We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
A complete list of Uq(sl2)-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed. The classical limits of the Uq(sl2)-module algebra structures are discussed.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
On New Root Finding Algorithms for Solving Nonlinear Transcendental EquationsAI Publications
In this paper, we present new iterative algorithms to find a root of the given nonlinear transcendental equations. In the proposed algorithms, we use nonlinear Taylor’s polynomial interpolation and a modified error correction term with a fixed-point concept. We also investigated for possible extension of the higher order iterative algorithms in single variable to higher dimension. Several numerical examples are presented to illustrate the proposed algorithms.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
A new derivation of the Black Scholes Equation (BSE) based on integral form stochastic calculus is presented. Construction of the BSE solution is based on infinitesimal perfect hedging. The perfect hedging on a finite time interval is a separate problem that does not change option pricing. The cost of hedging does not present an adjustment of the BS pricing. We discuss a more profound alternative approach to option pricing. It defines option price as a settlement between counterparties and in contrast to BS approach presents the market risk of the option premium.
equity, implied, and local volatilitiesIlya Gikhman
In this paper we present a critical point on connections between stock volatility, implied volatility, and local volatility. The essence of the Black Sholes pricing model is based on assumption that option piece is formed by no arbitrage portfolio. Such assumption effects the change of the real underlying stock by its risk neutral counterpart. Market practice shows even more. The volatility of the underlying should be also changed. Such practice calls for implied volatility. Underlying with implied volatility is specific for each option. The local volatility development presents the value of implied volatility.
In this paper, we present somewhat alternative point of view on early exercised American options. The standard valuation of the American options the exercise moment is defined as one, which guarantees the maximum value of the option. We discuss the standard approach in the first two sections of the paper. The standard approach was initially presented in the papers [3] - [7]. Our idea is that the exercise moment of the American call / put options is defined by maximum / minimum value of underlying. It was shown that at this moment exercise and sell prices are equal.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
Abstract. Regulations of the market require disclosure of information about the nature and extent of risks arising from the trades of the market instruments. There are several significant drawbacks in fixed income pricing modeling. In this paper we interpret a corporate bond price as a random variable. In this case the spot price does not a complete characteristic of the price. The price should be specified by the spot price as well as its value of market risk. This interpretation is similar to a random variable in Probability Theory where an estimate of the random variable completely defined by its cumulative distribution function. The buyer market risk is the value of the chance that the spot price is higher than it is implied by the market scenarios. First we quantify credit risk of the corporate bonds and then consider marked-to-market pricing adjustment to bond price. In the case when issuer of the corporate bond is the counterparty of the bond buyer counterparty and credit risks are coincide.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
What is greenhouse gasses and how many gasses are there to affect the Earth.moosaasad1975
What are greenhouse gasses how they affect the earth and its environment what is the future of the environment and earth how the weather and the climate effects.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
1. Stochastic Approach to Construction of the Schrödinger Equation Solution and its Applications to
Demolition Measurements.
Ilya Gikhman
6077 Ivy Woods Court
Mason, OH 45040 USA
Ph. (513)-573-9348
Email: ilyagikhman@mail.ru
This paper represents §4 of the 6 chapter of the book
STOCHASTIC DIFFERENTIAL EQUATIONS AND ITS APPPLICATIONS: Stochastic analysis of the
dynamic systems.
Paperback: 252 pages
Publisher: LAP LAMBERT Academic Publishing (July 13, 2011)
Language: English
ISBN-10: 3845407913
ISBN-13: 978-3845407913
The paper is presented in http://www.slideshare.net/list2do/stochastic-schrdinger-equations
The original idea to use complex coordinate space for interpretation of the quantum mechanics was first
represented by Doss, Sur une resolution stochastique de' l'equation de Schrödinger a coefficients
analytiques. Communications Mathematical Physics 73, 247-264, (1980).
Haba, J. Math. Phys.35:2 6344-6359, 1994, J. Math. Phys.39:4 1766-1787, 1998 extended Doss approach
for more general quantum mechanics problems.
Il. I. Gikhman, Probabilistic representation of quantum evolutions. Ukrainian Mathematical Journal,
volume 44, #10, 1992, 1314-1319 (Translation. Probabilistic representation of quantum evolution.
