Numerical Methods for the
Feynman’s path integrals
Assignment for the doctoral course “Feynman Integral in
Quantum Mechanics and Quantum Field Theory” held by prof. Saverio Pascazio
Michele Tuttafesta1
1
Università degli Studi di Bari
February 2011
Abstract
A numerical method freely published in 1995 by Eduardo Mendel et
al. [1] to calculate the Feynman path integrals has been re-implemented.
First the systems analyzed by Mendel, harmonic oscillator and double
well potential, have been tested in some different numerical conditions and after the method has been applied to a potential box and
to a Morse and Lennard-Jones potential. The code has been written
in C++ language and the computing time required by every single
test ranged from one minute to about two hours, using a Pentium(R)
Dual-Core CPU E5400 @ 2.7 GHz 2.7 GHz and a RAM of 4.00 GB.
Introduction
In order to numerically calculate the Feynman path integral many sophisticated methods have been developed since the early years of computer science.
Today the most effective among them are based on the PIMC (Path Integral
Monte Carlo) or using suitable strategies to find the most important class of
paths, e.g. stationary phase approximations or to damp the integrand (e.g.
imaginary time path integration) as used for finite temperature calculations.
In the present work we will show the results of our personal implementation of the numerical method, described in 1995 by Eduardo Mendel et al.
[1] which adopts a direct sum over all paths in configuration space and the
1
real time. So we first reproduce with some different numerical conditions
the same systems analyzed by Mendel (harmonic oscillator and double well
potential) and after we apply the method to calculate the energy spectrum
and the wave function of other systems such as box potential, Morse and
Lennard-Jones potential.
Theoretical basis
Let’s consider a quantum system described by a time-independent Hamiltonian H, then
−iH(t−t0 )
U (t, t0 ) = e ~
is the system time evolution operator. So if |α, t0 i is the state of the system
at the instant t0 then its evolution at the instant t is given by
|α, ti = U (t, t0 )|α, t0 i
In a one-dimensional system and in coordinate representation we can
write
R
|α, ti = R dx0 U (t, t0 )|x0 ihx0 |α, t0 i ⇒
⇒ hx|α, ti =
dx0 hx|U (t, t0 )|x0 ihx0 |α, t0 i
Defining the wavefunction Ψ(x, t) = hx|α, ti and the propagator G(x, t; x0 , t0 ) =
hx|U (t, t0 )|x0 i then the last equation becomes
Z
Ψ(x, t) = dx0 G(x, t; x0 , t0 )Ψ(x0 , t0 )
(1)
The trace (in the coordinate space) of the time evolution operator is
equivalent to the space integral of G, setting the initial and final states equal
and moreover it can be expressed in terms of the eigenvalues {En } of the
system energy; in fact, setting t0 = 0
R
trU (t, 0) =
dxG(x, t; x, 0)
R
P
−iEn t
=
dx n |hx|ni|2 e ~
(2)
P −iEn t
~
=
ne
where {|ni} is a complete set of energy eigenstates.
Since 1942, thanks to work [4] of the legendary american physicist R.P.
Feynman, it is demonstrated that (see also [3] p.7) (one particle system with
mass m)
2
0
G(x, t; x , t0 ) =
lim
N →∞
∆t→0
N ∆t=t−t0
C
N
Z
dx1 · · · dxN −1
N
Y
x
e
i∆t
L
~
n +xn−1 xn −xn−1
,
2
∆t
(3)
n=1
p m
and L(x, ẋ) = 21 mẋ2 − V (x) is the system lagrangian
where C = 2πi~∆t
using the prescription of “midpoint rule” (see [3] p.23).
An important condition we will have to take into account in our calculations is obtained by noting that the propagator is equal to the transition
amplitude, that is (see [2] p.114)
G(x, t; x0 , t0 ) = hx, t|x0 , t0 i
Consequently we must have
Z
dxdx00 G∗ (x, t; x00 , t0 )G(x, t; x0 , t0 ) = 1
(4)
for all x0 . The (4) meaning that the probability to go from one point to any
possible point at a later time has to be 1.
