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Summer Research Fellowship Programme of India's Science Academies

Generating -Bohmian trajectories for double slit experiment

Chitralekha Kumari

IIT Ropar, Birla seed farms, Rupnagar, Punjab 140111

Prof. Arvind

IISER Mohali, Sec-81 SAS Nagar, Mohali, Punjab 140306

Abstract

Bohmian mechanics, also known as pilot-wave theory or de Broglie-Bohm theory, is a formulation of quantum mechanics whose fundamental axioms are not about what observers will see if they perform an experiment but about what happens in reality. It is therefore called a “quantum theory without observers”. It follows from these axioms that in a universe governed by Bohmian mechanics, observers will see outcomes with exactly the probabilities specified by the usual rules of quantum mechanics for empirical predictions. Specifically, Bohmian mechanics asserts that electrons and other elementary particles have a definite position at every time and move according to an equation of motion that is one of the fundamental laws of the theory and involves a wave function that evolves according to the usual Schrodinger equation. Bohmian mechanics is named after David Bohm (1917–1992), who was, although not the first to consider this theory, the first to realize (in 1952) that it actually makes correct predictions. We can draw the trajectories in many cases like double slit experiment, hydrogen like atom etc. using the new casual interpretation.

Keywords: Schrodinger wave equation, guiding equation

Abbreviations

Abbreviations
SRFSummer Research Fellowship
IASIndian Academy of Science
QM Quantum Mechanics 
ODE Ordinary Diffrential Equation 
 PDEPartial Differential Equation 
EPR Einstein-Pdolsky-Rosen 
TDSE Time Dependent Schrodinger Equation 

INTRODUCTION

BACKGROUND

EPR Paradox

The EPR paradox (or the Einstein-Podolsky-Rosen Paradox) is a thought experiment intended to demonstrate an inherent paradox in the early formulations of quantum theory. It is among the best-known examples of quantum entanglement. The paradox involves two particles that are entangled with each other according to quantum mechanics. Under the Copenhagen interpretation of quantum mechanics, each particle is individually in an uncertain state until it is measured, at which point the state of that particle becomes certain.

At that exact same moment, the other particle's state also becomes certain. The reason that this is classified as a paradox is that it seemingly involves communication between the two particles at speeds greater than the speed of light, which is a conflict with Albert Einstein's theory of relativity.

The paradox's origin

The paradox was the focal point of a heated debate between Einstein and Niels Bohr. Einstein was never comfortable with the quantum mechanics being developed by Bohr and his colleagues. Together with his colleagues Boris Podolsky and Nathan Rosen, Einstein developed the EPR paradox as a way of showing that the theory was inconsistent with other known laws of physics. At the time, there was no real way to carry out the experiment, so it was just a thought experiment.

Several years later, the physicist David Bohm modified the EPR paradox example so that things were a bit clearer.In the more popular Bohm formulation, an unstable spin 0 particle decays into two different particles, Particle A and Particle B, heading in opposite directions. Because the initial particle had spin 0, the sum of the two new particle spins must equal zero. If Particle A has spin +1/2, then Particle B must have spin -1/2 (and vice-verse).

Again, according to the Copenhagen interpretation of quantum mechanics, until a measurement is made, neither particle has a definite state. They are both in a superposition of possible states, with an equal probability (in this case) of having a positive or negative spin.

The paradox's meaning

There are two key points at work here which make this troubling:

  • Quantum physics says that, until the moment of the measurement, the particles do not have a definite quantum spin but are in a superposition of possible states.
  • As soon as we measure the spin of Particle A, we know for sure the value we'll get from measuring the spin of Particle B.

To Einstein, this was a clear violation of the theory of relativity.

Statement of the Problems

‘Standard quantum mechanics’: the Copenhagen interpretation

  • System completely described by wave function ψ representing observer’s knowledge of the system, or ‘potentiality’. The wave function is all there is. ‘Hidden variables’ impossible. (Heisenberg)
  • Description of nature essentially probabilistic. Probability of event related to square of amplitude of wave function related to it (Born rule). ‘Measurement’ randomly picks out exactly one of the many possibilities allowed for by the state’s wave function through nonlocal ‘collapse process’.
  • Heisenberg’s uncertainty principle: observed fact that it is not possible to know values of all properties of system at same time; those properties not known with precision must be described by probabilities. Properties in fact supposed to be indeterminate not uncertain.
  • Complementarity principle: there is no logical picture (obeying classical propositional logic) that can simultaneously describe and be used to reason about all properties of a quantum system. Example: matter exhibits a wave-particle duality. An experiment can show the particle-like properties of matter, or wave-like properties, but not both at the same time. (Niels Bohr)
  • Measuring devices are essentially classical devices, and measure classical properties such as position and momentum.
  • The ‘correspondence principle’ of Bohr and Heisenberg: the quantum mechanical description of large systems should closely approximate to the classical description.