Ukrainian Mathematical Journal, Springer New York, Volume 44, Number 10, October, 1992, 1203-
2. 1208), A quantum particle under the forces of “white noise” type. Ukrainian Mathematical Journal,
Volume 45, #7, 1993, 907-914. (Translation. Ukrainian Mathematical Journal, Springer New York,
Volume 45, Number 7 / July, 1993, 1004-1011. I unfortunately did not know Doss paper at that time and
did not make reference on his original paper.
Relevant mathematical problems were discussed in: S.Albeverio, V. Kolokol'tsov, O. Smolyanov, C.R.
Acad. Sci. Paris, t. 323, Ser. 1, ( 1996 ), 661-664; V. Kolokoltsov, Lecture Notes in Mathematics v.1724,
(2000); I. Davis, O.Smolyanov, A. Truman, Representation of the solutions to Schrödinger Stochastic
Equations on compact Riemannian manifolds, Doklady Mathematics, v.62, No1, 2000, 4 – 7.
Bearing in mind a formal construction of the quantum trajectories one could study the problem related to
the quantum non-demolition continuous measurements. The correspondent problems was actively studied
in physics [M.B. Mensky, Decoherence and the theory of continuous measurements. Physics-Uspekhi 41
(9) 923-940 (1998) , 1998 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences].
Introduction.
In this paper we represent a class SPDE. It is stochastic Schrödinger equations. Two peculiarities
distinguish Schrödinger equation. The Schrödinger equation is linear with complex coefficients. The
linearity of the equation simplifies standard analytic methods are used but on the other hand, the complex
coefficients present mathematical difficulties which did not appear with the real valued partial differential
equations. We first introduce a stochastic representation of the Schrödinger equation solution. We will
show that ‘quantum’ particle associated with the Schrödinger equation represents a browning motion
in a complex coordinate space. The formula, which we present here for the solution differs from the
famous Feynman formula. It is clear that two different formulas for the wave equation should be equal.
We present next that the proof of equivalence of the complex space probabilistic representation and
Feynman formula. The last result of this section we show how choosing nonstandard type of Lagrangian
for the particle in the potential field to arrive at the probabilistic density of the particle. It looks as a
significant advantage with respect to the standard Lagrangian approach, which is used to use in quantum
mechanics.
On the first step, we recall the Feynman representation. The Lagrangian for a system of N particles with
masses m j , j =1 , 2 , .... N moving in a potential field V ( t, x )is
)t,
td
xd
,x(L =
N
1j2
1
m j (
td
xd j
) 2
– V ( x , t ) (6.4.1)
Here x = ( x ( 1 )
, x ( 2 )
, x ( 3 )
) is appoint in 3-dimensional Euclidean space. The function x j ( t ) represents
a path of the j-th particle and
td
xd j
is its velocity along the path.
3. Let ψ ( x , s ) denote a probability amplitude of finding the particle at the location x at the time s
. The function K ( x , s | y , t ) represents the kernel of the transformation of the wave function ψ over a
period [ t , s ]. In other words K is the transition of amplitude for a quantum particle emitted at ( x , s )
and then be detected at ( y , t ). Formally, this definition could be written in the form
ψ ( y , s ) = ∫ K ( x , s | y , t ) ψ ( x , t ) d x (6.4.2)
This equation (6.4.2) is equivalent to the Schrödinger equation. The problem is to find analytical
formula for the kernel. Following a Dirac idea, Feynman suggested to represent the kernel as
K ( x , s | y , t ) =
t
s
y)t(x
x)s(x
exp)(xd
h
i
L ( x ( r ) ,
rd
)r(xd
, r ) d r (6.4.3)
where the right hand side of (6.4.3) is interpreted as following
0ε
lim
… exp
h
i
S
A
xd
....
