Principle of Least Action in Classical and Quantum Mechanics, Study notes of Physics

Download Principle of Least Action in Classical and Quantum Mechanics and more Study notes Physics in PDF only on Docsity! Principle of Least Action Manoj K. Harbola Department of physics Acknowledgement: Varun LEAST ACTION HERO DOES A RAY OF LIGHT KNOW WHERE IT'S GOING? (Jim Holt, Lingua Franca vol. 9, No. 7,October 99) Suppose you are standing on the beach, at some distance from the water. You hear cries of distress. Looking to your left, you see someone drowning. You decide to rescue this person. Taking advantage of your ability to move faster on land than in water, you run to a point at the edge of the surf close to the drowning person, and from there you swim directly toward him. Your path is the quickest one to the swimmer − but it is not a straight line. Instead, it consists of two straight-line segments, with an angle between them at the point where you enter the water. What does the path of least action give us? Action = ( )2 2 22 1 2 )()(2)(2v hxdVEmhxmEds B A +−−++=∫ minimization of action with respect to x gives E VE hxd xd hx x )( )( )( 2 2 2 2 1 2 − = +− − + 1 v v)( sin sin 1 2 2 1 <= − = E VE θ θ which is equivalent to θ1 A B h1 h2 d P.E.=0 P. E.= V>0 θ2x What does the path of least time give us? Total time = )(2 )( 2v 2 2 22 1 2 VEm hxd mE hxds − +− + + =∫ minimization of time with respect to x gives )( )( )( 2 2 2 2 1 2 VE E hxd xd hx x − = +− − + which is equivalent to 1 v v )(sin sin 2 1 2 1 >= − = VE E θ θ θ1 A B h1 h2 d P.E.=0 P. E.= V>0 θ2x When particle strikes the surface, the component of velocity along the surface remains unchanged θ1 A B θ2 v1 v2 v1sinθ1 v2sinθ2 2211 sinvsinv θθ = 1 v v sin sin 1 2 2 1 <= ⇓ θ θ θ1 A B θ2 Trajectory of the particle is the path of least action Test case 2: A cricket ball hit so that it reaches a fielder x y ∆y Let actual path be y(x) Let a nearby path be y(x) ∫∫      +−== B A B A dx dx dymgyE m ds 2 1)(2vAAction for the actual path y(x) Change in action for a nearby path y(x) ∫∫     ′+−== B A B A dxymgyE m ds 21)(2vA δδδ must be zero if action for the actual path is minimum Change in action arises from: (i) Change in the speed )()()( xymgyE dy dmgyE δδ       −=− δy(x+∆x) δy(x) x x+∆x (ii) Change in the length of trajectory dxyy yd ddxy ′      ′+ ′ =′+ δδ 22 11 )()()( xy dx d x xyxxyy δδδδ = ∆ −∆+ =′ where Change in action therefore is ∫       ′+ ′ −+−′+= B A dxxy dx dy yd dmgyExymgyE dy dy )(1)()()(1A 22 δδδ ∫             ′+ ′ −−−′+= B A dxxyy yd dmgyE dx dmgyE dy dy )(1)()(1A 22 δδ Integration by parts leads to Since δy(x) is arbitrary, δA=0 implies 01)()(1 22 =      ′+ ′ −−−′+ y yd dmgyE dx dmgyE dy dy This simplifies to 0)1()(2 2 =′++−′′ ymgmgyEy Comparison with Newtonian approach: Given initial position and velocity of a particle, Newtonian method builds up its trajectory in an incremental manner by updating the velocity and position. Energy of the particle may or may not be fixed. Principle of least action says if a particle of fixed energy has to go from point A to point B, the path it takes is that which minimizes the action. But this can't be right, can it? Our explanation for the route taken by the light beam (particle in our case) − first formulated by Pierre de Fermat in the seventeenth century as the principle of least time (principle of least action in the present case) − assumes that the light (particle) somehow knows where it is going in advance and that it acts purposefully in getting there. This is what's called a teleological explanation. (Jim Holt) The idea that things in nature behave in goal-directed ways goes back to Aristotle. A final cause, in Aristotle's physics, is the end or telos toward which a thing undergoing change is aiming. To explain a change by its final cause is to explain it in terms of the result it achieves. An efficient cause, by contrast, is that which initiates the process of change. To explain a change by its efficient cause is to explain it in terms of prior conditions. One view of scientific progress is that it consists in replacing teleological (final cause) explanations with mechanistic (efficient cause) explanations. The Darwinian revolution, for instance, can be seen in this way: Traits that seemed to have been purposefully designed, like the giraffe's long neck, were re-explained as the outcome of a blind process of chance