Now well-known that Copenhagen cannot be reconstructed as a coherent philosophical framework - it is a collection of local, often contradictory, arguments embedded in changing theoretical and sociopolitical circumstances.. ..riddled with vaccillations, about-faces and inconsistencies.

Alternatives to Copenhagen

  • Ensemble interpretation, or statistical interpretation
  • Participatory Anthropic Principle (PAP)
  • Consistent histories
  • Objective collapse theories (GRW etc.)
  • Many worlds
  • Stochastic mechanics
  • The decoherence approach
  • Many minds
  • Quantum logic
  • Transactional interpretation
  • Relational quantum mechanics
  • Modal interpretations of quantum theory
  • Incomplete measurementsPilot-wave interpretation (a.k.a. Bohmian mechanics, deBroglie-Bohm interpretation, causal interpretation, ontological interpretation, hidden variables)
Capture.PNG
    Alternatives to copenhagen interpretation

    Objective of the Research

    Given a set of initial conditions - the theory allows us to calculate deterministic trajectories of particles obeying Newton’s laws. There are various equivalent mathematical formulations of this i.e. different equations leading to same trajectories:

    Fi(q1,q2,..........qN)=miq¨iF_i(q_1,q_2,..........q_N)=m_i\ddot{q}_i Newtonian mechanics

    q˙=Hp(q,p)p˙=Hq(q,p)\dot{q}=\frac{\partial{H}}{\partial{p(q,p)}}\dot{p}=-\frac{\partial{H}}{\partial{q(q,p)}}Hamiltonian dynamics → standard QM

    Solve a canonical system of 1st-order ODEs (2n equations for 2n functions of a parameter t in which all variables’ first derivatives are given by partial derivatives of the same function).

    δt0t1L(q(t),q˙(t))=0\delta\int_{t_0}^{t_1}L(q(t),\dot{q}(t))=0 Lagrangian dynamics → path-integral QM

    Solve the basic calculus of variations problem of finding n functions q1,...,qn of a parameter t that make stationary a line integral (i.e. solve n 2nd-order ODEs).

    St(q,t)+H(q,Sq)=0\frac{\partial{S}}{\partial {t(q,t)}}+H(q,\frac{\partial{S}}{\partial{q}})=0 Hamilton-Jacobi dynamics → pilot-wave theory

    Solve a single 1st-order PDE in which the unknown function does not occur explicitly.

    Main objective is to generate trajectories of The double slit experiment using bohmian mechanics.

    caputre 3_1.PNG
      Trajectories for two Gaussian slits

      LITERATURE REVIEW

      Information

      Bohmian mechanics

      (Pilot Wave Theory / De Broglie-Bohm Theory / Hidden VariableTheory)

      Bohmian mechanics is a theory about point particles moving along trajectories. It has the property that in a world governed by Bohmian mechanics, observers see the same statistics for experimental results as predicted by quantum mechanics. Bohmian mechanics thus provides an explanation of quantum mechanics.

      Bohmian mechanics is a version of quantum mechanics for non-relativistic particles: In Bohmian mechanics quantum particles have positions, always, and follow trajectories. These trajectories differ, however, from the classical Newtonian trajectories. As a consequence, the role of the wave function in Bohmian mechanics is to tell the matter how to move.

      Bohmian mechanics constitutes a quantum theory without observers, i.e., a theory that is formulated not in terms of what observers see but interms of objective events, regardless of whether or not they are observed. Bohmian mechanics provides a consistent resolution of all paradoxes of quantum mechanics, in particular of the so-called measurement problem. Bohmian mechanics is sometimes called a hidden variables theory because it involves variables besides the wavefunction.

      Defining equations

      Bohmian mechanics is a non-relativistic theory governing the behavior of a system of N point particles moving in physical space R3R^3 along trajectories. Let Qi(t)R3Q_i(t)\in{R^3} denote the position of the i-th particle of the system at time t, and Q(t)=Q1(t),Q2(t),.......,QN(t)R3N its configuration. The trajectories are governed by Bohm’s law of motion

      dQidt=miImψtiψtψtψt(Q(t)),\frac{dQ_i}{dt}=\frac{\hbar}{m_i}Im\frac{\psi_t^*\nabla_i\psi_t}{\psi_t^*\psi_t}(Q(t)),where mim_i (2.1.1)

      is the mass of particle i, Im denotes the imaginary part, ψt:R3NCk (i.e., a function of the configuration with k complex components) is the wave function at time t, ϕψ\phi^*\psi is the scalar product in CkC^k
      , and i\nabla_i is the gradient relative to the 3 coordinates of particle i. (In case k = 1, i.e., for complex-valued wave functions, a factor ψt\psi_t^*cancels on the right hand side of above equation.) The wave function evolves according to the Schrodinger equation

      iψtt=i=1N2mii2ψt+Vψti\hbar\frac{\partial{\psi_t}}{\partial{t}}=-\sum_{i=1}^N\frac{\hbar}{2m_i}\nabla_i^2\psi_t+V\psi_t (2.1.2)

      where V:R3NRV:R^{3N}\rightarrow{R}is the potential function. (The potential, while often assumed to be real-valued, may take values in the space of self-adjoint complex k × k matrices instead of R.)

