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从最小作用到概率：通过相位累积与矢量求和可视化光的传播
作者:
Flag137
简介:
用Python和Blender模拟光的传播
最后修改:
2025-06-19 17:34:24.000420
文章状态:
已发布
标签:
数学
编程
物理
✨实验海报
A
Two approaches w
e
r
e im
plemented:
1. A Python numerica
l
method computing path probabilities
us
ing wave
function superposi
tion.
2. A geometric si
mulati
on using Blender'
s
Geometry Node Editor for
The probabilistic na
t
ur
e
of quantum mechanics
di
f
f
ers
fundamentally
from cl
a
ssical d
e
t
e
r
m
i
ni
s
t
ic physics. This paper pres
ents
a vis
ualization
using Richard F
e
ynm
a
n
'
s pa
t
h int
e
gra
l
f
o
r
m
ul
a
t
i
on, s
howing that
particles contribute mos
t
strongly to path
s
ne
a
r
c
l
a
s
s
ical trajectories
,
Abstract
while non-
classical paths yield negligible
pr
oba
bi
l
i
ties
.
interact
ive visua
l
i
z
a
t
i
on.
Results show th
at t
he
P
ython simulation a
c
hi
e
ve
s
theoretical cons
is
tency
using vec
t
or
sum
mation (|v|² formulation
)
, w
hi
l
e
B
lender provides
enhance
d geometric int
ui
tion but faces lim
i
t
a
t
i
ons
in dynamic path
control and numer
i
c
a
l
pr
ecision.
These complementary a
pproaches highlight the importance of both
computational a
ccur
a
cy and visual interp
r
e
t
a
bi
l
i
t
y in unders
tanding
From Least Action to Probabilit
y:
Vis
ualizing
Light Propagation with Phase Accumulation and
Vector Summ
a
tion
Author: Jiaqi
D
e
ng
Date: June 18, 2025
This study ex
plor
es
the visualizati
on of qua
ntum proba
bility a
mplitude
for ph
ot
on propa
ga
t
i
on us
i
ng F
e
ynm
a
n'
s
pa
t
h i
nt
e
gra
l
formulation. It combines Python n
um
e
ric
a
l s
imula
tion a
nd B
le
nde
r ge
ome
try node
-ba
s
e
d s
i
m
ul
a
t
i
on t
o de
m
ons
t
ra
t
e
how
classical prin
ciples
e
volve into quantum me
c
ha
nic
a
l de
s
c
riptions
of light prop
a
ga
t
i
on.
quantum phenomena, pa
r
t
icularly for edu
c
a
t
i
ona
l
purpos
es
.
Quantum prob
a
bi
l
i
t
y a
m
pl
i
t
ude
i
s
c
a
l
c
ul
a
t
e
d us
i
ng F
e
ynm
a
n's
pa
t
h
Quantum
P
r
obabil
it
y Am
pl
i
t
ude
Calcula
t
ion
integral formu
l
a
t
i
on:
P
=
|
Σ
e
^
(iS
/
ℏ
)
|²
Key conce
pt
s
:
- P
has
e
φ
=
S
/
ℏ
- F
or practical
c
om
put
a
t
i
o
n, pha
s
e
i
s
br
oke
n i
nt
o c
os
i
ne
a
nd s
i
ne
components
:
R
e
l
a
t
ive P
roba
bi
l
i
t
y (
Σ
c
os
φ
)
²
+
(
Σ
s
i
n
φ
)
²
- Wa
ve numbe
r
k =
2
π
c
/
λ
- P
has
e
φ
=
kt
=
k * (
t
ot
a
l
pa
t
h l
e
ngt
h /
s
pe
e
d)
M
athemati
c
al N
ot
a
t
i
on:
- S
:
Action (o
pt
i
c
a
l
pa
t
h)
- n: R
e
frac
t
ive
i
nde
x
-
λ
: Waveleng
t
h
- c: S
peed of li
ght
- t: Tim
e
- h: P
lanc
k co
ns
t
a
nt
- : R
e
duced P
l
a
nc
k c
ons
t
a
nt
The appr
oach br
i
dge
s
c
l
a
s
s
i
c
a
l
l
e
a
s
t
-
a
c
t
i
on pr
i
nc
i
pl
e
w
i
t
h qua
nt
um
pa
t
h
probabilit
y c
o
m
put
a
t
i
on.
