An Introduction to the Path Integral Approach to

Luke Bovard

Department of Physics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (Dated: December 19, 2009) In 1948 reformulated using a sum-over-all paths approach. In this derivation he obtained a completely different 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 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 Schr¨odingerequation. Finally I will explain how this method can be used to solve a specific potential and discuss applications to more advanced topics.

HISTORY AND INTRODUCTION Schr¨odingerEquation[2]. Now we will show how to de- rive equation (1) following Shankar[3]. We first start with , , and the Schr¨odingerequation originally developed mechanics in 1925 and a year d later Erwin Schr¨odingercame up with the differential i¯h |Ψ(t)i = H|Ψ(t)i dt equation function approach. That same year, demonstrated that is equivalent From this we can obtain the time independent to the approach. These remained the stan- Schr¨odingerequation as H|ψni = En|ψni. We now dard approaches to quantum mechanics until 1948 when use an important property of Schr¨odingerequation in Richard Feynman developed the path integral formula- that we can expand the original wavefunctionP out in tion based off investigations by [1]. Feyn- terms of the eigenkets |ψni; |Ψ(t)i = an(t)|ψni where man’s formulation of quantum mechanics determines how an(t) = an(0) exp(−iEt/¯h) are obtained from the sepa- the wavefunction evolves in time by finding a propagator ration of variables and a0(0) = hψn|Ψ(0)i. Now substi- which describes how the system evolves in time and which tuting back into the expansion for |Ψ(t)i we obtain is independent of the initial wavefunction. Feynman’s key X −iEt/h¯ insight was to recognize the fact that this propagator is |Ψ(t)i = |ψnihψn|Ψ(0)ie directly related to the sum of all possible paths a particle X −iEt/h¯ can take between two points. = |ψnihψn|e |Ψ(0)i = U(t)|Ψ(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 finding 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/h¯ and the curious reader may con- X sult [2] to find a different derivation of (1). We also note −iEt/h¯ U(t) = |ψnihψn|e (1) that this derivation assumes discrete and non- but if we have a continuous spectrum, such where we can write the wavefunction in the different as the , we replace the sum with an integra- eigenstates evaluated in whatever basis happens to be tion over |ψni. 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 |Ψ(t)i = U(t)|Ψ(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 difficulty in working agator, U(t), determines how the wavefunction evolves out these 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? Effectively we can make an elementary guess that major result of the derivation is that between two points S[xcl(t)]/¯h ≈ π or otherwise if the action is π¯h 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)]/h¯ (2) of 1 erg sec ≈ 1027¯h[3] which is much greater then π¯h. In all paths this case we are well beyond the π¯h 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 , which has a very tiny , has an approxi- sical action corresponding to each path and is denoted mate action of ¯h/6[3] which is well within the π¯h 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 SCHRODINGER¨ knowledge that there are an infinite number of paths is EQUATION

Z x0 U(x, t; x0, t0) = eiS[x(t)]/h¯ 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 Schr¨odingerequation? 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 − ξ, 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 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 ∞ to represent a vector in the complex plane. Assume, with- Ψ(x, t) ≈ A dξeiS/h¯ Ψ(x − ξ, t − ²) out loss of generality, that each vector is of unit length. −∞ 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(ξ/²)2 − V (x)]² which is the interference as all the different non-classical paths tend action to the first order of epsilon. Now we have that Z to cancel each other out. As we move towards the classi- ∞ 2 imξ − i V (x)² cal path the picture changes. We know that Hamilton’s Ψ(x, t) ≈ A dξe 2¯h² e h¯ Ψ(x − ξ, t − ²) principle is −∞ Z with the intention of taking the limit ² → 0. Now as δS = δ Ldt = 0 the we take this limit the first exponential oscillates very rapidly unless the ξ is smaller then the ². Thus if we pick a small neighbourhood about ξ 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, ξ and hence follow from it. What it states is that the variation along Z ∞ imξ2 ∂Ψ i the classical path is at an extremum. Thus when we start Ψ(x, t) ≈ A dξe 2¯h² [Ψ − ² − V Ψ² + ··· moving towards the particles classical path we start hav- −∞ ∂t ¯h ing constructive interference because the variation in the ∂Ψ 1 ∂2Ψ − ξ + ξ2 + ··· ] path is zero and hence we are able to regain the classical ∂x 2 ∂x2 path. This means that the destructive interference effec- Now that we have expanded the terms, the Ψ is inde- tively cancels the contribution of all paths that are not pendent of ξ 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 2πi¯h² ∂Ψ i i¯h ∂2Ψ Ψ(x, t) ≈ A Ψ + − − V Ψ + ² + ··· is very important. m ∂t ¯h 2m ∂x2