variation and natural selection. (Least Action Hero, Jim Holt, Lingua Franca vol. 9, No. 7, October 99) Plan of the talk: Aristotle and the motion of planets Reflection of light and Hero of Alexandria Fermat’s principle of least time for light propagation; Descartes versus Fermat Wave theory and Fermat’s principle Maupertuis’ principle of least action Euler-Lagrange formulation Hamilton’s investigations Quantum connections REFLECTION OF LIGHT & HERO OF ALEXANDRIA (125 BC) Whatever moves with unchanging velocity moves in a straight line…. For because of the impelling force the object in motion strives to move over the shortest possible distance, since it has not the time for slower motion, that is, for motion over a longer trajectory. The impelling force does not permit such retardation. And so, by reason of its speed, the object tends to move over the shortest path. But the shortest of all lines having the same end points is a straight line……Now by the same reasoning, that is, by a consideration of the speed of the incidence and the reflection, we shall prove that these rays are reflected at equal angles in the case of plane and spherical mirrors. For our proof must again make use of minimum lines. Let a light ray start from point A and reach point B after reflection. The true path is AOB such that rays AO and BO make equal angles from the mirror. A B OO1 B1 C Draw an alternate path AO1B Drop a perpendicular BC on the mirror and extend it to B1 so that BC=B1C. Join O and B1 and O1 and B1. From congruency of ∆BOC and ∆B1OC and the fact that AO and BO make equal angles from the mirror, it follows that AOB1is a straight line. In ∆AO1B1: AO1+O1B1 > AB1( = AO+OB1=AO+OB) Path AOB is the shortest Proof: Descartes’ explanation (1637) of Snell’s law: Argument given by Descartes is a mechanical one, based on the fact that the component of velocity along the surface remains unchanged Descartes explained the constancy of the ration of sine of angles in terms of the ratio of the speed of light in the two media By Descartes’ explanation, light had to be moving faster in the denser medium 2211 sinvsinv θθ = 1 2 2 1 v vconstant sin sin == θ θ θ1 v1 v1sinθ1 v2sinθ2 v2 θ2 Descartes (1596-1650) Newton (1642-1727) Fermat’s principle of least time (1658): Generalization of Hero’s explanation of reflection to include refraction of light. During refraction a light ray does not take the path of least distance; that would be a straight line. Between two points, a light ray travels in such a manner that it take the least time. For reflection this leads to equal angles of incidence and reflection θ1 A θ2 B 2 1 2 1 v vconstant sin sin == θ θ For refraction this implies By Fermat’s principle of least time, light moves slower in the denser medium. Fermat (1601-1665) Newton (1642-1727) Descartes versus Fermat: Descartes (1596-1650) Fermat (1601-1665) Newton (1642-1727) Descartes believed that light traveled infinitely fast. Fermat on Descartes: 1. “of all the infinite ways to analyze the motion of light the author has taken only that one which serves him for his conclusion; he has therefore accommodated his means to his end, and we know as little about the subject as we did before.” 2. rejects Descartes’ assertion of infinite speed of light and therefore his illogical conclusion that light travels faster in water than in air . According to Fermat light traveled at finite speed in air and slowed down in water. Experimental verification of finiteness of speed of light – 1675 by Roemer Measurement of speed of light in water – 1850 by Fizeau and Foucault These observations CONFIRM Fermat’s principle of least time HUYGENS’ WAVE THEORY & FERMAT’S PRINCIPLE Huygens’ wave theory (1629-1695) and the principle of least time According to Huygens’ theory, light travels as a wave with path of light ray being in the direction perpendicular to the wavefront A B Consider the true ray path from A to B and also an alternate path. Because true path is perpendicular to the wavefronts, AB (true path) < AB (alternate path) Newton (1642-1727) Actual principle is the principle of stationary (minimum, maximum or saddle point) time Consider a light beam that starts in a diverging manner from point A and then converges to point B. A B All nearby paths of light should not have different time of arrivals implying stationarity of time of travel. Example 1: In an elliptical