Feynman Path Integrals in Quantum Mechanics

Feynman Path Integrals in Quantum MechanicsChristian EgliOctober 1, 2004AbstractThis text is written as a report to the seminar course in theoretical physics atKTH, Stockholm. The idea of this work is to show Quantum Mechanics from adifferent perspective: based on the Path Integral formalism, originally worked outby R.P. Feynman in 1948. The mathematical equivalence to the familiar formulationshall be shown. In addition some other applications and methods of Path Integralsin non-relativistic Quantum Mechanics are discussed.Contents1 Introduction 22 The Path Integral Formulation of Quantum Mechanics 22.1 Probability and Probability Amplitude . . . . . . . . . . . . . . . . . . . . 22.2 Probability Amplitude for a Path . . . . . . . . . . . . . . . . . . . . . . . 22.3 The Path Integral and the Wave Function . . . . . . . . . . . . . . . . . . 32.3.1 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Equivalence to Schrödingers Operator Notation 43.1 Derivation of the Schrödinger Equation . . . . . . . . . . . . . . . . . . . 43.2 Schrödinger Equation for the Kernel . . . . . . . . . . . . . . . . . . . . . 54 Pertubation Theory 55 Path Integrals in Phase Space 66 Path Integrals and Topology 76.1 The Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . 76.2 Path Integrals with Topological Constraints . . . . . . . . . . . . . . . . . 77 Statistical Mechanics 88 Summary 81

1 IntroductionThe standard formulations of Quantum Mechanics were developed by Schrödinger, Heisenbergand others in the 1920s. Already 1933 Dirac published a paper where he suggestsexp(iS/¯h) to correspond to the propagator, where S is the classical action. Feynmandeveloped this idea, concerning other paths than only the classical one and publisheda third complete formulation of Quantum Mechanics in 1948 ([1]). As we will see, thisformulation provides a much more intuitive introduction to the quantum theory. First ofall we will have to state the fundamental concepts of Quantum Mechanics. Since they arewell known to the reader I will do that very shortly but in a way which is useful for thedevelopment of the theory in later sections.2 The Path Integral Formulation of Quantum MechanicsThe goal of this section is to introduce Quantum Mechanics in a completely different wayfrom how it is usually done in textbooks.2.1 Probability and Probability AmplitudeLet’s start with the famous imaginary double-slit experiment: We have an electron sourcein some point A, a screen B with two holes through which the electrons might pass, anda detector plane C. We define P 1 (x) as the probability that, if we close hole number2, an electron is detected at the position x in C. Analogously for P 2 (x). It is a wellknown experimental fact that P (x), the probability to arrive at x with both holes open,is not equal to (P 1 (x) + P 2 (x)) as long as we do not detect which hole the electron hasgone through. We can state a correct law for P (x) mathematically as follows: there arecomplex numbers Φ 1 and Φ 2 such thatP = |Φ| 2 , where Φ = Φ 1 + Φ 2 , P 1 = |Φ 1 | 2 , P 2 = |Φ 2 | 2 (1)We call Φ(x) the probability amplitude for arrival at x.2.2 Probability Amplitude for a PathInstead of only having one screen and two holes as in 2.1, we can think about insertingmore screens and having several holes in each screen. The number of possible paths (eachof which will have some ”partial” amplitude Φ i (x)) from A to some point x on