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An Introduction to the Path Integral Approach to Quantum Mechanics

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An Introduction to the Path Integral Approach to Quantum Mechanics An Introduction to the Path Integral Approach to Quantum Mechanics Luke Bovard Department of Physics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (Dated: December 19, 2009) In 1948 Richard Feynman reformulated quantum mechanics using a sum-over-all paths approach. In this derivation he obtained a completely di®erent way of looking at quantum mechanics that demon- strates the close relationship between the classical action and the evolution of the wavefunction. In this paper I will give a short overview of the propagator and how it can be used to describe the system. Then I will discuss the results of the Feynman path integral formulation and its equivalence to the SchrÄodingerequation. Finally I will explain how this method can be used to solve a speci¯c potential and discuss applications to more advanced topics. HISTORY AND INTRODUCTION SchrÄodingerEquation[2]. Now we will show how to de- rive equation (1) following Shankar[3]. We ¯rst start with Werner Heisenberg, Max Born, and Pascual Jordan the SchrÄodingerequation originally developed matrix mechanics in 1925 and a year d later Erwin SchrÄodingercame up with the di®erential i¹h jª(t)i = Hjª(t)i dt equation wave function approach. That same year, Paul Dirac demonstrated that matrix mechanics is equivalent From this we can obtain the time independent to the wave function approach. These remained the stan- SchrÄodingerequation as HjÃni = EnjÃni. We now dard approaches to quantum mechanics until 1948 when use an important property of SchrÄodingerequation in Richard Feynman developed the path integral formula- that we can expand the original wavefunctionP out in tion based o® investigations by Eugene Wigner[1]. Feyn- terms of the eigenkets jÃni; jª(t)i = an(t)jÃni where man's formulation of quantum mechanics determines how an(t) = an(0) exp(¡iEt=¹h) are obtained from the sepa- the wavefunction evolves in time by ¯nding a propagator ration of variables and a0(0) = hÃnjª(0)i. Now substi- which describes how the system evolves in time and which tuting back into the expansion for jª(t)i we obtain is independent of the initial wavefunction. Feynman's key X ¡iEt=h¹ insight was to recognize the fact that this propagator is jª(t)i = jÃnihÃnjª(0)ie directly related to the sum of all possible paths a particle X ¡iEt=h¹ can take between two points. = jÃnihÃnje jª(0)i = U(t)jª(0)i OVERVIEW and hence we are able to obtain the expression for the propagator. We can now completely describe the system The Feynman path integral formulation depends upon since this propagator is independent of the initial position ¯nding the propagator for a given system. The propaga- of the wavefunction and can describe how the wavefunc- tor is a function that is independent of the initial con- tion evolves in time. We also note that this is entirely ditions and describes how a system evolves in time. In equivalent to another form of the propagator which we quantum mechanics the propagator is as follows write as U(t) = e¡iHt=h¹ and the curious reader may con- X sult [2] to ¯nd a di®erent derivation of (1). We also note ¡iEt=h¹ U(t) = jÃnihÃnje (1) that this derivation assumes discrete and non-degenerate energy levels but if we have a continuous spectrum, such where we can write the wavefunction in the di®erent as the free particle, we replace the sum with an integra- eigenstates evaluated in whatever basis happens to be tion over jÃni. If we have degeneracies we have to double convenient. To obtain this result we recall the following sum over the degeneracies. equation which describes how the wavefunction evolves in time. PATH INTEGRAL PROPAGATOR jª(t)i = U(t)jª(0)i As we have just seen above, if we are able to work out All this says is that given some initial wavefunction at the propagator we can completely describe the system at some time t0, chosen arbitrarily to be zero, the prop- any time. However there is a great di±culty in working agator, U(t), determines how the wavefunction evolves out these propagators due to the complicated nature of in time. Analysis of this equation can lead us to the the summations. Instead we turn to the path integral J. Phy334 4, 0014 (2009). JOURNAL OF PHYS 334 December 19, 2009 method due to Feynman to evaluate these propagators. Above the term 'moving towards the particles classical The