A
xd 1-N1
, A = 2
1
)
m
επ2
(
hi
(6.4.4)
and S =
t
s
L ( x ( r ) ,
rd
)r(xd
, r ) d r is the functional of action. In Schrödinger quantum mechanics
the complex-valued ψ ( x , t ) satisfies the Schrödinger equation
i h 2
2
j
N
1j x
ψ
m2t
ψ
h
+ V ( x , t ) (6.4.5)
and | ψ ( x , t ) | 2
is the probability density of the presence of the particles at the points
4. x = ( x 1 , x 2 , … , x N ) at time t. First, let us establish the probabilistic representation of the solution of
the Schrödinger equation (6.4.5) by using diffusion in complex space. Putting
z = x + i y , i = ; x , y E 3
define a random process
j ( s ; t , z j ) = z j + 2
1
j
)
m
(
hi
[ w j ( s ) – w j ( t ) ] =
(6.4.6)
= x j + 2
1
j
)
m2
(
h
[ w j ( s ) – w j ( t ) ] + i { y j + 2
1
j
)
m2
(
h
[ w j ( s ) – w j ( t ) ] }
where w j ( t ) are mutually independent Wiener processes. From the formula (6.4.6) it follows that for
given point z j = xj + i y j and t > 0 the coordinate space of the random process
j ( s ; t , z j ) is a linear manifold in the complex 3-dimensinal space. This linear manifold is the set of
direct lines in each coordinate plane x ( k )
, y ( k )
going through the point
x )k(
j , y )k(
j and having slope
4
π
, k = 1, 2, 3 ; j = 1, 2, … , N. It follows from (6.4.6) that each of the
functions j ( s ; t , z j ) is analytic with probability 1 with respect to the complex variable z j . We can call
the complex process j ( s ; t , z j ) by characteristics of the Schrödinger equation (6.4.5). On the other
hand the random function j ( s ; t , z j ) represent the quantum particle movement. The values of the
wave function ψ give us the complete information about the quantum particle system. Knowledge of
the function ψ in quantum mechanics is similar to the knowledge of the trajectory in classical
mechanics.
Theorem 6.4.1. Assume that nonrandom vector and scalar functions Ψ0 ( z ) , V ( t , z ) defined for ( t , z
) [ 0 , + ∞ ) Z3 N
= Z are analytic with respect to z and continuous in t . Then the function
Ψ ( z , t ) = E Ψ0 ( ( 0 ; t , z )) exp V ( ( s ; t , z ) , s ) d s (6.4.7)
1
N3
t
h
i
0
5. is a classical solution of the Cauchy problem
)t,z(
m2t
)t,z( 2
j
j
N
1j
hi
–
h
i
V ( x , t ) ( z , t )
(6.4.8)
( z , 0 ) = 0 ( z )
Recall that complex-valued function Ψ ( z , t ) is said to be analytic on an open area if it has derivative at
every point of the area. One possible way to justify analyticity is to test that the function does not
depend on z = x – i y . Other way to prove analyticity of the function is to verify the Cauchy-Riemann
equations. If the first partial derivatives of the real and imaginary parts of the complex function are
continuous at a point and satisfy the Cauchy-Riemann equations then the function is analytic at this
point. For illustrative simplicity consider a simple example to justify analyticity. Putting N = 1 , V = 0
and setting
Ψ ( z , t ) = P ( x , y , t ) + i Q ( x , y , t )
t 0 we see that
P ( x , y , t ) =
= E P ( x + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , y + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , 0 )
Q ( x , y , t ) =
= E Q ( x + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , y + (
m2
h
) 2
1
[ w ( s ) – w ( t ) ] , 0 )
6. One can easy to see that first order partial derivatives of the functions P ( x , y , t ) , Q ( x , y , t ) , t > 0
are equal to the correspondent derivatives of these functions at t = 0. Hence, Cauchy-Riemann
equations
x
)t,y,x(P
=
y
)t,y,x(Q
,
y
)t,y,x(P
= –
x
)t,y,x(Q
are valid. Applying Cauchy-Riemann equations one can easy justify that
m4
h
(
x
+
y
) 2
P ( x , y , t ) = –
m2
h
2
2
x
)t,y,x(Q
m4
h
(
x
+
y
) 2
Q ( x , y , t ) = –
m2
h
2
2
x
)t,y,x(P
Hence,
t
[ P + i Q ] =
m2
hi
[ P + i Q ]
Putting z = x + i 0 in (6.4.7) we verify that ψ ( x , t ) = Ψ ( x + i 0 , t ) is a solution of the Schrödinger
equation with initial condition ψ ( x , 0 ) = Ψ0 ( x ). Thus, Schrödinger equation can be interpreted as
the complex trace of backward Kolmogorov equation on the real subspace of the complex-valued
quantum diffusion. Recall that the wave function representing solution of the Schrödinger equation
admits the probabilistic interpretation. The feasibility of this interpretation stems from normalization
condition
7. | ψ ( x , t ) | 2
d x = 1
taking place for each t 0. The initial condition of the Schrödinger equation is a complex-valued function
ψ0 ( x ), x ( – ∞ , + ∞ ) sa sfying normaliza on condi on. On the other hand recall that the function Ψ
0 ( z ) in (6.4.8) was assumed analytic in the complex space. Next is a formal result that states a
possibility of a continuation of a complex-valued function defined on the real space onto complex
extension.