C increases.The total amplitude Φ(x) for an electron starting in A to reach this point x on C in acertain time T will be the sum of all partial amplitudes Φ i (x). To formulate this moreformally, we shall limit ourselves to a one-dimensional problem. The generalization toseveral dimensions is obvious.We define P (b, a) as the probability to go from a point x a at the time t a to the point x b att b . We now know that we can write the probability P (b, a) = |K(b, a)| 2 of an amplitudeK(b, a) to go from a to b. This amplitude is the sum of all the partial amplitudes 1 , onefor each possible path from (x a , t a ) to (x b , t b ). We now write the partial amplitudes asΦ[x(t)].∑K(b, a) =Φ[x(t)] (2)all paths from a to bThe question is how each path x(t) contributes to the total amplitude. It is so to speakthe central point in the Path Integral theory of Quantum Mechanics that Feynman postulates2 : The paths contribute equally in magnitude, but the phase of their contribution1 This corresponds to Feynmans postulate I in [1].2 Postulate II in [1]2

is the classical action S in units of the quantum of action ¯h , i.e.where the classical action S is defined by:Φ[x(t)] = const e (i/¯h)S[x(t)] (3)S[x(t)] =∫ tb2.3 The Path Integral and the Wave Functiont aL(ẋ, x, t) dt (4)The main question is, how to perform the sum over all paths. We divide the time interval(t b − t a ) in N intervals of length ɛ. For each path x(t) we can write x i = x(t i ) [Nɛ =t b − t a , ɛ = t i+1 − t i , t 0 = t a , t N = t b , x 0 = x a , x N = x b ] and get as a result for”the sum over all paths”:1K(b, a) = limɛ→0 A∫ ∫∫· · ·e (i/¯h)S[b,a] dx 1Adx 2A · · · dx N−1Awith some normalization factor A, which we need for the convergence of the whole expression.Its value shall be determined later (but it will naturally depend on ɛ). 3K(b, a) is called the Kernel of the motion. We will denote it asK(b, a) =∫ ba(5)e (i/¯h)S[x(t)] Dx(t) (6)and call it in this notation the configuration space path integral or Feynman path integral.For K(b, a) there is a rule for combining amplitudes for events occuring in successionin time:∫K(b, a) = K(b, c)K(c, a)dx c , if t a < t c < t b (7)x c[this rule follows from the fact S[b, a] = S[b, c] + S[c, a] (which is true from the definitionof the action. It holds for any point c lying on the path from a to b). To prove (7), simplyinsert the relation above in Eq. (6).]This procedure can be extended from only having one to N −1 intermediate steps betweena and b. We get the expression∫ ∫K(b, a) = · · K(b, N−1)K(N−1, N−2) · · · K(i+1, i) · · · K(1, a)dx 1 dx 2 · · · dx N−1 .∫x 1 x 2·x N−1(8)If we compare now to (5), we find for the kernel for two points separated by an infinitesimaltime interval ɛ, which is correct to first order 4 in ɛ:K(i + 1, i) = 1 [ ( iɛA exp ¯h L xi+1 − x i, x i+1 + x i, t i+1 + t iɛ 2 2Dealing with Quantum Mechanics we are however more used to have some wave functionΨ(x, t) rather than the kernel 5 . We can derive a expression for Ψ(x, t) from (7):Ψ(x 2 , t 2 ) =∫ ∞−∞)](9)K(x 2 , t 2 ; x 1 , t 1 )Ψ(x 1 , t 1 )dx 1 (10)In physical terms: The total amplitude to arrive at (x 2 , t 2 ) is equal the sum over allpossible values of x 1 of the amplitude to be at x 1 (at a fixed time t 1 ) multiplyed by theamplitude to go from 1 to 2.3 To be precise: In (5) the action is defined by S = P i S(x i+1, x i ), where S(x i+1 , x i ) =min R t i+1t iL(ẋ(t), x(t))dx, which obviously corresponds to the classical action on the interval (x i , x i+1 ).4 Detailed calculations and Feynmans own ”justification” are found in [1]5 Actually the kernel K(x 2 , t 2 ; x 1 , t 1 ) = Ψ(x 2 , t 2 ) is a wave function as well.3

