (t + h)y (t + h) = Th*y * (t)Thy (t) = Th*Thy * (t)y (t)
r(t + h) = [T T ] r(t)
*
h h
Unitarity implies:
Th*Th = I
Th* = Th-1 = T-h
This is guaranteed if T = exp(iR), where R is a real symmetric matrix
Unitarity must be preserved by the numerical discretization!
15
Forward Euler
y (t + h) = y (t)+ihLy (t)
Formally:
T Euler = (1+ ihL)
This implies
T *T = (1-ihL)*(1+ihL) = (1+ h2 L2 ) >1
Unconditionally UNSTABLE for any timestep!!!
Note that this is INDEPENDENT of the specific discretization of
the Liouville operator: unlike classical ADR, no space discretization
can cure the problem. Genuinely quantum effect: diffusion in imaginary
16
time is qualitatively different from diffusion in real time.
Forward Euler: tridiagonal form
y (t + h) = y (t)+ihLy (t)
Forward Euler leads to the usual tridiagonal form:
n
n
n
y n+1
=
a
y
+
c
y
+
b
y
j
j-1
j
j+1
With the following complex coefficients:
a = id
b = id
c =1- 2id - if
Where:
d = Dh / d 2
f = Wh
Q: Do they recover the continuum Dispersion Relation?
17
Dispersion Relation
The continuum dispersion relation reads as:
wr = Dk 2 + W(x)
g =0
WKB approximation
Slow potentials…
Pure propagation, no growth/decay: oscillations
Euler forward:
eg h cos(wr h) =1+ f
eg h sin(wr h) = 2d[cos(kd)-1]
e2g h = (1+ f )2 + 4d 2 [cos(kd) -1]2 >1
Unconditionally unstable! q.e.d.
18
What to do?
1. Implicit methods: Crank-Nicolson
2. Explicit: leapfrog time marching
19
Transfer Operator: back to ADR
Let us discuss Implicit methods from a general perspective.
It is then convenient to go back to classical systems.
Back to ADR in Liouville form (no imaginary unit):
¶tj = Lj
L = -U¶x + D¶ + R
2
x
Formal solution:
j (x;t + h) = e j (x;t) º Th × j (x)
Lh
Q: how do we compute the propagator in practice?
20
Transfer Operator: classical
Simplest: power expansion in L*h
j (x;t + h) = [1+ Lh + L h / 2 +...]j (x;t)
2 2
First order: good old Euler forward:
j (x;t + h) = [1+ Lh]j (x;t)
This first-order accurate in time and stable if : ||1+Lh||<1.
This require all eigenvalues of L to obey:
|1+ lh |<1Û -2 < lh < 0
Which is the general form of the well-known CFL conditions!
21
Time Marching: Quadrature
t+h
ò ¶ j (t ')dt ' = j (t + h) - j (t)
Exact!
t'
t
j (t + h) = j (t) +
t+h
ò Lj (t ')dt '
Must approximate
t
On the lattice x_j=j*d:
j j (t + h) = j j (t) +
t+h
òL
j k (t ')dt '
jk
t
Perform the integral numerically
22
Explicit vs Implicit
Using a (generalized) trapezoidal rule:
j (t + h) = j (t)+ h[(1- q )Lj (t)+ q Lj (t + h)]
Instead of taking high order derivatives at present time t (predictors),
Jump ahead into the future!
q =0
Fully Explicit
q =1/ 2
Semi-Implicit (Crank-Nicolson)
q =1
Fully Implicit
t
t+h
23
Theta=0: Euler, no good
y (x;t + h) = e y (x;t) º Th × y
*
-iLh *
y (x;t + h) = e y (x;t)
iLh
First order in time:
e
iLh
=1+ iLh
(1+ iLh)*(1-iLh) =1+ L h >1
2 2
Unconditionally unstable!