- Sim
ulates path
probabili
ty using quantum phas
e
s
uperpos
ition
Python Sim
ulation
- Interactive GU
I
w
ith s
liders for:
·
Number of pa
t
hs (de
f
ault: 10)
·
Path perturbation (ra
ndom
nes
s
)
·
Draw
ing spe
ed
- Calculates pro
babil
i
ty via:
Probability (
Σ
cos
φ
)
² + (
Σ
sin
φ
)²
(where
φ
= kt
=
pha
s
e
f
rom action S)
Key Results:
Randomi
z
ed pa
t
hs ca
nc
el via destructive
i
nt
e
r
f
e
r
e
n
ce
Higher path co
unts r
educe overall probability
Mathe
m
a
t
ically
r
i
gor
ous implementation
Blender
Simula
t
i
on (
G
e
ometry Node Wo
r
kf
l
ow
)
:
- Creates 30 ran
domized paths with adjustable curvature
- Embedded node
l
ogi
c
:
1. Com
putes pa
t
h l
e
ngt
h time phase
φ
2. Vector summ
a
t
i
on of
e^(i
φ
) phases
3. Final probab
i
l
ity from |
Σ
e^(i
φ
Blender Simulation
)|²
Visualiza
t
ion Fe
atures
:
Real-tim
e
3D p
a
t
h e
di
ting
Geometric intui
tion for phase accumulat
i
on
Limitations:
Fixed path count
(
30 pa
t
hs only)
Floating-point
prec
i
s
i
on issues
Manual node dupli
c
ati
on instead of loops
This
r
eport com
pa
r
e
s
t
w
o i
m
pl
e
m
e
nt
a
t
i
ons
of
qua
nt
um
pa
t
h pr
oba
bi
l
i
t
y
Conclu
s
ion
vis
ua
l
iza
t
ion:
- P
yt
hon S
im
ul
a
t
i
on:
- High nume
r
i
c
a
l
pr
e
c
i
s
i
o
n a
nd c
ons
i
s
t
e
nt
w
i
t
h t
he
or
y
- Accur
ately de
m
ons
t
r
a
t
e
s
i
nt
e
r
f
e
r
e
nc
e
e
f
f
e
c
t
s
w
i
t
h i
nc
r
e
a
s
i
ng pa
t
h
variabilit
y
- S
ui
table f
or
s
c
i
e
nt
i
f
i
c
a
na
l
ys
i
s
a
nd t
e
a
c
hi
ng qua
nt
um
pr
oba
bi
l
i
t
y
- B
l
e
nder S
imul
a
t
i
on:
- S
t
r
ong geo
m
e
t
r
i
c
i
nt
ui
t
i
on a
nd r
e
a
l
-
t
i
m
e
i
nt
e
r
a
c
t
i
on
- Vis
ua
l
a
ppe
a
l
a
nd f
l
e
xi
bi
l
i
t
y i
n pa
t
h e
di
t
i
ng
- Limit
e
d by f
i
xe
d pa
t
h c
ount
, pr
e
c
i
s
i
on i
s
s
ue
s
, a
nd i
nt
e
r
f
a
c
e
c
ons
t
r
a
i
nt
s
F
indings
:
- C
l
a
s
s
ical pat
hs
dom
i
na
t
e
pr
oba
bi
l
i
t
y a
m
pl
i
t
ude
- Non-clas
s
i
c
a
l
pa
t
hs
c
a
nc
e
l
out
vi
a
de
s
t
r
uc
t
i
ve
i
nt
e
r
f
e
r
e
nc
e
- Numerical p
r
e
c
i
s
i
on a
nd
ge
om
e
t
r
i
c
vi
s
ua
l
i
z
a
t
i
on pl
a
y c
om
pl
e
m
e
nt
a
r
y r
ol
e
s
in teaching qua
nt
um
c
onc
e
pt
s
F
uture
Work:
- Hybrid approa
c
he
s
i
nt
e
gr
a
t
i
ng P
yt
hon's
num
e
r
i
c
a
l
s
t
r
e
ngt
h w
i
t
h B
l
e
nde
r
's
vis
ua
l
iza
t
ion
- Expans
ion int
o r
e
l
a
t
i
vi
s
t
i
c
or
m
ul
t
i
-
pa
r
t
i
c
l
e
s
ys
t
e
m
s
- Educa
t
ional
r
e
s
our
c
e
s
ha
r
i
ng t
o pr
om
ot
e
e
xpe
r
i
m
e
nt
a
t
i
on a
nd l
e
a
r
ni
ng
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