      The wave function is postulated to belong to the Hilbert space H=L2(R3N,Ck)H=L^2(R^{3N},C^k) of square-integrable functions (and to be sufficiently smooth).

      Fundamental laws of Bohmian mechanics

      While Bohmian mechanics has been considered as a tool for visualization, for the efficient numerical simulation of the Schrodinger equation and other applications the main interest in it arises from the fact that Bohmian mechanics provides a possible way how our world might be and might work. In its non-relativistic form, the theory asserts the following: N material points (“particles”) move in 3-dimensional Euclidean space (denoted for simplicity as R3NR^{3N}) in a way governed by a field-like entity that is mathematically given by a wave function ψ (as familiar from standard quantum mechanics). More precisely, the position Qk(t)Q_k(t) of particle number k at time t obeys Bohm’s equation of motion

      dQkt=mkImψkψψψ(Q(t),t)\frac{dQ_k}{t}=\frac{\hbar}{m_k}Im\frac{\psi^*\nabla_k\psi}{\psi^*\psi}(Q(t),t) (2.1.3)

      where Q(t)=Q1(t),Q2(t),.......,QN(t)R3N denotes the configuration of the particle system at time t, mkm_k is the mass of particle k, Im the imaginary part of a complex number, ψ.

      Deterministic

      The wave function ψ(q,t)=ψ(q1,.....,qN,t)\psi(q,t)=\psi(q_1,.....,q_N,t) evolves with time t according to the usual Schrodinger equation

      iψ(q,t)t=k=1N22mkk2ψ(q,t)+V(q)ψ(q,t)i\hbar\frac{\partial\psi(q,t)}{\partial{t}}=-\sum_{k=1}^N\frac{\hbar^2}{2m_k}\nabla_k^2\psi(q,t)+V(q)\psi(q,t) (2.1.4)

      where V:R3NRV:R^{3N}\rightarrow{R} is the potential function, for example the Coulomb potential

      V(q1,...,qN)=12jkej.ekqjqkV(q_1,...,q_N)=\frac{1}{2}\sum_{j\neq{k}}\frac{e_j.e_k}{|q_j-q_k|} (2.1.5)

      with ej the electric charge of particle j. The state of the system at time t is described by the pair Q(t), ψ(t), and Equations (2.1.1) and (2.1.3) together determine the state at any other time; thus, Bohmian mechanics is a deterministic theory.

      Equivariance

      If the initial configuration Q(t0)Q(t_0) is chosen at random with probability density ψ(t0)2|\psi(t_0)|^2 then the configuration Q(t) at any other time t is random with probability density ψt2|\psi_t|^2. (Whenever speaking of probabilities, we assume that Ψ has been normalized, by multiplication by a suitable constant, so that <ψψ>=ψ(q)2dq=1<\psi|\psi>=\int|\psi(q)|^2dq=1.) This fact, known as equivariance, follows from the continuity equation and with the Bohmian velocity vector field v = v Ψ. The continuity equation is in turn a consequence of the Schrodinger equation; it is usually written in terms of the quantum probability current J = ρ v.

      Quantum equilibrium hypothesis

      This is the assertion that whenever a system has wave function ψ then its configuration is random with probability distribution ψ2|\psi|^2. Equivariance implies that this hypothesis is consistent with the time evolution of isolated systems, and it is not hard to show that it is also consistent with the time evolution if the system is not isolated, provided we take ψ to mean the conditional wave function. An important consequence of the quantum equilibrium hypothesis is the empirical equivalence between Bohmian mechanics and quantum mechanics: For every conceivable experiment, whenever quantum mechanics makes anunambiguous prediction, Bohmian mechanics makes exactly the same prediction. Thus, the two cannot be tested against each other.

      Summary

      Basic idea of pilot-wave theory

      De Broglie: “A freely moving body follows a trajectory that is orthogonal to the surfaces of an associated guiding wave.” Used a proto-pilot-wave theory to predict ‘diffraction’ of electrons by small holes.

      1. System has wave function Ψ(x,t) and definite particle configuration x(t) at all times. To define trajectory need velocity vector, which can be deduced from the probability current density vector of the standard theory:

      x˙(t)=j(x,t)ψ(x,t)2\dot{x}(t)=\frac{j(x,t)}{|\psi(x,t)|^2}= ratio of quantum probability current j to probability density ρ.

      2. Recall probability ‘fluid’ has current j=i2m(ψψψψ)=mIm(ψψ)j=\frac{i\hbar}{2m}(\psi\nabla\psi^*-\psi^*\nabla\psi)=\frac{\hbar}{m}Im(\psi^*\nabla\psi) which satisfies the quantum-mechanical continuity equation, ρt+.j=0\frac{\partial\rho}{\partial{t}}+\nabla.j=0 or tVψ2dV+sj.dA=0\frac{\partial}{\partial{t}}\int_V|\psi|^2dV+\int_{s}j.dA=0 i.e. the conservation law for probability density in QM.