24
Theta=1/2: Crank-Nicolson, good
Implicit methods: Crank-Nicolson: Break Causality but gain Unitarity.
Doable because the quantum dynamics is reversible, no causal arrow.
y (t + h) - y (t) = iLh[y (t)+ y (t + h)] / 2
(1-ihL / 2)* y (t + h) = (1+ ihL / 2)* y (t)
This is a MATRIX problem A*x=b:
A* y (t + dt) = B* y (t) = b
|| T ||=1
The scheme is unitary:
-1
T = (1-ihL / 2) *(1+ihL / 2)
*
-1
-1
T = (1+ihL / 2) *(1-ihL / 2) = T
Hence:
TT * = T *T =1
q.e.d.
For any timestep!
25
Theta=1, no good …
-1
Th = (1- ihL)
-1
T-h = (1+ ihL)
Hence
T-h ×Th =1/ (1+ h2 L2 )
The scheme is stable, but non-unitary and only first order accurate.
Since it still involves a matrix problem, no point versus Crank-Nicolson
26
General Theory of Propagators
Pade’ approximants of the exponential
e
ihL
PN (+h / N )
» TN,M (h) =
PM (-h / M )
Where P_N, P_M are polynomials of degree N and M, respectively
Explicit: M=0, Implicit: M>0
Euler:
N=1, M=0
Crank-Nicolson: N=M=1
Fully Implicit: N=0,M=1
Very Important for Path-Integral
formulation of quantum mechanics!
Unitarity requires N=M: Past and Future are symmetric
27
Geometrical summary
eihLy
Continuum: y (t + h) = e y (t) ºUhy (t)
ihL
Discrete:
(1+ ihL)y
y º y (t)
y (t + h) = Thy (t)
(1+ ihL)-1y
y (t + h) =| Th | e * | y (t) | e =| Th | e
*
-iJ
-is
-i(J +s)
y (t + h) =| Th | e * | y (t) | e =| Th | e
iJ
i(J +s)
is
| y (t + h) | =| Th | | y (t) |
2
2
2
The unitary operator “rotates” the wavefunction by an angle hL (orange).
Euler (black) amplifies the “vector” , Fully Implicit compresses it,
Crank-Nicolson (green) rotates by a slighly different angle, 2*tan(hL/2),
28
but still rotates it without changing the amplitude = 1
Crank-Nicolson:
Solving the matrix problem
29
1D: tridiagonal matrices
{a, c, b}
Physical Stencil
(Sparse matrix)
a = id; b = id, c = -i(2d + f )
B = 3; Bandwidth
L jk =
You can use the Dispersion
Relation formalism to
assess the numerical properties …
Nx
30
1d Matrix Problem
Physical Stencil
h
[1+ i L jk ]
2
y (t + h)
h
[1- i L jk ]
2
y (t)
=
This can be solved by any Linear Algebra Solver
In d=1 this can be done algebraically, but in d=2,3
requires O(N^2) operations, unless iterative methods
are used. An extra lecture on this is probably healthy…
31
Back to Explicit
Crank-Nicolson is robust and popular, it allows large dt
because it is free from CFL constraints, but each dt requires
the solution of a linear system, which
scales like N^2 or more unless special methods are used.
If possible, we wish to stay with lightweight explicit methods
which scale like N^1 (but march in much smaller steps).
Doable?
Yes, let’ see how.
32
Visscher method
First, separate the Real and Imag parts:
y = A +iB
¶t A = -LB
¶t B = +LA
é Aù
é0 - L ù é Aù
¶t ê ú = ê
úê ú
ë Bû
ë+L 0 û ë B û
This is a Hamiltonian system, wave equation:
¶tt X = -L X,
2
X = A, B
This is a Hamiltonian system: wave equation.