2.3.1 The Free ParticleIt might be good to see a kernel once in its very explicit form. Maybe it will even behelpfull in later sections. For the free particle case we simply have to put L = mẋ 2 /2 in(9) and insert this in (8). I do not write down the calculations, since it is basically solvinga set of Gaussian integrals and can certainly be found in any of the references.[ ] −1/2 2πi¯h(tb − t a )K(b, a) =exp im(x b − x a ) 2m2¯h(t b − t a )(11)2.4 Classical LimitIn the classical limit, i.e. when ¯h → 0, only paths that lie very close to the classical onegive significant contribution to the probability amplitude.One can see this in the following way: Let’s look at any two neighbouring paths x(t) andx ′ (t) which contribute to the path integral (6). We can write x ′ (t) = x(t) + η(t), withη(t) small. For the action we get:∫S[x ′ (t)] = S[x(t) + η(t)] = S[x(t)] +η(t) δS[x(t)]δx(t) dt + O(η2 ) (12)The contribution of the two paths to the path integral is:(A = e iS[x(t)]/¯h + e iS[x′ (t)]/¯h ≃ e iS[x(t)]/¯h 1 + exp ī ∫hη(t) δS[x(t)]δx(t) dt ), (13)where we have neglected terms of order η 2 . The phase difference between the contributionsis approximately ¯h −1 ∫ (η(t)δS[x(t)]/δx(t))dt. Even if δS gets very small, in the limit¯h → 0 the phase difference is huge and consequently on average destructive interferencebetween neighbouring paths occurs! There is an exception for the classical path. SinceδS at x c (t) vanishes, the action is of the form S[x c + η] = S[x c ] + O(η 2 ) and even for verysmall values of ¯h constructive interference results.3 Equivalence to Schrödingers Operator NotationIn the classical formulation of Quantum Mechanics the Schrödinger equation is postulated.In this section we will show, that it is possible to derive the Schrödinger equation fromthe postulates made in part 2.3.1 Derivation of the Schrödinger EquationLet’s start with Eq. (10) for the wave function. We set t 1 = t and t 2 = t + ɛ and use forthe kernel the expression derived in (9). In words, for a short time interval ɛ the actionis approximately ɛ times the lagrangian for this interval. We getΨ(x, t + ɛ) =∫ ∞−∞(1A expɛ ī ( x − yh L ɛ, x + y )), t Ψ(y, t)dy (14)2Observe that this equation is not exact. But it needs only be true to first order to ɛ, since(5) still holds 6 ; i.e. we shall derive the Schrödinger equation assuming (14) to be true infirst order to ɛ. This can be done in the following way 7 : write the Lagrangian in the formL = mẋ 2 /2 − V (x, t) and substitute y = x + η. The integration over y becoms an integral6 For a finite time interval T the number of factors is T/ɛ. If an error of order ɛ 2 is made in each factor,the resulting error is of order ɛ and vanishes in the limit7 Intermediate steps are left out here. The are found in [3]4