Numerical calculation of the propagator
If we consider an equidistant discretization of time t0 , · · · , tN , ∆t = T /N ,
T = t − t0 and space x0 , · · · , xD , ∆x = (x − x0 )/D and defining the “key of
the method” (D + 1) × (D + 1) matrix
i∆t
xm + x n xm − xn
L
,
Kn,m (∆t) := C exp
(5)
~
2
∆t
then from equation (3) we can approximate G for a fixed N (T = N ∆t
defined as a “characteristic time”) as follows
G(x
Z r , tk + N ∆t; xs , tk ) ≈
≈ dx1 dx2 · · · dxN −2 dxN −1 Kr,1 K1,2 · · · KN −2,N −1 KN −1,s ≈
Z
≈ ∆x dx2 dx3 · · · dxN −2 dxN −1 (K 2 )r,2 K2,3 · · · KN −2,N −1 KN −1,s ≈
Z
2
≈ (∆x)
dx3 dx4 · · · dxN −2 dxN −1 (K 3 )r,3 K3,4 · · · KN −2,N −1 KN −1,s ≈
≈ · · · (∆x)N −1 (K N (∆t))r,s := Gr,s (N ∆t) = Gr,s (T )
3
(6)
noting that the final discretized expression is independent from tk .
So the propagator on a finite time T has became a (D + 1) × (D + 1)
matrix, the scheme of such propagation is shown in Figure 1(a).
In addition with a minimal computational effort one can obtain a propagator at a time nT multiple of T , recursively by the composition law
G(nT ) = G(T )G((n − 1)T )∆x
(7)
whose conceptual scheme is shown in Figure 1(b).
(a) Single path in a finite time transition
described by the propagator Gr,s (T ). The
initial and the final positions xs and xr
are fixed.
(b) Single path in a finite time transition described as a multiple of a base one.
The propagator is obtained by the recursion G(nT ) = G(T )G((n − 1)T )∆x.
Figure 1: Schemes of finite propagations
In order to compensate for numerical errors we have to apply the constrain
(4) which in our matrix approximation translates to
P=
D
X
(∆x)2 Gij∗ Gik −−−−→ 1
D→∞
∆x→0
i,j=0
(8)
for all k.
Energy levels and FFT
Through a Fourier transform of the time evolution operator trace, as one
can deduce from the last right expression in (2), we are able to calculate the
P −iEn t
energy spectrum of the system. Setting f (t) = trU (t, 0) = n e ~ we will
calculate the transform as
Z ∞
1
ˆ
f (ω) =
dtf (t)eiωt
2π −∞
4
that is using the “positive convention” for the sign of the exponent in eiωt
so we should find energy level values as peaks in the fˆ(ω) graph where ω =
En /~. Actually we calculate the discretized time development of the trace
in the interval [T, (NT + 1)T ] (T is our characteristic time, see equation 6)
trU (nT, 0) =
D
X
Gii (nT )∆x;
n = 1, · · · , NT
i=0
using the composition law (7). We then perform an FFT (Fast Fourier Transform) using the appropriate routine of the Gnu Scientific Library named
gsl fft complex radix2 backward (which adopts the above required “positive convention”).
The FFT is an optimized algorithm for the Discrete Fourier Transform
(DFT) that, neglecting aliasing, provides N samples of fˆ(ω) in its major
amplitude interval [0, ωp ] taking in input N samples of f (t) in its major amplitude interval [t0 , t0 + Tp ] so that Tp ωp = 2πN ; the samples are equidistant
∆ω in the ω domain and ∆t in the t domain so that ∆t = TNp = ω2πp and
∆ω = ωNp = 2π
.
Tp
Therefore our recipe to obtain the energy spectrum is summarized as
follows.
1. Set ~ and m to 1 in order to simplify the formalism and the results
validation.
2. Choose the maximum value Emax of the system energy up to which
perform the calculation, therefore must be ωp = Emax .
3. According to the chosen Emax we should be able to evaluate a spatial
limitation for the quantum particle position, that is the interval [x0 , xD ].
4. Set the numbers (quite arbitrarily): N for the evaluation of (6) (N
should be a power of 2 in order to make recursive matrix multiplication possible), D for the spatial discretization, NT for the trace time
sampling (NT should be a power of 2 in order to make FFT possible).
Wavefunctions
Through our numerical method the equation (1) can be discretized as follows
Ψ(xi , nT ) =
D
X
∆xGij (nT )Ψ(xj , 0)
j=0
with n = 1, · · · , NT and the (D + 1) vector Ψ(xj , 0) given as input.