Not very convenient, though, since L^2 contains
fourth order derivatives…
33
Leapfrog-Visscher method
Time-staggered: 2 time levels, step dt=h
A
B
n+1
j
- A = -hL × B
n+3/2
j
n
j
-B
Euler start-up:
n+1/2
j
n+1/2
j
= hL × A
0
B1/2
B
j
j =+
n+1
j
Lh 0
× Aj
2
0 1 2 …
A-timeline
0
1
B-timeline
1/2
3/2
0
This is the leapfrog scheme for classical Hamiltonian systems.
Stable because it is a second order “classical” system in time.
Undesirable feature: A and B are staggered, therefore
The diagnostics is awkward, for instance how to define
Solution:
r nj = (Anj )2 + (Bnj )2 ?
n+1/2
r nj = (Anj )2 + Bn-1/2
B
j
j
But we can do better…
34
Modified Visscher method
Time-aligned: 3 time levels step 2h
A
n+1
j
B
n+1
j
-A
n-1
j
= -2hL × B
-B
n-1
j
= +2hL × A
Euler start-up:
n
j
n
j
A1j - A0j = -hL B0j
B1j - B0j = +hL A0j
This is also a leapfrog scheme for Hamiltonian systems.
Desirable feature: A and B are aligned, therefore
the diagnostics is well-defined:
r nj = (Anj )2 + (Bnj )2
35
Quantum Mechanics in Fluid form
(Quantum tornado…)
36
Quantum Mech in Fluid form
Eikonal form:
Let:
r(x;t) = log R(x;t)
Simple algebra yields:
yt = {rt +ist }y
yx = {rx + isx }y
yxx = [{rx + isx }2 +{rxx +isxx }]y
The SCE becomes:
izt = -Dzxx +V(x)
Real part:
Imag part:
-st = -D(rx2 - sx2 + rxx )+V
rt = -2D(rx sx + sxx )
37
Quantum Mech in Fluid form
Imag part:
rt = -2D(rx sx + sxx )
Multiply by R^2:
RRt = -2D(RRx sx + R2 sxx )
rt = -2D(rx sx + rsxx ) = -2D¶x (rsx )
Continuity Equation!
The quantum-fluid correspondence is:
1
r = R , u = ¶x S
m
2
38
Quantum Mech in Fluid form
Real part:
-st = -D(rx2 - sx2 + rxx )+V
Take the gradient * 2D
-ut = -D¶x (r - s + rxx )+¶xV
u2
-ut = -¶x ( ) - 2D 2¶x [(rx2 + rxx ) +V ]
2
2
2
2 Rxx
Define the quantum potential: Q = -2D [(rx + rxx ) = -2D
R
This is a perfect fluid with zero pressure
u2
subject to the classical potential V(x)
¶t u +¶x ( ) = -¶x (V + Q)
plus the self-consistent
2
2
x
2
x
quantum potential Q(x)!
39
Quantum potential
The quantum potential is configuration-dependent, hence
It can take very complex shapes in space and time.
It is held responsible for the non-locality of quantum
Mechanics (Bohm’s formulation)
One can solve the SCE
using the methods of
Fluid Dynamics.
The quantum potential
requires special care…
Hence, this is not a mainstream.
40
Time-Independent SE
The Schroedinger equation is linear, hence suitable to spectral
decomposition and superposition of basic EIGENfunctions
H jky = Ely
(l)
k
(l)
k
This a classical matrix eigenvalue problem of size k,l=1,N
Note that eigenvalues are real because the Hamiltonian is hermitian
Very often only the first few eigenvalues are required:
Ground state + low-lying excited modes.
This is a systematic but expensive route: Exact Diagonalization
scales like N^3… (very fast diagonalization methods in d=1).
Very important for quantum many-body problems
41
Major Extensions
Non-linear SCE: optics, Bose-Einstein condensates (Gross-Pitaevski)
V(y ) = g | y |
2
Random potentials: Anderson Localization
N-body Schroedinger: a world of its own (Computational Chemistry)
42
Summary
The Schroedinger equation can be viewed as diffusion-reaction equation in imaginary
time and also like a peculiar fluid equation subject to the quantum potential.