Theorem ( Polya, Plancherel ). In order to a complex-valued function φ ( z ) be an integer function of the
exponential type and | φ ( x + i 0 ) | 2
dx < , it is necessary and sufficient that following
representation takes place
( z ) = ( 2 ) 2
n
n
E
( q ) exp – i ( q , z ) d q
where is a function from the Lebesgue space L2 ( E n
) having a compact support.
Introduce inverse time Wiener process ( s ; t , x ) starting at the moment t from the point x
( s ; t , x ) = x + [ w ( s ) – w ( t ) ]
Then the complex process describing quantum particle evolution admits representation
( s ; t , x ) = x + (
m2
hi
) 2
1
[ ( s ; t , x ) – x ]
Using inverse time Markov property of the process the formula (6.4.7) can be representated as
( x , t ) = V ( x , t ) ×
h
i
explim
0
8. ×
exp V ( x + (
m2
hi
) 2
1
( x N – 1 – x ) , t N – 1 ) p ( t , x ; t N – 1 , x N – 1 ) d x N – 1 ×
(6.4.9)
×
exp V ( x + (
m2
hi
) 2
1
( x 1 – x ) , t 1 ) p ( t2 , x 2 ; t 1 , x 1 ) d x 1 ×
×
0 ( x + (
m2
hi
) 2
1
( x 0 – x ) ) p ( t 1 , x 1 ; t 0 , x 0 ) d x 0
where = t j + 1 – t j , j = 0 , 1 , … , N – 1 , s = t0 < t1 < …< tN = t and
p ( t , x ; s , y) = [ 2 π ( t – s ) ] exp –
)st(2
)yx( 2
Remarkably, that the wave function (6.4.9) does not coincide with Feynman path integral
(6.4.2-4). Next theorem states the equivalence of these two representations. Using this theorem, we
could see that non-relativistic quantum mechanics can be interpreted as the real-world trace of the
complex-valued dynamics. We make some technical assumptions that technically will simplify the proof.
Theorem 6.2.2. Assume that V = 0 , N = 1 and let the dimension of the coordinate space is 1. Then
representations (6.4.2-4) and (6.4.9) are equivalent.
Proof. Taking into account conditions the formula (6.4.9) could be rewritten
( x , t ) = E 0 ( ( s ; t , x ) ) =
(6.4.10)
= [ 2 π ( t – s ) ] 0 ( x + (
m2
hi
) 2
1
λ ) exp –
)st(2
λ 2
d λ
h
i
h
i
2
3 N
2
1
9. Introdice Fourier transforms
( y ) = ( 2 π ) – 1
e – i x y
0 ( x ) d x
0 ( x ) =
( y ) e i x y
d y
and suppose that the function ( y ) has a compact support. Then changing the order of the integration
in (6.4.10) we note that
( x , t ) = [ 2 π ( t – s ) ] 2
1
exp –
)st(2
λ 2
d λ
( y ) ×
× exp { [ x + (
m2
hi
) 2
1
λ ) ] y } d y = [ 2 π ( t – s ) ] 2
1
( y ) d y ×
×
exp { –
)st(2
λ 2
+ i [ x – (
m2
hi
) 2
1
λ ] y } d λ = [ 2 π ( t – s ) ] 2
1
×
×
( y ) ei x y
d y
exp { –
)st(2
λ 2
+ (
m2
hi
) 2
1
λ y } d λ =
=
( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y [ 2 π ( t – s ) ] 2
1
×
10. ×
exp – =
=
( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y
Taking into account equality
( y ) exp – i
m2
y)st( 2
h
= ( 2 π ) – 1
0 ( λ ) exp – i [ λ y +
+ ( 2 m ) – 1
h ( t – s ) y 2
] d λ
we arrive at the formula
( y ) exp i [ –
m2
y)st( 2
h
+ x y ] d y =
= ( 2 π ) – 1
0 ( λ ) d λ
exp i [ y ( x – λ ) – ( 2 m ) – 1
h ( t – s ) y 2
] d y
Calculation of the inner integral results
( 2 π ) – 1
exp i [ y ( x – λ ) – ( 2 m ) – 1
h ( t – s ) y 2
] d y =
d
m
sthi
y
st
2
]
)(
[
2
1
11. = ( 2 π ) – 1
exp –
2
1
{ [
m
)st( hi
] 2
1
y – (
)st(
m
h
i
) 2
1
( x – ) } 2
exp –
)st(2
)λx(m 2
h
i
d y = 2
1
]
)s-t(
mπ2
[
π2
1
hi
exp –
)st(2
)λx(m 2
h
i
Hence,
( x , t ) =
2
1
]
)st(π2
m
[
h
exp –
)st(2
)yx(m 2
h
i
0 ( y ) d y
This representation of the wave function is identical to the Feynman formula. Thus, the probabilistic
representation of the Schrödinger equation solution takes place in the complex coordinate space. The
quantum measurements are the real-world actions of the complex quantum distributions of the
quantum particles.