      Theory based on this “completely avoids all the quantum paradoxes, all the mysticism of Bohr and Heisenberg, and replaces it with sharp mathematics.”

      Basic equations - de Broglie form (1927)

      Focusing for simplicity on non relativistic quantum system of N (spin less) particles with positions xix_i(i = 1,2,...,N) it is now generally agreed that, with appropriate initial conditions, quantum physics may be accounted for by the deterministic dynamics defined by two differential equations, i.e. the Schrodinger equation

      iψt=i=1N22mii2ψ+Vψi\hbar\frac{\partial{\psi}}{\partial{t}}=\sum_{i=1}^N-\frac{\hbar^2}{2m_i}\nabla_i^2\psi+V\psi for a pilot wave ψ(x1,x2,....,xN,t)\psi(x_1,x_2,....,x_N,t) in configuration space and the guidance equation midxidt=iSm_i\frac{dx_i}{dt}=\nabla_iS for particle trajectories xi(t)x_i(t). Phase S(x1,x2,.....xN,t)S(x_1,x_2,.....x_N,t) defined by S=ImlnψS={\hbar}Im {ln} \psi of complex polar form

      ψ=ψexp[iS]\psi=|\psi|exp[i\frac{S}{\hbar}]. Compare jψ2\frac{j}{|\psi|^2}. Particles ‘pushed along’ by wave along trajectories perpendicular to surfaces of constant phase.

      • Can also guess guidance equation from de Broglie relation p=kp=\hbar{k} (connects particle and wave properties). Wave vector k defined only for plane wave. For general wave, obvious generalization of k is local wave vector S(x)\frac{\nabla{S(x)}}{\hbar}. Hence v = ∇S/m.
      • Operators on Hilbert space and all that play no fundamental role, but are exactly right mathematical objects to provide compact representation of the statistics in a de Broglie universe.

      Basic equations - Bohm form (1952)

      Bohm’s 1952 presentation: take first time derivative of guidance equation mx˙=Sm\dot{x}=\nabla{S}, then use TDSE to get second-order theory analogous to Newton’s second law:

      tiS(x,t)=itImlnψ=iIm[ψtψ]=iIm[iψ(22mii2ψVψ)]=i[1ψ(22mii2ψ+Vψ)]=i[V+Q]=mix¨i\frac{\partial}{\partial{t}}\nabla_iS(x,t)=\nabla_i\frac{\partial}{\partial{t}}\hbar{Im}ln\psi=\nabla_iIm[\frac{\hbar}{\psi}\frac{\partial}{\partial{t}}\psi]=\nabla_iIm[\frac{i}{\psi}(\frac{\hbar^2}{2m_i}\nabla_i^2\psi-V\psi)]=-\nabla_i[\frac{1}{\psi}(-\frac{\hbar^2}{2m_i}\nabla_i^2\psi+V\psi)]=-\nabla_i[V+Q]=m_i\ddot{x}_i

      where Qi22mii2ψψQ\equiv-\sum_i\frac{\hbar^2}{2m_i}\frac{\nabla_i^2|\psi|}{|\psi|} is the so-called ‘quantum potential’. Bohm regarded this as the law of motion, with the de Broglie guidance equation added as a constraint on the initial momenta. For de Broglie, in contrast, law of motion for velocity had a fundamental status, and for him represented a unification of the principles of Maupertuis and Fermat.

      Usually stated de Broglie did one-particle theory only, and that Bohm generalized it to N particles (as claimed in Bohm’s book). However, de Broglie gave the full N-particle theory at the 1927 Solvay conference and also indicated that he knew about above second-order theory (see Valentini book). De Broglie’s priority clearly established - thus ‘Bohmian mechanics’ is probably the wrong name.

      METHODOLOGY

      Concepts

      Superposition of plane waves

      (Interference: the two slit experiment)

      The principle of superposition allows one to construct solution to the wave equation whose properties qualitatively distinct from the interfering partial waves. Consider

      ψ(x,t)=A[ψ1(x,t)+cψ2(x,t)]\psi(x,t)=A[\psi_1(x,t)+c\psi_2(x,t)] (3.1.1)

      Where ψj(x,t)=e(kj.xωkj.t),ωkj=kj22m\psi_j(x,t)=e^{(k_j.x-\omega_{k_j}.t)} ,\omega_{k_j}=\frac{\hbar{k_j^2}}{2m} (3.1.2)

      J=1,2 and A=Aeiϕ,c=ceiδA=|A|e^{i\phi},c=|c|e^{i\delta} are complex constants. c|c| determines the relative amplitude of waves. We assume throughout that k1k2k_1\neq{k_2}; if k1=k2k_1=k_2 we recover a simple plane wave.