Numerics:
Explicit: Euler is unconditionally unstable
Explicit: Leapfrog cures the problem, but small timesteps
Implicit: Crank-Nicolson is unitary and allows large time-steps, but
it may become expensive unless specialized linear algebra is used
Matter of taste somehow…
Fluid methods must handle the quantum potential with great care
Eigenvalue solvers constantly in progress (Time-Independent SCE)
43
Going Quantum Relativistic!
44
Relativistic Mechanics
E 2 = p2 c 2 + m2c 4
Negative energy allowed!
E
N
E
R
G
Y
PARTICLES
ANTIPARTICLES
MOMENTUM
45
Klein-Gordon Equation
For spinless bosons, the correspondence p=-i*hbar*d/dx, E=i*hbar*d/dt gives
Klein-Gordon equation:
ytt - c Dy = -w y
2
2
c
where
Is the Compton frequency
Traveling oscillations: superposition of left and right movers
y (x;t) = å[A+k ei(kx-w t ) + A-k ei(kx+w t ) ] º y> + y<
k
k
k
dw
kc
V(k) =
= ±c
dk
w (k)
In KGE the L/R modes are permanently mixed, hence |psi|^2
is NOT a pdf because of interference!
46
Klein-Gordon: Disp Relation
Klein-Gordon dispersion relation flows directly from energy-momentum:
w 2 = k 2 c2 + wc2
Withe the standard identification:
It is a statement of Lorentz-invariance, which must be
preserved by the numerical approximation.
The KGE leads to notorious problems with the definition of a suitable
density rho(x): formally possible (thru time-derivatives), but lacks
positive-definiteness (due to antiparticles).
That’s why Schroedinger turned it down!!!
47
It was for Dirac to fix it all…
Klein-Gordon: Numerics
Plain leapfrog is fine: 3° order unitary and accurate
n
n-1
y n+1
2
y
+
y
j
j
j
h2
= c2 [
y nj+1 - 2y nj + y nj-1
d2
]+ w c2y nj
With the standard plane-wave representation, the Discrete DR reads as:
F(wh) = a 2 F(kd)+ m 2
Where:
F(wh) º 2[cos(wh) -1]
F(kd) º 2[cos(kd) -1]
CFL numbers
ch
aº
d
m º wc h
48
Klein-Gordon: continuum limit
Massless particles
Taking
a =1
F(wh) = a 2 F(kd)
gives
wh = kd
namely:
w = kc
This is EXACT for any finite lattice spacing and time-step!
In the continuum limit (massless or massive):
F(wh) º 2[cos(wh) -1] ® -w 2h2
F(kd) º 2[cos(kd) -1] ® -k 2 d 2
m º wc h
whence
Take again
w 2h2 = a 2 k 2 d 2 + wc2 h2
a =1« c = d / h
w 2 = c2 k 2 + wc2 +O(k 3d 3 )
Which is stable and unitary up to third order
49
Full Dirac Equation
Dirac looked for a first order PDE compatible with quantum mechanics
He realized that this is impossible for scalar wavefunctions, but
becomes possible for generalized wavefunctions with internal
degrees of freedom: spinors
In the above:
y ºyi = [y1, y2 … y4s+2 ]
is a spinor of rank 2*(2s+1), where s is the spin of the particle.
Hence alfa and beta are matrices of rank 2*(2s+1).
The extra factor 2 accounts for antiparticles.
The Dirac equation contains KG as a special case and permits
to define a positive-definite probability density.
It looks like a transport equation for “spinning particles”, i.e.
particles which mix their internal state while they
50
propagate. Very useful analogy for numerical methods.