Let us consider a classical mechanic system with Lagrangian in the form (6.4.1). Let potential
function in (6.4.1) represents the ‘white noise’ external forces
V ( t , x , ω ) = F ( x , t ) + G ( x , t )
β ( t )
where
F ( x , t ) = ( f ( t ) , x ) , gj k ( t ) x j
β k ( t ) = G ( x , t )
β ( t )
N
j
d
k1 1
12. Here ( t ) is a d-dimensional Wiener process independent on the Wiener process w ( t ). This type of
the potential functions could be used for description of the continuous time quantum measurements.
Using the probabilistic representation, we derive the Schrödinger equation that corresponds to ‘white
noise’ potential function. Bearing in mind the complex interpretation of the quantum evolutions, we
introduce the functional
Ψ ( z , t ) = E Ψ0 ( ( 0 ; t , z )) exp
h
i
t
0
[ F ( ( s ; t , z ) , s ) d s +
(6.4.11)
+
t
0
G ( ( s ; t , z ) , s ) d
β ( s ) | F ] }
Filtration F is generated by the increaments of the Wiener process up to the moment t . Let us
briefly comment the formula (6.4.11). In this formula the Wiener process ( s ) is ndependent on the
Wiener process w ( t ). Recall that the process w ( t ) is a characteristic of the quantum particle system,
i.e. relates to the quantum world. The random process ( s ) is the characteristic of the interaction of
the external world with the quantum system. This interaction does not be quantized and therefore
preserves its the classical mechanics form. Such a construction can be formally represented by the
conditional expectation with respect to -field generated by the observations on the external random
potential. Consider a derivation of the correspondent Schrödinger equation.
Theorem 6.3.3. Suppose that nonrandom functions Ψ0 ( z ) , F ( t , z ) , G ( t , z ) are continuous in t and
analytic with respect to z. Then the function ( x , t ) = Ψ ( x + i 0 , t ) is a classical solution of the
Cauchy problem of the Schrödinger equation
t
)t,x(ψ
=
m2
hi
2
( x , t ) + h – 1
[ i F ( x , t ) – h – 1
G ( x , t ) ] ( x , t ) –
– i h – 1
G ( x , t ) ] ( x , t )
β ( t ) (6.4.12)
β
t
β
t
13. ( x , 0 ) = 0 ( x + i 0 )
The classical solution of the Cauchy problem (6.4.12) is a random function ( x , t ) twice continuously
differentiable in x in the sense of mean convergence and measurable for each t with respect to F and
satisfies with probability 1 the equality
( x , t ) – ( x , 0 ) =
t
0
{
m2
hi
2
( x , s ) + h – 1
[ i F ( x , s ) –
(6.4.13)
– h – 1
G ( x , s ) ] ( x , s ) d s – i h – 1
t
0
G ( x , s ) ( x , s ) d ( s )
with probability 1 for all t at once. In the equation (6.4.13) the stochastic integral with respect to is
interpreted as the forward time Ito integral.