      Formula for amplitude and phase of ψ\psi:

      R=A(1+c2+2ccosξ)1/2R=|A|(1+|c|^2+2|c|\cos{\xi})^{1/2} (3.1.3)

      tan(Sϕ)=[sin(k1xωk1t)+csin(k2xωk2t+δ)].[cos(k1xωk1t)+ccos(k2xωk2t+δ)]1\tan(\frac{S}{\hbar}-\phi)=[\sin(k_1x-\omega_{k_1}t)+|c|\sin(k_2x-\omega_{k_2}t+\delta)].[\cos(k_1x-\omega_{k_1}t)+|c|\cos(k_2x-\omega_{k_2}t+\delta)]^{-1} (3.1.4)

      Where ξ=(k1k2).x(ωk1ωk2)tδ\xi=(k_1-k_2).x-(\omega_{k_1}-\omega_{k_2})t-\delta (3.1.5)

      is a relative phase. Clearly, ψ\psi is not a stationary state (unless k1=k2k_1=-k_2). The amplitude function displays a set of interference fringes whose maxima (minima) occur where cosξ=1\cos\xi=-1). The difference between the maxima and minima is proportional to c|c|. The wave function has strict Nodes only in case c=1|c|=1 (the partial waves have the same amplitude)

      Capture 4.PNG
        A snapshot of amplitude function in one dimension as time passes pattern move to right.

        and cosξ=1\cos\xi=-1. In this case nodes are set of propagating planes:

        (k1k2)x(ωk1ωk2)tδ=(2n+1)π(k_1-k_2)x-(\omega_{k_1}-\omega_{k_2})t-\delta=(2n+1)\pi (3.1.6)

        where nZn\in{Z} labels which surface. To find momentum field p=Sp=\nabla{S}

        S=[k1+c2k2+c(k1+k2)cosξ](1+c2+2ccosξ)\nabla{S}=\hbar\frac{[k_1+|c|^2k_2+|c|(k_1+k_2)\cos\xi]}{(1+|c|^2+2|c|\cos\xi)} (3.1.7)

        Which is a well defined function of x and t everywhere except at the nodes (where the denominator vanishes).

        In the one-dimensional case we can give an explicit expression for trajectories as a function of time, for mx˙=Sxm\dot{x}=\frac{\partial{S}}{\partial{x}}

        sin[(k1k2)x(ωk1ωk2)tδ]+[(k1k2)2c][(1+c2)xtm(k1+c2k2)]=constant=c\sin[(k_1-k_2)x-(\omega_{k_1}-\omega_{k_2})t-\delta]+[\frac{(k_1-k_2)}{2|c|}][(1+|c|^2)x-\frac{\hbar{t}}{m}(k_1+|c|^2k_2)]=constant=c (3.1.8)

        this relation we rewrite it in the form ξ+ϵsinξ=ωt+a\xi+\epsilon\sin\xi=\omega{t}+a (3.1.9)

        Where ξ=(k1k2)x(ωk1ωk2)tδ\xi=(k_1-k_2)x-(\omega_{k_1}-\omega_{k_2})t-\delta (3.1.10)

        ϵ=2c(1+c2),0ϵ1\epsilon=2\frac{|c|}{(1+|c|^2)},0\leq\epsilon\leq1 (3.1.11)

        ω=(k1k2)22m1c21+c2\omega=\frac{\hbar(k_1-k_2)^2}{2m}\frac{1-|c|^2}{1+|c|^2} (3.1.12)

        Which may be positive, negative or zero, and a is a constant defined in terms of the initial position x0x_0 by

        a=(k1k2)x0δ+ϵsin[(k1k2)x0δ]a=(k_1-k_2)x_0-\delta+\epsilon\sin[(k_1-k_2)x_0-\delta] (3.1.13)

        x0x_0arbitrary except in case c=1|c|=1 it can not lie on nodes(the particle remains between the given pair of nodes for all the time).

        If the axes is (ξ,ωt)(\xi,\omega{t}) then particles oscillates symmetrically about line ξ=ωt+a\xi=\omega{t}+awith period 1ω\(ωtωt+2π,ξξ+2π)
        and amplitude ϵ\epsilon.

        x=vt+(δ+a)(k1k2)x=vt+\frac{(\delta+a)}{(k_1-k_2)} (3.1.14)

        v=(k1+k2c2)m(1+c2)v=\frac{\hbar(k_1+k_2|c|^2)}{m(1+|c|^2)} (3.1.15)

        Amplitude ϵ(k1k2)\frac{\epsilon}{(k_1-k_2)} when (tt+2π/ω,xx+2πv/ω)(t\rightarrow{t+2\pi/\omega},x\rightarrow{x+2\pi{v}/\omega}). Different choices of ‘a’ correspond to different x0x_0. If we consider two lines from equation (14) characterized by the constants a and a’, their separation along x axis is .

