Outline introduction basic existence theory regularity end of rst part existence in the lipschitz class. Schrodingers equation may be developed from the timeindependent hamiltonjacobi equation, which simply expresses the conservation of energy. The schrodinger equation is a linear partial differential equation that describes the wave function or state function of a quantummechanical system 12 it is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. Schr odingers own published derivations of quantum wave equations, the first of which was also based on the hamiltonjacobi equation, are. For a free particle in 3 dimensions, v 0 and the hje reduces to 1 or, by assuming uniform energy, 2 where weve replaced with e as discussed at the end of chapter 1. Simple derivations of the hamiltonjacobi equation and the. For the next example, well be working in spherical coordinates, with a single particle under the influence of a central potential, vr 1r. Hamiltonjacobi equation article about hamiltonjacobi. Even if the quantum world did not exist, it would make. Curiously, one might look upon the schrodinger equation as a linearized. Hamilton jacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials a, similar to schroedinger equation. What is the relation between hamilton jacobi and the.
Apr 02, 20 it is this mutual coupling between amplitude and phase that defines a quantum matter wave and ensures the linearity of the wave equation. First of all the hamiltonjacobi partial differential equation has nothing to do with quantum theory. That the wavefunction satisfies the correct schrodinger equation can be seen from the above definition and the qhj equation. We derived the aharonovbohm effect from hamilton jacobi equation thereby having proved that this effect is of classical origin. May, 2011 the hamiltonjacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the point of closest approach between classical and quantum mechanics. The hamiltonjacobi equation also represents a very general method in solving mechanical problems. The course webpage, including links to other lectures and. Physics stack exchange is a question and answer site for active researchers, academics and students of physics. Apr 03, 2012 it is shown how the timedependent schrodinger equation may be simply derived from the dynamical postulate of feynmans path integral formulation of quantum mechanics and the hamiltonjacobi equation of classical mechanics. Most textbook treatments of hamiltonjacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. The spaces satisfying the stronger condition that the corresponding schrodinger equation is separable are isolated in a. There is a parameter in quantum mechanics, called planks constant, that determines how quantum mechanical your system is.
Books like goldstein explain the relationship hamiltonjacobi eq. For this reason, bohm 1 suggested that s is the classical action function, which relates to the actual velocity, vrsm, of the particle. The complete integral defines an integral surface on which there are characteristics that are solutions to a set of first order coupled odes. About the derivation of the hamiltonjacobi equation. In the hamiltonjacobi theory we look for a function wqi, pi, which generates the desired cannonical transformation making the hamiltonian a constant. It is purely classical and a powerful and elegant method to solve classical equations of motion of hamiltonian systems. A transition from newtons second law to the hamiltonjacobi equation can be achieved with the help of the algorithm for transforming a system of ordinary di erential equations into a partial di erential equation. Hamiltonjacobi and schrodinger equations quantum mechanics is a theory significantly larger than its standard textbook formulation. The motivation of this paper is to improve the general knowledge of the path integral formulation of quantum mechanics, and illustrate its importance, by exploring the connections of this approach with the more conventional one based on the schrodinger wavefunction. Direct derivation of schrodinger equation from hamiltonjacobi equation using uncertainty principle article in romanian journal of physics 569. Q vanishes and 4 reduces to the hamiltonjacobi equation. The lhs is the classical hamiltonjacobi equation for complex s, therefore complex x and p complex trajectories.
Of course, a harmonic oscillator can be easily solved using the conventional equation of motion, but this excercise would be useful to understand the basic method. Dec 08, 2014 lecture 5 of my quantum theory course at mcgill university, fall 2012. One of their consequences is the schrodinger equation for stationary states of the molecule. We begin with its origins in hamiltons formulation of classical mechanics. Hamiltonjacobi and schrodinger separable solutions. In this way 3 simply expresses the conservation of probability. Since only appears differentiated, if we have a solution to the equation, we can always add an arbitrary constant term, to give an equally valid solution.
The postulates of the quantum theory constitute the foundation of quantum mechanics. The hamilton jacobi equation is therefore a third complete description of the dynamics, equivalent to lagranges equations and to hamilton s equations. Solution of the timedependent schrodinger equation method 1. Sep 05, 2017 first of all the hamilton jacobi partial differential equation has nothing to do with quantum theory. The rhs is the quantum potential which is now complex. Stefano bianchini an introduction to hamiltonjacobi equations. The similarity between the schrodinger equation and the hamiltonjacobi hj equation of classical mechanics is very apparent and well known 18. The latter is a partial di erential equation of the rst order. When that parameter is zero, your system behaves classically. The wavefunction tpx, e satisfies the correct schrodinger equation and the appropriate physical boundary conditions.
The hamiltonjacobi equation is also used in the development of numerical symplectic integrators 3. Generation of solutions of the hamiltonjacobi equation. Schrodingers own published derivations of quantum wave equations, the first of which was also based on the hamiltonjacobi equation, are also. Pdf schrodinger and the hamiltonjacobi equation jacques. The hamilton jacobi equation also represents a very general method in solving mechanical problems.
Deriving time dependent schrodinger equation from wavemechanics, schrodinger time independent equation, classical and hamiltonjacobi equations nilesh p. Bardapurkar3 1department of physics, badrinarayan barwale mahavidyalaya, jal na, maharashtra, india. The hamiltonjacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the point of closest approach between classical and quantum mechanics. We will show that under suitable conditions on, the hamiltonjacobi equation has a local solution, and this solution is in a natural way represented as a lagrangian. With regard to the foundations of quantum mechanics, recent progress in the generalized hamiltonjacobi. Let us apply hamiltonjacobi method to a harmonic oscillator. Solution of the timedependent schrodinger equation method 2. D the hamiltonjacobi equation for a particle of mass m in a realvalued3 potential v r, t is. There still is a widely held perception that the generalized hamiltonjacobi representation is at best only equivalent to the schr.