Proof. We briefly outline the derivation of the equation (6.4.13). Let s = t 1 < t 2 < …< t N = t be a
partition of the interval [ s , t ] and λ = max t j . Taking into account that the processes
( s ; t , z ) and ( t ) are independent and filtration F is continuous in t we note that
Ψ ( z , t k + 1 ) – Ψ ( z , t k ) = E { Ψ ( ( t k ; t k + 1 , z ) , t k ) –
– Ψ ( z , t k ) exp
h
i
[
N
1j
f j ( t k ) z j t k –
N
1j
d
1l
g j l ( t k ) z j l ( t k ) ] +
+ Ψ ( z , t k ) [ exp
h
i
[
N
1j
f j ( t k ) z j t k –
–
N
1j
d
1l
g j l ( t k ) z j l ( t k ) ] | F } – 1 ] + o ( λ )
β
t
β
t
1k
14. where F β
1k = F β
t 1k
. Bearing in mind that -algebras F β
)1k,k[ and F β
)1k,k[ F ξ
)k,0[ are
conditionally independent, we have
Ψ ( z , t k + 1 ) – Ψ ( z , t k ) = E { [ ( t k ; t k + 1 , z ) – z ] z Ψ ( z , t k ) +
+ [ ( t k ; t k + 1 , z ) – z ] z Ψ ( z , t k ) [ ( t k ; t k + 1 , z ) – z ] *
+
+ Ψ ( z , t k )
h
i
[
N
1j
f j ( t k ) z j – 2
2
1
h
N
1j
d
1l
g j l ( t k ) z j l ( t k ) ] t k –
– g j l ( t k ) z j l ( t k ) + o ( λ ) | F }
Passing to the limit in probability as λ 0 we arrive at the equation
Ψ ( z , t ) – Ψ ( z , s ) =
t
s
[
m2
hi
( z , r ) +
h
i
F ( z , r ) ( z , r ) –
(6.4.14)
– 2
2
1
h
G 2
( z , r ) ( z , r ) ] d r –
h
i
t
s
G ( z , r ) ( z , r ) d ( r )
Setting in this equation z = x + i 0 and taking into account that
z Ψ ( z , t ) | z = x + i 0 = x Ψ ( x , t )
d
l
N
jh
i
11
1k
15. one could note that equation (6.4.14) transforms into (6.4.13).
Theorem 6.4.4. Assume that the conditions of the Theorem 6.4.3 are fulfilled. Then for
each t 0
| ( x , t ) | 2
d x = | ( x , 0 ) | 2
d x
Proof. Rewrite the equality (6.4.14) in the differential form
s Ψ ( z , s ) = [
m2
hi
( z , s ) +
h
i
F ( z , s ) ( z , s ) –
– 2
2
1
h
G 2
( z , s ) ( z , s ) ] d s –
h
i
G ( z , s ) Ψ ( z , s ) d ( s )
where s denotes the partial differentiation with respect to s. Taking the complex conjugation in latter
equality we see that
s )s,z(Ψ = [ –
m2
hi
)s,z(Ψ –
h
i
F ( z , s ) )s,z(Ψ –
– 2
2
1
h
G 2
( z , s ) )s,z(Ψ ] d s +
h
i
G ( z , s ) )s,z(Ψ d ( s )
Applying the integration by parts formula, we note that
16. s | Ψ ( z , s ) | 2
= s [ )s,z(Ψ Ψ ( z , s ) ] = [ s )s,z(Ψ ] Ψ ( z , s ) +
+ )s,z(Ψ [ s Ψ ( z , s ) ] + s < )s,z(Ψ Ψ ( z , s ) > =
= [
m2
hi
)s,z(Ψ Ψ ( z , s ) +
h
i
F ( z , s ) | ( z , s ) | 2
– 2
2
1
h
G 2
( z , s )
| ( z , s ) | 2
] d s –
h
i
G ( z , s ) | Ψ ( z , s ) | 2
d ( s ) – [
m2
hi
Ψ ( z , s ) )s,z(Ψ +
+
h
i
F ( z , s ) | ( z , s ) | 2
+ 2
2
1
h
G 2
( z , s ) | ( z , s ) | 2
] d s –
h
i
G ( z , s )
| Ψ ( z , s ) | 2
d ( s ) + 2
1
h
G ( z , s ) G ( z , s ) | ( z , s ) | 2
d s
Setting in this equality z = x + i 0 and putting ( x , t ) = ( x + i 0 , t ) the right hand side could
be simplified. Indeed
s | ( x , s ) | 2
=
m2
hi
[ )s,(ψ x ( x , s ) – ( x , s ) )s,(ψ x ]
Integrating this equality in x over the sphere S R = { x : | x | R } and applying Green’s formula we
see that
RS
t
| ( x , t ) | 2
d x =
17. =
m2
hi
RS
div [ ψ ( x , t ) ( x , t ) – ψ ( x , t ) ( x , t ) ] d x =
=
m2
hi
RS
[ ψ ( x , t ) ( x , t ) – ψ ( x , t ) ( x , t ) ] n d S R
where [ q ] n denotes projection of the vector q onto external normal to the S R and
d S R is differential element of the surface of the sphere S R . The right hand side has the standard form
and it is usually assumed in quantum mechanics that it converges to 0 when R + . This remark
completes the proof.