        1) When c|c| very large or very small (disparity in relative amplitudes very large): the amplitude of oscillation of particle is small & path approaches uniform motion motion implied by a pure plane wave

        x=k1tm+x0 or x=k2tm+x0x=\frac{\hbar{k_1}t}{m}+x_0  or  x=\frac{\hbar{k_2}t}{m}+x_0 (3.1.16)

        from (3.1.13) x0=(δ+a)(k1k2) whenϵ=0x_0=\frac{(\delta+a)}{(k_1-k_2)}  when{} \epsilon=0

        2) When ( c=1ϵ=1|c|=1\Rightarrow{\epsilon=1}) we expect maximum oscillation when amplitude of sub waves equal but in fact we obtain uniform motion from (3.1.12) if ( c=1ω=0|c|=1\Rightarrow{\omega=0}) so period is infinite.

        3) Trajectory to parallel to line (3.1.14) but in general coincide with x=(k1+k2)t2m+x0x=\frac{\hbar(k_1+k_2)t}{2m}+x_0 (3.1.17)

        4) From equation (3.1.6) nodes ravel at same speed

        x=(k1+k2)t2m+[δ+(2n+1)π](k1k2)1x=\frac{\hbar(k_1+k_2)t}{2m}+[\delta+(2n+1)\pi](k_1-k_2)^{-1} (nodal eq.)(3.1.18)

        Particle remains at constant distance from nodes.

        5) if vary relative phase (k1k2)orδ(k_1-k_2) or \delta: Increases (k1k2)(k_1-k_2) then amplitude decreases and frequency increases.

        6) Equation (3.1.9) is similar to the Kepler equation of celestial mechanics with the difference that we have negative eccentricity, then solution of (3.1.9) and (3.1.10) is

        x=vt+(δ+a)(k1k2)+1(k1k2)n=12n(1)nJn(nϵ)sin[n(ωt+a)]x=vt+\frac{(\delta+a)}{(k_1-k_2)}+\frac{1}{(k_1-k_2)}\sum_{n=1}^{\infty}\frac{2}{n}(-1)^nJ_n(n\epsilon)\sin[n(\omega{t}+a)] (3.1.19)

        Where Jn are bessel functions. Invert equation (3.1.13)

        x0=(δ+a)(k1k2)+1(k1k2)n=12n(1)nJn(nϵ)sin(na)x_0=\frac{(\delta+a)}{(k_1-k_2)}+\frac{1}{(k_1-k_2)}\sum_{n=1}^{\infty}\frac{2}{n}(-1)^nJ_n(n\epsilon)\sin(na) (3.1.20)

        In all the other cases x(t) is complicated function of x0x_0.

        7) Equal amplitude case c|c| an example where energy and momentum are conserved even ψ\psi is not stationary state and not eigen function of momentum. To see this evaluate quantum potential.

        Q=22R2mR=2A2c(k1k2)22mR4(R2cosξ+A2csin2ξ)Q=-\hbar^2\frac{\nabla^2R}{2mR}=\frac{\hbar^2|A|^2|c|(k_1-k_2)^2}{2mR^4}(R^2\cos\xi+|A|^2|c|\sin^2\xi) (3.1.21)

        Which is function of x and t. when c=1|c|=1,

        Q=2(k1k2)28mQ=\frac{\hbar^2(k_1-k_2)^2}{8m} (3.1.22)

        from (3.1.7) kinetic energy is also a constant,

        (S)22m=2(k1k2)28m\frac{(\nabla{S})^2}{2m}=\frac{\hbar^2(k_1-k_2)^2}{8m} (3.1.23)

        From (3.1.22) and (3.1.23)

        E=[(k1)2+(k2)2]/4mE=[(\hbar{k_1})^2+(\hbar{k_2})^2]/4m (3.1.24)

        This is simply the kinetic energy associated with the component plane waves.

        c=1|c|=1 equation (3.1.4)

        S=12(k1+k2).x2(k12+k22)t4mS=\frac{1}{2}\hbar(k_1+k_2).x-\hbar^2\frac{(k_1^2+k_2^2)t}{4m} (3.1.25)

        The wave function is a plane wave with time and spacedependent amplitude and Q satisfies our energy conservation criterion ( Qt=0\frac{\partial{Q}}{\partial{t}}=0) and momentum conservation criterion (Q=0\nabla{Q}=0).

        Methods

        Interference by division of wavefront

        Particle Trajectories in Electron Interferometer

        We will assume that this analysis has been carried out and the true electron slit interaction, which in a more complete analysis is likely to be a quite involved many body problem, may be approximated by a single body in an external force.

        We also assume that electron wave emitted by the source in each trial is identical, although in practice there will be some drift from this ideal and true distribution is a sharply peaked mixed state.

        Capture 5_1.PNG
          True electron slit interaction

          Electrons are emitted by source S1, pass trough two slits B,B’ in a barrier P and arrive at screen S2. The detection process at screen is nontrivial but it plays no casual role in the basic phenomenon of interference of electron waves.