mirror, light starting from one focus of the ellipse and reaches the other focus after reflection from the mirror. There are many possible paths for this and all of them are equal. F1 F2 Does light really go through all possible paths? Experiment : Take a point different from the image. No light from the source reaches the that point. Light source No light Now blacken parallel strips on the mirror to remove light of opposite phase. It forms a grating and light reaches many different points. Light source Image PRINCIPLE OF LEAST ACTION MAUPERTUIS, LAGRANGE & EULER Principle of Least Action Nature acts in a way so that it renders a quantity called action a minimum Maupertuis in “The agreement between different laws of Nature that had, until now, seemed incompatible” read on April 15, 1744 to Académie des sciences. Newton (1642-1727) Action is defined as the product of the mass, the velocity and the distance. sm ××= vAction Comment: Maupertuis’ attempts was to explain the propagation of light and movement of a particle by a single principle. Example 3: Refraction of light θ1 A θ2 B O OBAO ×+×= 21 vvAction Minimization of action leads to constant v v sin sin 1 2 2 1 == θ θ Snell’s law is verified through the principle of least action, and agrees with Descartes’ conclusions. Lagrange (1736-1813) on Maupertuis’ principle in Mécanique Analytique, 1788 “This principle, looked at analytically, consists in that, in the motion of bodies which act upon each other, the sum of the product of the masses with the velocities and with the distances travelled is a minimum. The author deduced from it the laws of reflection and refraction of light, as well as those of the impact of bodies. But these applications are too particular to be used for establishing the truth of a general principle. Besides, they have somewhat vague and arbitrary character, which can only render the conclusions that might have been deduced from the true correctness of the principle unsure……………… But there is another way in which it may be regarded, more general, more rigorous, and which itself merits the attention of the geometers. Euler gave the first hint of this at the end of his Traité des isopérimètres, printed at Lausanne in 1744.” Principle of Least Action in Mechanics Proper mathematical foundation is provided by Euler (1744) Before paying attention to this problem, Euler had already developed Calculus of Variations and given the Euler condition for making the variation of an integral of the form ∫ ′ ),( ),( 22 11 ),,( yx yx dxxyyf between two fixed points (x1,y1) and (x2,y2) vanish with respect to arbitrary variations in y(x) Euler condition 0= ∂ ∂ −      ′∂ ∂ y f y f dx d ( ) 2 yx 232 2 11 v y FyF y ym ′+ +′− = ′+ ′′ ( ) y yr ′′ ′+ = 21curvatureofradius Fx Fy X Y 2 yx 2 1 v y FyF r m ′+ −′ = MAKING ACTION STATIONARY LEADS TO THE CORRECT FORCE BALANCE EQUATION What does the minimum action principle imply for one-dimensional motion? ∫= dxxA )(v )()(v)(v ∫∫ += dxxdxxA δδδ t x δx If the total energy is constant ( ) ( )           ∂ ∂      −= ∂ ∂ − −= − ∂ ∂ = x x xU m x x xU xUEm xxUE mx x δ δ δδ )( v 1 )( )(2 1 )(2)(v { } { } ( ) )( )()( )()( xd xx xx xxxxxx xxxdx δ δδδ = −∆= ∆−∆= −−∆+−∆+= −∆+= δx δ(x+∆x) ∆x ∆x δ(∆x) ∆(δx) ∆t x ∫∫ ∫∫ + ∂ ∂      −= + ∂ ∂      −= xdxdx x xU m xdxdx x xU m A δδ δδδ v)( v 1 )(v)( v 1 If the end points of the trajectory are kept fixed HAMILTON’S PRINCIPLE OF VARYING ACTION PARALLEL BETWEEN GEOMETRIC OPTICS AND MECHANICS Consider the action integral dsxA ∫= )x,x,v(x)( 321∆s as a function of the end points of the true path. The integral is obviously taken along the true path. As the path is increased by ∆s to the next point, we have )(vOR)(v x ds dAsxA =∆=∆ Can this equation be used instead to find the path taken by light? Function v(x1,x2,x3;α1,α2,α3) is considered to be a function of the directional cosines {αi} of the ray of light. Making action stationary with respect to variations in {xi; αi} gives the equation for {αi} . Conventional approach (Hamilton): Recall that in an earlier minimization, the integrand was taken to be function of y(x) and y′(x). Thus ∫ ′+= dxyyA 21)(v y(x) is then found by making the variation of the action vanish with respect to variations δy(x). Now the independent variables are taken to be {xi; αi} instead. Thus ∫= dsxA ii });({v α {αi(x)} are found by making the variation of the action vanish with respect