over η. The resulting integrals are Gaussian and can be solved explicitely. Expanding 8 tofirst order in ɛ and second order to η leeds then to the equationΨ + ɛ ∂Ψ∂t = Ψ − iɛ¯h V Ψ − ¯hɛ ∂ 2 Ψ2im ∂x 2 . (15)(15) is true in first order to ɛ, if Ψ satisfies [defining H = − ¯h22m ∂x+ V ] the differential2Equation−¯h ∂Ψ= HΨ , (16)i ∂twhich is the Schrödinger equation!3.2 Schrödinger Equation for the KernelThe kernel K(2, 1) introduced by (6) is only defined for t 2 > t 1 . Since K(2, 1), consideredas a function of the variables 2, is a special wave function it is clear that (16) is satisfied:−¯hi∂Ψ∂t 2K(2, 1) − H 2 K(2, 1) = 0 for t 2 > t 1 (17)Where H 2 acts on the variables 2 only.If we define K(2, 1) = 0 for t 2 < t 1 , (17) is also valid for t 2 < t 1 but K(2, 1) getsdiscontinous at t 2 = t 1 . One can show, that the kernel satisfies−¯hi∂Ψ∂t 2K(2, 1) − H 2 K(2, 1) = −¯h i δ(x 2 − x 1 )δ(t 2 − t 1 ) (18)Remark:We introduced K(2, 1) as the probability amplitude. Another possibility is to start fromthe usual formulation of Quantum Mechanics. The kernel can then be defined as theGreen’s function for the Schrödinger equation. This means: one defines K(2, 1) to bethe solution of Eq. (18). This is equivalent 9 to the following definition in the betterknown Dirac representation: K(2, 1) = 〈x 2 |Û(t)|x 1〉, where Û(t) = e−iĤt/¯h as long as theHamiltonian is time-independent.Finally we remark that K(2, 1) only depends on the time difference (t 2 − t 1 ) [as longas the Hamiltonian is time-independent].4 Pertubation TheoryThe idea of Pertubation theory in the path integral formulation of Quantum Mechanicsis quite simple. We look at a particle moving in some potential 10 V (x, t). If we insert (4)in (6) and expand the exponential, we getK V (2, 1) = K 0 (2, 1) + K 1 (2, 1) + K 2 (2, 1) + · · · , where (19)∫ 2[ ( ∫ īt2mẋ 2 )]K 0 (2, 1) = exp1 h t 12 dt Dx(t) (20)K 1 (2, 1) = − ī ∫ 2[ ( ∫ īt2mẋ 2 )] ∫ t2exph 1 h t 12 dt V [x(s), s]ds Dx(t) (21)t 1K 2 (2, 1) = − 1 ∫ 2[ ( ∫ īt2mẋ 2 )] [∫ t2] 22¯h 2 exp1 h t 12 dt V [x(s), s]ds Dx(t) (22).8 Demanding this expansion to be true in the limit ɛ → 0, the promised condition in A appears, with the result A = 2πi¯hɛ 1/2.9mThis follows from ”normal” Quantum Mechanics. Discussed in [4], [5]10 The index V in Eq. (19) shall just symbolize that the particle is moving in some potential V (x, t)t 1∂ 25

K 0 is just the free-particle kernel (11). If we interchange the order of integration over xand x(t) we get for K 1 :∫ t2K 1 (2, 1) = − ī F (s)ds , where (23)h t 1∫ 2[ ( ∫ īt2mẋ 2 )]F (s) = exp1 h t 12 dt V [x(s), s] Dx(t) (24)This expression F (s) can be interpreted as free propagation from t 1 until some intermediatetime s and again from s until t 2 , only weighting each path with a characteristic factorV [x(s), s]. Consequentely we can write F (s) in the formF (s) =∫ ∞−∞and with (23) the pertubation term givesK 1 (2, 1) = − ī h∫ t2∫ ∞t 1K 0 (x 2 , t 2 ; x s , s)V (x s , s)K 0 (x s , s; x 1 , t 1 )dx s (25)−∞K 0 (x 2 , t 2 ; x s , s)V (x s , s)K 0 (x s , s; x 1 , t 1 )dx s ds (26)The interpretation is the following: K 0 (2, 1) gives us the amplitude that the particle ismoving all the way from 1 to 2 without being affected by V (x, t) at all. K 1 (2, 1) is theamplitude that the particle is scattered once at any time s between 1 and 2. K 2 (2, 1)will then be the amplitude to be scattered twice, and so on. This can be developed bysimilar steps as above. A very interesting feature of the kernel K V (2, 1) is that it fulfillsthe following integral equation:K V (2, 1) = K 0 (2, 1) − ī h∫ t2∫ ∞t 1−∞K 0 (x 2 , t 2 ; x s , s)V (x s , s)K V (x s , s; x 1 , t 1 )dx s ds (27)We will not further look at that. It shall just be said, that this has some applications inscattering theory.5 Path Integrals in Phase SpaceAs we have seen in the end of 3.2, an equivalent definition for the kernel is K(2, 1) =〈x 2 |Û(t)|x 1〉. With this definition and making use of Eq. (8) (Dividing (t 1 − t 2 ) in Nintervals