Klein-Gordon vs Dirac
Dirac: Co-evolution of left and right movers (chiral representation)
A sort of sqrt of the Klein-Gordon equation:
ytt - c Dy = -w y
2
2
c
Traveling oscillations: superposition of left and right movers (first order in spacetime):
¶>y> = -wcy<
¶<y< = +wcy>
¶> º ¶t + c¶x
¶< º ¶t - c¶x
Right and Left
cov-derivatives
Apply the left covariant derivative to the right mover:
¶<[¶>y> ] = -¶<[wcy< ] = -wc¶<y< = -w y>
2
c
Since
¶<[¶>y> ] = ¶2y> = -w c2y>
¶2y> = -w 2cy>
qed
51
KG as a relativistic two-fluid
KG in chiral form:
¶>y> = -wcy<
¶<y< = +wcy>
The R/L movers are taken to be real (Particles only)
The total density is the sum of Left and Right “fluids” :
r(x, t) = r< (x;t)+ r> (x;t)
r<,> º| y<,> |2
The chiral representation is more transparent:
y *>¶ty> = -cy *>¶xy> - wcy *>y<
y *<¶ty< = +cy *<¶xy> + wcy *<y>
Sum up:
Since the R/L movers are real
¶t (r< + r> ) = -c¶x (r> - r< )+ -wc (y *>y< - y *<y> )
52
This again the continuity eq. For L/R species moving at \pm c !!!
Klein-Gordon vs Schroedinger
Note that both movers must be accounted for norm conservation
¶>y> = -wcy<
¶<y< = +wcy>
The R/L movers are real
Complexification of the wavefunction: unitary transformation
y± = (
y> ± iy<
2
) e-iwct
It can be shown that the PLUS mode (symmetric) is slow
while the MINUS (antisymmetric) is fast
The slow mode obeys Schroedinger in the limit V/c to zero!
The fast, albeit much smaller in amplitude,
never dies completely out (Zitterbewegung)
53
Quantum Lattice Boltzmann
Start from Left/Right chiral representation, and use “upwind” like discretization:
¶>y> = -wcy<
¶<y< = +wcy>
t +1
t
QLB rule: (in lattice units h=d=c=1)
y> (x +1, t +1) - y> (x, t) = y< (x -1, t +1) - y< (x, t) = -
wc
2
wc
2
x -1
x
[y< (x, t)+ y< (x -1, t +1)]
[y> (x, t) + y> (x +1, t +1)]
It is locally implicit: can be reorganized in explicit form:
54
x +1
Quantum Lattice Boltzmann
By solving the simple 2x2 linear system:
y> (x +1, t +1) = ay> (x, t)- by< (x, t)
y< (x -1, t +1) = ay< (x, t)+ by> (x, t)
1- m 2 / 4
a=
1+ m 2 / 4
m
b=
1+ m 2 / 4
t +1
t
x -1
x
The QLB scheme is unconditionally unitary and second order accurate!
It is an efficient solver of the full Dirac equation, which morphs naturally
Into a Schroedinger solver in the v<<c limit!
55
x +1
Assignements
1. Write an explicit solver of the Schroedinger eq. (d=1 and d=2) for
a. Free particle, b. Harmonic oscillator, c. Potential barrier
d. Random potential (for the brave)
2. Like 1, using the fluid-dynamic formulation
3. Solve the Klein-Gordon equation in classical form
and check the dispersion relation by using plane waves
as initial conditions. d=1 is ok,.
4. Like 3, using Quantum Lattice Boltzmann
56
Free particle
Ballistic loss of coherence:
Proton:
d 2 (t) = d02 (1+
D 2t 2
d
4
0
)
d0 » 10-15 m
Decoherence time:
57
Harmonic oscillator
Natural scales:
t=
1
w
Coherent motion:
d0 = l
A gaussian wavepacket
of width
0
d =l
oscillates according without
any spreading or shrinking
(beware sqrt 2 factors…)
58
Test case 3: scattering barrier
E < V(x) «ik
Spatial Decay
59
End of lecture
60