5
(9)
Harmonic oscillator
We apply the numerical method described in the previous sections to the harmonic oscillator, that is at the system described by the lagrangian L(x, ẋ) =
1
mẋ2 − 12 kx2 . Actually setting m = k = 1 we consider
2
1
1
L(x, ẋ) = ẋ2 − x2
2
2
(10)
We suppose (remember the setting ~ = 1) the system spatially confined
with En ≤ Emax = ωp = 8. Furthermore supposing Emax = Vmax = 12 x2max
then it must be −4 ≤ x ≤ 4.
The results are grouped in Figure 2. The location of the peaks in the
energy spectrum is seen to be in excellent agreement with the theoretical
energies
1
n = 0, 1, · · ·
En = n +
2
Double well potential
Another interesting application for our method is describing a quantum particle in the following potential
V (x) = α(x − xmin )2 (x + xmin )2
also known as “mexican hat potential”. From the calculated energy spectrum
in Figures 3(a,b) one can infer that the system has two lowest and closed
energy levels, named ES and EA , that are probably associated to a symmetric
and antisymmetric states respectively. Such levels are below the threshold in
the potential center (Figure 3(c)) then a classical particle with that energy
is not able to leave one of the wells. But this is possible in the quantum case
through the so called “tunnel effect” as shown in Figure 3(d).
The results in Figure 4 show that in this test case enhancing computational effort does not provides additional significant informations.
6
(a) Energy spectrum - linear scale
(b) Energy spectrum - log scale
(c) Transition amplitude
(d) Time evolution of wavefunction norm |Ψ(x, t)|2
and potential energy V (x)
Tp
NT
xmin
xmax
D
ωp
N
NT
T =
-4
4
100
8
4
256
0.785
=
2π
ωp
V (x)
1 2
x
2
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs ) ; α = 2; xs = 0.8
π
Figure 2: Harmonic oscillator.
7
(a) Energy spectrum - linear scale.
(b) Energy spectrum - log scale.
(c) Potential energy
(d) Time evolution of wavefunction norm
xmin
xmax
D
ωp
N
NT
T
-5
5
150
7.5 2
128
0.838
V (x)
α(x − xm )2 (x + xm )2
α = 0.02; xm = 2.5
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs )
π
α = 2; xs = 2
Figure 3: Double well potential - Test 1: minimal computational effort.
8
(a) Energy spectrum - linear scale.
(b) Energy spectrum - log scale.
(c) Potential energy
(d) Time evolution of wavefunction norm
xmin
xmax
D
ωp
N
NT
T
-6
6
200
16
2
512
0.393
V (x)
α(x − xm )2 (x + xm )2
α = 0.02; xm = 2.5
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs )
π
α = 2; xs = 2
Figure 4: Double well potential - Test 2: enhanced computational effort.
9
Particle in a box
As a further validation of the method we consider a particle in a one-dimensional
box of length a. The potential energy of the system is
0
|x| < a/2
V (x) =
+∞ |x| ≥ a/2
The theoretical energy levels are given by En =
and a = 32 π then we have
~2 n2 π 2
.
2ma2
Setting ~ = m = 1
2
En = n2 n = 1, 2, 3, · · ·
9
so the first five values are 0.22, 0.89, 2.0, 3.55, 5.55. We chose to perform
our calculations up to ωp = 70. The results are grouped in Figure 5. The
location of the calculated energy peaks in Figure 5(a) are in good agreement
with the theoretical ones.
Morse potential
It is a convenient model for the potential energy of a diatomic molecule and
has the following expression
2
V (r) = d 1 − e−a(r−re )
where r is the distance between the
atoms, re is the equilibrium bond distance, d is the well depth (defined relative to the dissociated atoms), and a
controls the “width” of the potential
(the smaller a the larger the well).
In order to a comparison with the
harmonic oscillator we set r − re = x
so we have
V (x) = d 1 − e−ax
2
Expanding the last potential expression up to second order then one obtains V (x) ≈ da2 x2 , which suggests us to set d = 8 and a = 1/4 so that
V (x) ≈ 21 x2 . Thus our Morse potential is
x 2
V (x) = 8 1 − e− 4
10
Moreover it is known that the theoretical energy levels for such system are
2
p
[~ω0 (n+ 21 )]
1
En = ~ω0 (n + 2 ) −
2d/m. In our case, setting
,
with
~ω
=
a
0
4d
~ = m = 1, we have
2
(n + 12 )
1
En = n + −
2
32
n
En
0
1
2
3
4
5
6
0.49 1.43 2.3 3.12 3.87 4.55 5.18
7
5.74
···
···
Table 1: Energy levels for Morse potential
The results of our numerical method applied to this case are shown in
Figure 6 and for instance we note an excellent agreement between the locations of the computed energy peaks (Figure 6(b)) and the theoretical ones in
Table 1.