          In the 2-D system of (x,y) coordinates whose origin O is shown, the centers of the slit lies at (0,+Y) and (0,-Y). The wave incident on the slits will be taken as plane, ψ=aei(k1x+k2y)\psi=ae^{i(k_1x+k_2y)}where a is constant and u1=k1mu_1=\frac{\hbar{k_1}}{m} and u2=k2mu_2=\frac{\hbar{k_2}}{m} are x- and y-components of velocity. To avoid mathematical complexity of Fresnel diffraction at a sharp edge slit, we assume that the slits have soft edges that generate waves having identical Gaussian profiles in the y-direction. The plane wave in the x-direction is unaffected. The instant at whic the packets are formed will be our zero of time. The waves emerging from the slits are initially

          ψBO=a(2πσ02)14e(yY)2/4σ02+i[k1x+k2(yY)]\psi_{BO}=a(2\pi\sigma_0^2)^{\frac{-1}{4}}e^{-(y-Y)^2/4\sigma_0^2+i[k_1x+k_2(y-Y)]} (3.2.1)

          ψBO=a(2πσ02)14e(y+Y)2/4σ02+i[k1xk2(y+Y)]\psi_{B'O}=a(2\pi\sigma_0^2)^{\frac{-1}{4}}e^{-(y+Y)^2/4\sigma_0^2+i[k_1x-k_2(y+Y)]} (3.2.2)

          Where σ0\sigma_0 is the half width of each slit. Wave B is a packet moving in y-direction with velocity u2u_2 and a plane wave in the x-direction, and wave B’ is similar except it has group velocity - u2u_2. Subsequently the packets move with relative velocity 2 u2u_2 and spread into one another. This is how we recombine the beams, although they are not strictly non overlapping at t=0 due to Gaussian tails. The interference comes about from both effects (relative motion and dispersion) or just one of them if the other is negligible.

          At time t the total wave function at a space point (x,y) is given by

          ψ(x,y,t)=N[ψB(x,y,t)+ψB(x,y,t)]\psi(x,y,t)=N[\psi_B(x,y,t)+\psi_{B'}(x,y,t)] (3.2.3)

          Where

          ψB(x,y,t)=a(2πst2)1/4e(yYu2t)2/4σ0st.ei[k1x+k2(yY12u2t)E1t]\psi_{B}(x,y,t)=a(2\pi{s_t^2})^{-1/4}e^{-(y-Y-u_2t)^2/4\sigma_0s_t}.e^{i[k_1x+k_2(y-Y-\frac{1}{2}u_2t)-\frac{E_1t}{\hbar}]} (3.2.4)

          ψB(x,y,t)=a(2πst2)1/4e(yY+u2t)2/4σ0st.ei[k1xk2(y+Y+12u2t)E1t]\psi_{B'}(x,y,t)=a(2\pi{s_t^2})^{-1/4}e^{-(y-Y+u_2t)^2/4\sigma_0s_t}.e^{i[k_1x-k_2(y+Y+\frac{1}{2}u_2t)-\frac{E_1t}{\hbar}]} (3.2.5)

          With E1=12mu12E_1=\frac{1}{2}mu_1^2 and N=(2+2eY2/2σ02)1/2N=(2+2e^-Y^2/2\sigma_0^2)^{-1/2} is a normalization constant. For a source places far from the slits the extra terms are negligible and we obtain essentially our wave functions. It is evident that the total wave function is factorizable in orthogonal directions:

          ψ(x,y,t)=f1(x,t)f2(y,t)\psi(x,y,t)=f_1(x,t)f_2(y,t) (3.2.6)

          Where f1 is a plane wave factor. Hence the components of the particle motion in the x- and y-direction are independent and the quantum potential is a function of y just:

          Q=22mR(2Rx2+2Ry2)=22mf22f2y2Q=-\frac{\hbar^2}{2mR}(\frac{\partial^2R}{\partial{x^2}}+\frac{\partial^2R}{\partial{y^2}})=-\frac{\hbar^2}{2mf_2}\frac{\partial^2f_2}{\partial{y^2}} (3.2.7)

          The trajectory obtained by integration m mx˙=Sm\dot{x}=\nabla{S}, where S is the phase of (3.2.3) and x0 must be specified has a uniform and rectilinear x-component, x=u1tx=u_1t (3.2.8)

          ( x0=0x_0=0 since all the particles pass trough slits at t=0), and a plot of y vs t will look the same as a space plot (y vs x) apart from a difference in scale.