actual derivation of the propagator is fairly involved path' was used, but what exactly do we mean by 'mov- and too long to be derived in this short introduction but ing towards'? How far away from the particle's classical can be read in either [3] or [4]. Instead I will discuss path must we go before we encounter destructive interfer- the consequences of the path integral formulation. The ence? E®ectively we can make an elementary guess that major result of the derivation is that between two points S[xcl(t)]=¹h ¼ ¼ or otherwise if the action is ¼¹h away from (x; t) ! (x0; t0) the classical path we have destructive interference. The X classical action of a macroscopic particle is on the order U(x; t; x0; t0) = A eiS[x(t)]=h¹ (2) of 1 erg sec ¼ 1027¹h[3] which is much greater then ¼¹h. In all paths this case we are well beyond the ¼¹h and hence the clas- sical path completely dominates the sum. However for where A is a normalisation factor and S[x(t)] is the clas- an electron, which has a very tiny mass, has an approxi- sical action corresponding to each path and is denoted mate action of ¹h=6[3] which is well within the ¼¹h needed by and thus we cannot think of the electron moving along Z it's classical path since many of the non-classical paths S = Ldt contribute to the summation. An alternate way to write this formulation to better ac- EQUIVALENCE TO THE SCHRODINGERÄ knowledge that there are an in¯nite number of paths is EQUATION Z x0 U(x; t; x0; t0) = eiS[x(t)]=h¹ D[x(t)] (3) Feynman has now told us the propagator is equiva- x lent to summing over the action of all the possible paths between two points. But does this give us the correct where it is implied by the D[x(t)] that we sum over all result as predicted by the SchrÄodingerequation? To 0 0 the paths connecting (x; t); (x ; t ). show the equivalence imagine a system evolving from An immediate consequence of this formula is that all (x ¡ »; t ¡ ²) ! (x; t)[5]. We know that the propaga- paths are equally weighted. Thus a question arises is as tor tells us exactly how this system evolves and we can to how the classical path is favoured in the classical limit approximate this as a straight line and constant velocity and by what mechanism is the classical path chosen over between the two points. Thus we are only summing over the other paths if they are equally weighted? a single path One way to think of the above formula is for each path Z 1 to represent a vector in the complex plane. Assume, with- ª(x; t) ¼ A d»eiS=h¹ ª(x ¡ »; t ¡ ²) out loss of generality, that each vector is of unit length. ¡1 Here the action determines the phase of the vector. As we where A is a normalisation constant and the action is move far away from the classical path there is destructive determined by S = [1=2m(»=²)2 ¡ V (x)]² which is the interference as all the di®erent non-classical paths tend action to the ¯rst order of epsilon. Now we have that Z to cancel each other out. As we move towards the classi- 1 2 im» ¡ i V (x)² cal path the picture changes. We know that Hamilton's ª(x; t) ¼ A d»e 2¹h² e h¹ ª(x ¡ »; t ¡ ²) principle is ¡1 Z with the intention of taking the limit ² ! 0. Now as ±S = ± Ldt = 0 the we take this limit the ¯rst exponential oscillates very rapidly unless the » is smaller then the ². Thus if we pick a small neighbourhood about » such that it is smaller which serves as the basis for analytical classical mechan- then ² we can expand out terms into Taylor Series about ics as both the Hamiltonian and Lagrangian formulation ²; » and hence follow from it. What it states is that the variation along Z 1 im»2 @ª i the classical path is at an extremum. Thus when we start ª(x; t) ¼ A d»e 2¹h² [ª ¡ ² ¡ V ª² + ¢ ¢ ¢ moving towards the particles classical path we start hav- ¡1 @t ¹h ing constructive interference because the variation in the @ª 1 @2ª ¡ » + »2 + ¢ ¢ ¢ ] path is zero and hence we are able to regain the classical @x 2 @x2 path. This means that the destructive interference e®ec- Now that we have expanded the terms, the ª is inde- tively cancels the contribution of all paths that are not pendent of » and hence all the integrals can easily be near the classical path and constructive interference near performed as they are all Gaussian integrals. the classical path allows us to regain the classical path. r · µ ¶ ¸ Thus despite all being equally weighted the classical path 2¼i¹h² @ª i i¹h @2ª ª(x; t) ¼ A ª + ¡ ¡ V ª + ² + ¢ ¢ ¢ is very important. m @t ¹h 2m @x2 2 J. Phy334 4, 0014 (2009).
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