2 J. Phy334 4, 0014 (2009). JOURNAL OF PHYS 334 December 19, 2009

Now to make the terms in the order of ² vanish we set we instead look at the propagator from Feynman’s point A = (m/2πi¯h²)1/2 and hence the terms in front of the ² of view, all we need to do is obtain the classical action of are the oscillator and we are within a function A(t) from the sum. The classical action is easily derived as ∂Ψ ¯h2 ∂2Ψ i¯h = 2 + V Ψ (4) mω ∂t 2m ∂x S = [(x2 + x02) cos ωt − 2x x0] cl 2 sin ωt 0 0 which is the time-dependent Schr¨odinger equation. In this case the function A(t) can be evaluated using Thus the Feynman path integral formulation is entirely [4] and works out to be equivalent to the Schr¨odingerwave equation. ³ m ´1/2 A(t) = 2π¯hit APPLICATION Thus the propagator for the is · ¸ ³ ´1/2 To demonstrate the Feynman path integral formula- m imω 2 02 0 U = [(x + x ) cos ωt − 2x0x ] tion, we will look at the case of the general potential 2π¯hit 2¯h sin ωt 0 V = a + bx + cx2 + dx˙ + exx˙, of which the well-known (6) harmonic oscillator is a special case. The evaluation of path integrals of this form can be found in both [4] and This expression for the propagator can in fact be used [2] and hence I will discuss only the results. It can be to find the energy levels and the wave functions[4] and found that for the above potential the propagator is we are able to completely describe the harmonic oscilla- tor without resorting to solving a complicated PDE but U(x, t; x0, t0) = eiScl/h¯ A(t) (5) instead the equation of for a harmonic oscillator, which is much easier. This is a very interesting result because it says that the propagator depends only on the classical path and some function A(t) which has to be determined through other CONCLUSION means. Consider the case of the free particle. As we ex- plained above it makes sense that the classical path plays While this method looks much more complicated, the an important role in the evaluation of the path integral path integral formulation has a much deeper application and we even reasoned that it dominates the calculation. in quantum mechanics. One such example is that it al- It turns out that it completely dominates the summation lows us to formulate driven quantum harmonic oscilla- and we can easily work out the propagator for the free tors which play an important role in quantum electrody- particle. namics since we can represent the E-M field as quantum Now consider the harmonic oscillator. Suppose we driven harmonic oscillators [4]. Finally the path integral wanted to write it down in position space in the form method leads to perturbation calculations through Feyn- (1). We can derive all the wave functions and energy man diagrams and plays an important role in relativistic levels and then plug them in to obtain the propagator in quantum mechanics alongside the . position space X U(x, t; x0, t0) = |ψihψ|e−iEt/h¯ X∞ ³ mω ´ = A exp − x2 H (x)× [1] R.P. Feynman, Rev. Mod. Phys 20, 367 (1948). n 2¯h n n=0 [2] J.J Sakurai, Modern Quantum Mechanics (Pearson Edu- ³ ´ cation, 1994) mω 02 0 An exp − x Hn(x ) [3] R. Shankar, Principles of Quantum Mechanics (Plenum 2¯h 0 Press, 1980) × exp[−i(n + 1/2)ω(t − t )] [4] R.P Feynman and A. R. Hibbs, Path Integrals and Quan- tum Mechanics (McGraw-Hill Book Company, 1965) where Hn(x) are the Hermite polynomials[6]. This is a [5] M.G. Calkin, Lagrangian and very complicated sum and once evaluated can completely (World Scientific Publishing, 1998) describe the system. Evaluating the sum is no easy task [6] D. Griffiths, Introduction to Quantum Mechanics (Pearson and no attempt to do so will be made here. However if Education, 2005)

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