to variations {δxi} and {δαi}. Using make v({xi;αi}) homogeneous Of degree 1 in {αi} 12 3 2 2 2 1 =++ ααα ∑ ∂ ∂ = i i i α α vv since v is a homogeneous function of {αi} of degree 1 Demand that δA vanish for arbitrary variations with the end points of the path fixed i.e. δx1/0=0. This gives ∫ ∑ ∫∑∑       ∂ ∂ −+       ∂ ∂ − ∂ ∂ +      ∂ ∂ −      ∂ ∂ = )(vv vvvv 01 ds xdds x xxA i i i i iii i ii i i δα α δ α δ α δ α δ Differential equation for the path of light ray ii dds x α∂ ∂ = ∂ ∂ vv 2 3 2 2 2 1321 ),,(v);(v αααα ++= xxxxThus Note: 12 3 2 2 2 1 =++ ααα ∑∑∫       ∂ ∂ −      ∂ ∂ = i i ii i i xxds 01 vvv δ α δ α δ Now consider the integral with the initial point fixed and δx1 non zero and in the direction of the path. )0(1 == iii sx δαδαδ ssxds i i ii i i δδα α δ α δ vvvv 1 =      ∂ ∂ =      ∂ ∂ = ∑∑∫Then The action is a function of the end points of the true path∫ dsv CONCLUSION: Stationary action implies existence of a characteristic function A(x) such that iix xA α∂ ∂ = ∂ ∂ v)( 1 How to find the path if A(x) and v(x,α) are given? From the equation solve for as a function of (x1,x2,x3) iix xA α∂ ∂ = ∂ ∂ v)( 1 ),,( 321 ααα Differential equation for the characteristic function A(x) vv)( 1 i iix xA α α = ∂ ∂ = ∂ ∂ 12 3 2 2 2 1 =++ ααα )(v)()()( 2 2 3 2 2 2 1 x x xA x xA x xA =      ∂ ∂ +      ∂ ∂ +      ∂ ∂ v(x) is the refractive index of the medium Mechanical systems: In going from point A to point B, a particle also satisfies the principle of least action 0v =∫ dsδ 2 3 2 2 2 1321 2 3 2 2 2 1 ),,(vvvvv ααα ++=++= xxx is the speed of the particle Thus there exists a characteristic function A(x) for a mechanical system also such that })({v})({v i i i i xAOR x xA ∇= ∂ ∂ =  And the path of a particle can be determined if we know the characteristic function Equation for the characteristic function )(v)()()( 2 2 3 2 2 2 1 x x xA x xA x xA =      ∂ ∂ +      ∂ ∂ +      ∂ ∂ From the energy conservation equation })({v 2 1 2 ixUmE += Thus the equation for the characteristic function is mExmU x xA x xA x xA i 2})({2)()()( 2 3 2 2 2 1 =+      ∂ ∂ +      ∂ ∂ +      ∂ ∂ Example: A projectile thrown with initial speed v0 at an angle φ0 in a gravitational field mgyyU =)( The equation for the characteristic function )2v( 2 0 22 gym y A x A −=      ∂ ∂ +      ∂ ∂ Solve the equation by separation of variables to get 2322 0 2322 0 )2v( 3 1)(v 3 1),( gyk g xkk g yxA −−−+−= Values of A(x,y) are obtained by substituting for x and y, the coordinates of a trajectory Question: Can we associate the action of a particle with a phase? Trajectories of the projectile Mechanical motion of a particle is like the motion of ray of light and therefore equivalent to Geometric optics. QUANTUM CONNECTIONS Fast forward to 1920s: When it was discovered that particles have a wave associated with them, Hamilton’s theory became the natural choice to account for it and develop the quantum-mechanical wave equation. How Schrödinger obtained the wave equation (Ist paper by Schrödinger) mExmU x xA x xA x xA i 2})({2)()()( 2 3 2 2 2 1 =+      ∂ ∂ +      ∂ ∂ +      ∂ ∂ Start with the equation for the action Treating action like phase, take the wavefunction Ψ as Ψ==Ψ log)(OR)/)(exp( KxAKxA Substitute this wavefunction in the equation for action to obtain a quadratic form in Ψ, which is equal to zero 0)( =∆−∆=∆ tExAmφ xx x AxA particle∆=∆ ∂ ∂ =∆ v)( This gives )(2mv UEm EE t xu particle phase − == ∆ ∆ = particle phase particlephase m h UEm h hE u u v)(2 ANDv = − ==≠ λ Group velocity of the waves particlegroup UEm Eh E k u v )(21 = −∂ ∂ =      ∂      ∂ = ∂ ∂ = λ ω And finally the wave equation 01 2 2 2 2 =Ψ∇− ∂ Ψ∂ tu phase leads to the Schrödinger equation Ψ=Ψ+Ψ∇− EU m h 2 2 2 8π      −Ψ=Ψ − = t h Eixtx UE Euphase π2exp)();(AND )(2 Substituting Classical Mechanics: Path of a particle is that of least action and therefore normal to the surfaces of constant action Quantum Mechanics: Because of the waves associated with a particle, it does not know which path to take. It takes all possible paths with certain probability amplitude and phase and these probability amplitudes interfere. The phase depends on the action. Path of least action is where the interference is constructive to the largest extent possible. Classically we see only those result when amplitudes interfere constructively giving a large final amplitude A comparison between Classical and Quantum Mechanics (Feynman):