The Lagrangian form of the classical mechanics is the initial step of the Feynman approach and
we have followed this line. Nevertheless, it is known another form of Lagrangian that also leads to the
same classical dynamic equations. Let us recall this construction by considering a system
m 2
2
td
xd
+ r
td
xd
+ k x + a = 0
(6.4.15)
m 2
2
td
yd
– r
td
yd
+ k y + a = 0
Here m, r, k , are known constants though the next contstuctions remain correct when these constant
are functions on t. The Lagrangian of the system (6.4.15) can be written in the form
L ( x ,
td
xd
, y ,
td
yd
, t ) = m
td
xd
td
yd
+
2
r
[ y
td
xd
– x
td
yd
] –
(6.4.16)
18. – k x y – a ( x + y )
The correspondent action S then is defined as
S =
t
s
L ( x ,
td
xd
, y ,
td
yd
, l ) d l
The variables x and y in S are assumed to be independent. One can verify that variations of the action
with respect to the variables x and y lead to the motion equations (6.4.15). Bearing in mind that the
same mechanics motion equations could be presented by the different Lagrangian functions (6.4.1) or
(6.4.16) it looks essential to justify that Feynman’s interpretation of the wave function does not depend
on a choice of the Lagrangian. For illustrative simplicity we assume that r = a = 0. Denote
I ( x , y ) = exp
h
i
t
s
L ( x ,
td
xd
, y ,
td
yd
, l ) d l
where
L ( x ,
td
xd
, y ,
td
yd
, t ) = m
td
xd
td
yd
– k x y (6.4.16)
Then the expression P corresponding to I ( x , y ) could be written as following
P =
2
1
x
x
2
1
y
y
I ( x , y ) D x ( * ) D y ( * ) (6.4.17)
t
s
19. We will show that the propagator P has another interpretation than the wave function presented by
Feynman’s approach dealing with Lagrangian (6.4.1). Note that the right hand side of (6.4.17) is the limit
of the correspondent discrete time approximation. That is
P =
0ε
lim
2
A
1
… exp
h
i
{
N
1j ε
)yy()xx(m 1jj1jj
–
(6.4.18)
– k x j – 1 y j – 1 }
A
xd 1
A
yd 1
…
A
xd 1N
A
yd 1N
where A = ( 2 i h m – 1
) is the normalized factor, and x 0 = x s , x N = x t . For calculation
expression in the right hand side of (6.4.18) introduce the new coordinates
x = ( 2 ) 2
1
( x l + y l )
y = ( 2 ) 2
1
( y l – x l )
l = 1, 2, … , N – 1. The inverse transformations could be presented as
x l = ( 2 ) 2
1
( x – y )
y l = ( 2 ) 2
1
( x + y )
l = 0 , 1, … N. Bearing in mind that the transition Jacobian equal to 1 we note that substitution of the
new variables into right hand side of (6.4.18) leads us to the formula
2
1
*
l
*
l
*
l
*
l
*
l
*
l
20. P ( x s , y s , s | x t , y t , t ) =
0ε
lim
2
A
1
... exp
h
i
2
{
N
1j ε
m
( x j – x j – 1 ) 2
–
–
ε
m
( y j – y j – 1 ) 2
+ k ( x 2
1j – y 2
1j ) }
A
xd 1
A
yd 1
…
A
xd 1N
A
yd 1N
In this formula for the writing simplicity the *-symbol is omitted under the integrals for the variables x l
, y l , l = 1, ... , N – 1. Thus
P ( x s , y s , s | x t , y t , t ) = K ( x s , s | x t , t ) )t,y|s,y(K ts
Here K ( | ) is the Feynman’s kernel. Putting x s = y s = x and x t = y t = y we arrive at the result
that starting with the Lagrangian (6.4.16) Feynman’s approach presents distribution density in the form
P ( x , s | y , t ) = P ( x , x , s | y , y , t ) = | K ( x , s | y , t ) | 2
(6.4.19)
The Lagrangian (6.4.16) corresponds to the motion equations (6.4.15) in which the first equation
describes the motion of the particle with external power given that r > 0 while the second one
describes the motion of damped particle. It looks unexpectedly that the quantum dynamics either
motions nonseparable from its time inverse counterpart.