of length ɛ; we then assume Ĥ to be time independent on such an interval, i.e.K(i, i + 1) = 〈x i |e −iĤɛ/¯h |x i+1 〉) one finds the following expression for K(2, 1) [see [7]]:∫ N−1 ∏∫ N−1)∏ dp jK(2, 1) = lim dx i(ɛɛ→0 2π¯h exp ī N−1∑(p k ẋ k − H(p k , ¯x k )) , (28)hi=1 j=0k=0where ẋ k = 1 ɛ (x k+1 − x k ) and ¯x k = 1 2 (x k + x k+1 ). Similar to (6) we write that asK(b, a) =∫ ba[ īexph∫ tbt a](pẋ − H(p, x))dtDx(t) Dp(t)2π¯h(29)and call it the phase space path integral.The phase space path integral is somehow more general than the Feynman path integral.For the case where the Hamiltonian has its standard form (H = p 2 /2m + V (x, t))it is easy to show the equivalence to the configuration space path integral. One inserts 11the standard form of the Hamiltonian in (28) (the proper definition of (29)) and gets (5)by straightforward calculation, which then can be written in the form of (6).11 For the explicit calculations look again in [7].6

6 Path Integrals and Topology6.1 The Aharonov-Bohm effectLet’s ”abuse” the Aharonov-Bohm effect to illustrate a first example of a non-simplyconnectedconfiguration space. Take the double slit experiment from section 2.1 and puta perfectly shielded magnetic flux between the holes. Classically, because of the perfectshields, an electron does not feel any force. Due to the magnetig flux, we have to replacethe Lagrangian L by L ′ :L(ẋ, x) → L ′ (ẋ, x) = L(ẋ, x) − e v · A(x) , where B = ∇ × A (30)c∫ ∫This changes the action S of a path by the amount − e c dt v·A(x) = −ec dtdxdt ·A(x(t)),which is the line integral of A taken along the path. For any two paths x 1 (t) and x 2 (t),which go through hole 1 and 2 respectively we get∫∫∮dx · A(x) − dx · A(x) = dx · A(x) = Φ , (31)x 2(t)x 1(t)where Φ is the flux inside the closed loop. Note that the value of the integral does notdepend on the details of x 1 (t) and x 2 (t)! We might write the total amplitude K ′ =K 1 ′ + K 2, ′ where K 1 ′ (K 2) ′ is the sum over all paths through slit 1 (2). K = K 1 + K 2denotes the case, when the magnetic field ist turned off (Φ = 0). We can write for K 1:′∫K 1 ′ = Dx e i(S[x]−(e/c) R dx·A)/¯h = e −ie R 1 dx·A/¯hc K 1 , (32)slit 1where we have pulled out the line integral factor since it is the same for all paths throughslit 1. Thus for the total amplitude:K ′ = e −ie R 1 dx·A/¯hc K 1 + e −ie R 2 dx·A/¯hc K 2 (33)= e −ie R ( 1 dx·A/¯hc K 1 + e −ie H )dx·A/¯hc K 2 (34)= e −ie R ( )1 dx·A/¯hc K 1 + e −ieΦ/¯hc K 2 (35)The first factor is just an irrelevant overall phase. The interference pattern is determinedby the phase −eΦ/¯hc and can be changed by varying the magnetic flux.6.2 Path Integrals with Topological ConstraintsThere are many examples for path integrals for spaces with topological constraints, suchas a particle in a box, on a half-line or generally in a half-space. Here we look at a particleon a cirle. On a circle our coordinate is ϕ, 0 ≤ ϕ ≤ 2π with ϕ = 0 and ϕ = 2π identified.A path in this system is a continuous function ϕ(t) with the identification above. We candivide the set of all paths into subsets of paths with the same winding number. We cantherefore write (take ¯h = 1 for this section) the kernelK(ϕ 2 , t 2 ; ϕ 1 , t 1 ) =∑e iS[ϕ(t)] =ϕ(t 1)=ϕ 1ϕ(t 2)=ϕ 2∞∑n=−∞∑g n = {ϕ(t)| t 1 ≤ t ≤ t 2 , ϕ(t 1 ) = ϕ 1 , ϕ(t 2 ) = ϕ 2 ,ϕ(t)∈g ne iS[ϕ(t)] , where (36)ϕ(t) is continuous and has winding number n} (37)We now assume that each term in the sum (36) individually satisfies the Schrödingerequation 12 . We see that in this case∑ ∑A n e iS[ϕ(t)] , A n ∈ C (38)n ϕ∈g n12 A justification for that can be found in [5]7

