Feynman Path Integrals in Quantum Mechanics

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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
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Page 9: References[1] R.P. Feynman: Space-T

Feynman Path Integrals in Quantum Mechanics

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