Lennard-Jones potential
Now we test the Lennard-Jones potential (also referred to as the L-J potential, 6-12 potential, or 12-6 potential) that is a mathematically simple
model that approximates the interaction between a pair of neutral atoms or
molecules.
The base expression from which
we start is (see Figure on the right)
x 6 xm 12
m
−2
V (x) = d
x
x
where x is the distance between the
atoms, xm is the equilibrium bond
distance, d is the well depth (defined
relative to the dissociated atoms);
1
moreover one can write xm = 2 6 σ,
where σ is a “characteristic” dimension of the atoms.
Such model is empiric and (we know) nobody has a finite expression for
the associated propagator. In order to compare the results of the simulation
we will perform on this system with the ones we obtained about harmonic
oscillator and Morse potential then we transform the above expression of the
11
potential so to have the minimum in the origin of our reference frame. So we
will refer to the following form
"
12
6 #
xm
xm
V (x) = d
−2
+d
x + xm
x + xm
1
1 72 d 2
Moreover V (x) ≈ V (0) + V 0 (0)x + V 00 (0)x2 =
x.
2
2 x2m
Now we require that V (x) → 12 x2 for small x and the well depth d be the
same of the one we have just used
√ for the Morse potential.
d
=
1
⇒
x
=
So d = 8 ∧ 72
576 = 24.
m
x2m
In the results shown in Figure 7 we emphasize the comparison with the
ones obtained in the Morse potential case.
Conclusions
The numerical method applied in this work is certainly not “optimized” in
the matter of required computational resources. In particular there is a severe
limitation in order of the spatial dimension of the system one can consider.
However the results obtained from the examined simple quantum systems
are accurate enough, then we hope that in a not so far future an improving
in the computer capability and parallel architectures (GPU with CUDA for
simple personal computers) will give new importance to such method.
References
[1] Mendel E. Dullweber A., Hilf E.R. Simple quantum mechanical phenomena and the feynman real time path integral. quant-ph/9511042,
november 1995.
[2] Sakurai J.J. Modern Quantum Mechanics. Addison-Wesley Publishing
Company, Inc., 1994.
[3] Schulman L.S. Techniques and Applications of Path Integration. John
Wiley & Sons, New York 1981.
[4] Feynman R.P. A New Approach to Quantum Theory. PhD thesis, Princeton University, 1942.
12
(a) Energy spectrum - log scale. The vertical arrows
show theoretical peaks at 0.22, 0.89, 2.0, 3.55, 5.55.
(c) First steps in the time evolution of
wavefunction norm.
(b) Transition amplitude
(d) Time evolution of wavefunction norm up to a
nearly stationary state starting at t = 435 T
xmin
xmax
D
ωp
N
NT
T
− 34 π
3
π
4
100
70
2
512
0.0897
V (x)
0
|x| < 34 π
+∞ |x| ≥ 43 π
Figure 5: Particle in a box.
13
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs )
π
α = 10; xs = 0.5
(a) Energy spectrum - linear scale.
(b) Energy spectrum - log scale. Theoretical numeric
data are on graph top.
(c) Morse potential compared with the harmonic one.
(d) Time evolution of wavefunction norm
xmin
xmax
D
ωp
N
NT
T
V (x)
-2.5
8
100
6
2
256
1.05
8 1 − e− 4
x
2
Figure 6: Morse potential.
14
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs )
π
α = 2; xs = 0.8
(a) Energy spectrum - linear scale.
(b) Energy spectrum - log scale. Dash line and numeric data on graph top refer to Morse potential.
(c) Lennard-Jones potential compared with the harmonic one.
(d) Time evolution of wavefunction norm
xmin
xmax
D
ωp
N
NT
-2.37
9.55
100
6
2
256
T
V (x)
d [p(x)12 − 2p(x)6 ] + d
1.05
xm
p(x) = x+x
; d = 8; xm = 24
m
Figure 7: Lennard-Jones potential.
15
Ψ(x, 0)
1
2
α 4 −α
e 2 (x−xs )
π
α = 2; xs = 0.8