can be a solution for the kernel as well. From periodicity follows the condition A n+1 = e iδ A n ,δ ∈ R. With A 0 = 1 we get A n = e inδ and can then write:∞∑K(ϕ 2 , t 2 ; ϕ 1 , t 1 ) = A n K n (ϕ 2 , t 2 ; ϕ 1 , t 1 ), (39)n=−∞where K n (ϕ 2 , t 2 ; ϕ 1 , t 1 ) is the kernel for all paths with n loops. For each K n we can carrythe Lagrangian 13 from S 1 to R and, for a free particle, use 14 (11). Inserting this in (39)we obtain the kernel for a free particle moving on a circle:K(ϕ 2 , t 2 ; ϕ 1 , t 1 ) =∞∑n=−∞(I2πi(t 2 − t 1 )7 Statistical Mechanics) 1/2exp[inδ +]iI2(t 2 − t 1 ) ((ϕ 2 − ϕ 1 ) − 2nπ) 2In this last section we just want to see, how Statistical Mechanics and path integralsin Quantum Mechanics are related. The central object in Statistical Mechanics is thepartition function. It is defined by(40)Z := ∑ je −βEj= ∑ j〈j|e −βĤ|j〉 = Tre −βĤ, (41)where β = 1/k B T and E j is the energy of the state |j〉. Recall the alternative definitionfor the kernel: K(x 2 , t 2 ; x 1 , t 1 ) = 〈x 2 |e −iĤ(t2−t1)/¯h |x 1 〉. Let’s see what happens, if wemake the substitution (t 2 − t 1 ) := −iβ¯h, where β is real:K(x 2 , −iβ¯h; x 1 , 0) = 〈x 2 |e −iĤ(−iβ¯h)/¯h |x 1 〉 = 〈x 2 |e −βĤ ∑ j|j〉〈j||x 1 〉 (42)= ∑ je −βEj 〈x 2 |j〉〈j|x 1 〉 = ∑ je −βEj 〈j|x 1 〉〈x 2 |j〉 (43)where we used the completenessrelation ∑ j |j〉〈j| = 1. Putting x = x 1 = x 2 and integratingover x, we get:∫K(x, −iβ¯h; x, 0)dx = ∑ ∫e −βEj 〈j| dx|x〉〈x| |j〉 = Z (44)j } {{ }=1This equation tells us how the propagator evaluated at negative imaginary time isrelated to the partition function! This can be of practical use: It gives us a tool to easilycalculate the partition function of a physical system of which we know the path integral(kernel).8 SummaryThis was a short introduction in the path integrals in Quantum Mechanics. I hope it gavea vague overview over the field even if the work is not complete at all. Especially thetreatment of measurements and operators in the path integral formalism was completelyleft out. But still the basic ideas and some applications were presented. In quantum fieldtheory path integrals will play an even more important role and I hope (for myself) thatthe acquired knowledge will be of use! In the last section it was shown how different fieldsof physics, such as statistical mechanics and Quantum Mechanics, can be linked by pathintegrals. A similar relation can be made between statistical field theory and quantumfield theory.13 We can do that, since we can define a smooth mapping from R to S 1 . For details consult [5]!14 Since L = I ˙ϕ 2 /2, we just set m = I.8

References[1] R.P. Feynman: Space-Time Approach to Non-Relativistic Quantum MechanicsRev.Mod.Phys. 20 (1948) 367[2] R.P. Feynman: Statistical Mechanics: A set of lectures McGraw-Hill, 1965[3] R.P. Feynman and A.R. Hibbs: Quantum Mechanics and Path Integrals McGraw-Hill, 1965[4] H. Kleinert: Path Integrals in Qunatum Mechanics, Statistics and Polymer PhysicsWorld Scientific, Singapore, 1990[5] L.S. Schulman: Techniques and Applications of Path Integration John Wiley andSons, 1981[6] M. Chaichian: Path integrals in physics. Vol. 1, Stochastic Processes and QuantumMechanics Institute of Physics, Bristol, 2001[7] R. MacKenzie: Path Integral Methods and Applications quant-ph/0004090[8] C. Grosche: An Introduction into the Feynman Path Integral hep-th/93020979