          We write down the relevant differential equation (for u2=0u_2=0) which must be solved numerically:

          dydt=F[Asin[Byt(1+Ct2)1]+Dytcos[Byt(1+Ct2)1]+1/2Dt(yY)eEy(1+Ct2)1+1/2Dt(y+Y)eEy(1+Ct2)1]×(1+Ct2)1(eEy(1+Ct62)1+eEy(1+Ct2)1+2cos[Byt(1+Ct2)1])\frac{dy}{dt}=F[-A\sin[Byt(1+Ct^2)^{-1}]+Dyt\cos[Byt(1+Ct^2)^{-1}]+1/2Dt(y-Y)e^{Ey(1+Ct^2)^{-1}}+1/2Dt(y+Y)e^{-Ey(1+Ct^2)^{-1}}]\times(1+Ct^2)^{-1}(e^{Ey(1+Ct62)^{-1}}+e^{-Ey(1+Ct^2)^{-1}}+2\cos[Byt(1+Ct^2)^{-1}]) (3.2.9)

          Where A,B,C,D,E and F are constants and we choose a range of initial positions y0 in the slits. Notice that S(y,t)y=S(y,t)y\frac{\partial{S(y,t)}}{\partial{y}}=-\frac{\partial{S(-y,t)}}{\partial{y}}which expresses the symmetry of the velocity field along x-axis. To see the nature of the interference pattern, Amplitude square of the total wave function:

          R2(y,t)=a2N2(2πσ2)1/2e[y2+(Y+u2t)2]/2σ2×(ey(Y+u2t)/σ2+ey(Y+u2t)/σ2+2cos[2k2y(Y+u2t)yt/2mσ02σ2])R^2(y,t)=a^2N^2(2\pi\sigma^2)^{-1/2}e^{[y^2+(Y+u_2t)^2]/2\sigma^2}\times(e^{y(Y+u_2t)/\sigma^2}+e^{-y(Y+u_2t)/\sigma^2}+2\cos[2k_2y-(Y+u_2t)y\hbar{t}/2m\sigma_0^2\sigma^2]) (3.2.10)

          The term outside the curly brackets defines the (Gaussian) envelope of the pattern and the terms inside the fringes. The wavefunction has nodes only at space-time points defined by

          t=Y/u2t=-Y/u_2 y=(n+1/2)π/k2y=(n+1/2)\pi/k_2 (3.2.11)

          Where n is an integer and we must put u2<0u_2<0 (the packets approach one another) so that t>0. In the case u20,ψu_2\geq{0,\psi} has no strict nodes at all and there is no region of space that a particle may not potentially visit if y0 is suitably chosen. The Interference structure is evidently a dynamic one that remains symmetric about the x-axis and ultimately evolve into the familiar pattern.

          RESULTS AND DISCUSSION

          Notebook_Doubleslit_Research_Gate_final_picture_1_1.png
            Double slit interference

            The images are generated in Mathematica using the code available at:

            https://demonstrations.wolfram.com/CausalInterpretationOfTheDoubleSlitExperimentInQuantumTheory/

            Notebook_Doubleslit_Research_Gate_final_picture_2.png
              Bohmian trajectroies for double slit experiment

              CONCLUSION AND RECOMMENDATIONS

              The Best Things About Pilot Waves

              • Minimum benefit of pilot-wave theory, independent of debate about whether preferable formulation, is disproof of claim that QM implies particles can’t exist before being measured. Since based on familiar concepts of realism, pilot waves give simple, intuitive, non-mystical, and natural answers to what in standard theory are perplexing philosophical questions.

              • Pilot-wave approach gives better formulation of QM since defined more precisely than Copenhagen, which is based on theorems not expressed in precise mathematical terms but in words. Pilot waves subsume quantum concepts of measurement, complementarity, decoherence, and entanglement into mathematically precise guidance conditions on position variables.

              • Like standard QM it implies existence of EPR type of nonlocality, but it avoids need for positing other types of alleged nonlocality, such as wave collapse; and also avoids need for splitting universes as in many-worlds.

              • Can derive time-dependent wave function from trajectories. Implies we are not dealing just with another interpretation or ‘superfluous ideological superstructure’, but rather with an alternative mathematical representation or picture of QM, equivalent in status to, say, Feynman’s path-integral method.

              • Can treat e.g. dwell/tunneling times, escape times/escape positions, scattering theory, quantum chaos. Difficult or impossible to analyze in pure wave model.

              REFERENCES

              Capture 6.PNG

                ACKNOWLEDGEMENTS

                I have taken efforts in this project. However, it would not have been possible without the kind support and help of many individuals and organizations. I would like to extend my sincere thanks to all of them.

                I am highly indebted to Prof. Arvind, Director, IISER Mohali for their guidance and constant supervision as well as for providing necessary information regarding the project & also for their support in completing the project.

                I would like to express my gratitude towards my parents & member of IISER Mohali for their kind co-operation and encouragement which help me in completion of this project.

                I would like to express my special gratitude and thanks to Institution persons for giving me such attention and time.

                My thanks and appreciations also go to my colleague in developing the project and people who have willingly helped me out with their abilities.

                I also thank IAS for their support throughout the programme and for giving me this great opportunity. It would have not been possible without this programme. I would also like to thank IISER Mohali, Punjab for providing accommodation and other requirements hence making my stay comfortable.

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