The hamiltonjacobi equation hj equation is a special fully nonlinear scalar rst order pde. The hamiltonjacobi equation university of virginia. The hamilton jacobi equation hj equation is a special fully nonlinear scalar rst order pde. Solutions of the schrodinger equation given by solutions. This paper contains an investigation of spaces with a two parameter abelian isometry group in which the hamilton jacobi equation for the geodesies is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. The hamiltonjacobi equation is a partial nonlinear differential equation. In classical mechanics, a change of variables from q i,p itoq i,p. On the origins of the schrodinger equation 8 april 20, by lisa zyga. This fact does not sound very strange if we consider the relationship between the hj equation and the schrodinger equation. Let say we are able to find a canonical transformation taking our 2n phase space variables directly to 2 qp ii, n constants of motion, i. Quantum mechanics is categorical in nature and we need to go beyond parochial points of view and understand the links between different formulations and the categorical approach. Derivation of the schrodinger equation from the hamiltonjacobi equation 65.
The selfadjoint form of the classical equation of motion of the harmonic oscillator is used to derive a hamiltonianlike equation and the schrodinger equation in quantum mechanics. The transformation equation relating the old and new cannonical variables are 2. In modern language, a solution to this hamiltonjacobi equation is a generating function 1 for a symplectomorphism that maps the hamiltonian vector. Next, we show how the equation can fail to have a proper solution. This procedure leads to a condition for the equivalence of the two equations and shows the possibility. To make the transition from the nonlinear classical. We derived the aharonovbohm effect from hamiltonjacobi equation thereby having proved that this effect is of classical origin.
One can formulate quantum mechanics in the semiclassical ap. Coherent states, schrodinger equation, and the hamilton. The hamilton jacobi equation is a partial nonlinear differential equation. Derivation of the schrodinger equation from the hamiltonjacobi. Hamiltonjacobi equation with time dependent hamiltonian. It is this mutual coupling between amplitude and phase that defines a quantum matter wave and ensures the linearity of the wave equation. An introduction to hamilton jacobi equations stefano bianchini. Also we give a short introduction into the control theory and dynamic programming, thus also deriving the hamiltonjacobibellman equation. This paper is a survey of the hamiltonjacobi partial di erential equation. In the timeindependent schrodinger equation stationary states can be produced as solutions of the equation. Lecture 5 of my quantum theory course at mcgill university, fall 2012. Hamilton jacobi theory december 7, 2012 1 free particle thesimplestexampleisthecaseofafreeparticle,forwhichthehamiltonianis h p2 2m andthehamiltonjacobiequationis.
Hj equation as an evolution equation, which determines the function sqi. Generic hjb equation the value function of the generic optimal control problem satis es the hamiltonjacobibellman equation. Hamilton jacobi eq examples visual quantum physics. Hamiltonjacobi equation which governs classical mechanics and electrodynamics explicitly depends on the electromagnetic potentials a, similar to schroedinger equation. The timedependent schrodinger equation is a cornerstone of quan tum physics. An overview of the hamiltonjacobi equation alan chang abstract. Im not going to go through the motions of solving this thing, as its been done elsewhere usually using an additive separation of variables techniquedetails can be found here and elsewhere. Indeed, in the classical matter wave, this coupling is broken. In mathematics, the hamiltonjacobi equation hje is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations, and is a special case of the hamiltonjacobibellman equation. The spaces satisfying the stronger condition that the corresponding schrodinger equation is separable are isolated in a canonical. Wavefunctions and the hamiltonjacobi equation sabrina gonzalez pasterski dated. This paper contains an investigation of spaces with a two parameter abelian isometry group in which the hamiltonjacobi equation for the geodesies is soluble by separation of variables in such a way that a certain natural canonical orthonormal tetrad is determined. Derivation of the schrodinger equation from the hamilton.
The derivation of the timedependent equation is based on an a priori assumption equivalent to feynmans dynamical postulate. Solutions to the hamiltonjacobi equation as lagrangian. The hamiltonjacobi equation in one dimension for a particle of mass m and momentum p can be written as 2 1 2, 22 ssp vxt vxt tm mx. Hamiltonjacobi theory december 7, 2012 1 free particle thesimplestexampleisthecaseofafreeparticle,forwhichthehamiltonianis h p2 2m. Schrodinger equation an overview sciencedirect topics. Hamiltonjacobi and schrodinger separable solutions of. Deriving time dependent schrodinger equation from wave. It is named for william rowan hamilton and carl gustav jacob jacobi. Last time i presented a concrete problem in the phase space formulation, and. The hamilton jacobi equation is 4 where weve assumed uniform energy again. In turn, the action s obeys the hamiltonjacobi equation. Recall that the qcm general wave equation derived from the general relativistic hamiltonjacobi equation is approximated by a schrodingerlike wave equation and that a qcm quantization state is completely determined by the systems total baryonic mass m and its total angular momentum h. Schrodinger s equation and the hamilton jacobi equations are cousins and they both are equations in the configuration space. Sep 10, 2015 conversely, if we plug the ansatz into the schrodinger equation we recover the hamilton jacobi equation and an additional term proportional with \\hbar\.
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