The Path Integral Formulation of Quantum Mechanics

Transcription

1 Based on Quantum Mechanics and Path Integrals by Richard P. Feynman and Albert R. Hibbs, and Feynman s Thesis The Path Integral Formulation Vebjørn Gilberg University of Oslo July 14, 2017

2 Contents 1 Introduction 2 2 Fundamental Concepts Operators and the Inner Product The Schrödinger Picture The Heisenberg Picture The Quantum Mechanical Law of Motion The Classical Action The Quantum Mechanical Amplitude The Classical Limit The Sum Over All Paths Analogy with the Riemann Integral Constructing the Sum The Path Integral Example: The Free Particle Events Occurring in Succession The Rule for Two Events Extension to Several Events

3 1 Introduction Richard Feynman and Albert Hibbs have made significant remarks on the path integral formulation of quantum mechanics that I find hard to overlook. Therefore, i have borrowed a lot of their explanations while subsidising with some of my own, or at least my understanding of what they are saying. I have, however, not done so in the explanation of the fundamental concepts of quantum mechanics as this is a subject which is more accessible to an undergrad whose knowledge of quantum field theory and the sum over all paths formulation is virtually non-existent. Perhaps the latter is the reason for my apparent theft of their explanations. In any case, someone in my position usually stands on the shoulders of giants, as I too am doing in writing this text. 2 Fundamental Concepts 2.1 Operators and the Inner Product 2.2 The Schrödinger Picture 2.3 The Heisenberg Picture 3 The Quantum Mechanical Law of Motion 3.1 The Classical Action When an objects travels from a to b it moves in some particular path. But what determines the particular path of the object? How does the object know which path to take? This is where the principle of least action comes in. This principle determines the path x(t) out of all the possible paths an object might take. We can find a quantity denoted by S, and the extremum 1 of this quantity determines the path x(t). The extremum of S simply means that S has to be constant under some small variation of the path x(t). The quantity S is named the action and is expressed as S = L(x, ẋ, t)dt, (1) 1 Usually a minimum. 2

4 where L is the Lagrangian of the system. The Lagrangian for a particle of mass m and a time-dependent potential V (x, t) is L = 1 mẋ(t) V (x, t). (2) 2 Suppose now that we apply to the path x(t) a small variation δx(t) with fixed endpoints such that δx(t a ) = δx(t b ) = 0. The condition that S must be an extremum for the physical path taken by the object can be stated by δs = S[ x + δx] S[ x] = 0. (3) If we now insert this into the definition of the action given in (1), we get S[x + δx] = L(x + δx, ẋ + δẋ, t)dt [ = L(x, ẋ, t) + δx L ] dt = S[x] + x + δẋ L ẋ ] dt [ δx L x + δẋ L ẋ (4) If we perform normal integration by parts focusing on δẋ and L ẋ δs = [ δx L ] tb tb ( δx d ẋ t a t a dt we get ( ) L δx L ) dt = 0 (5) ẋ x The requirement that x(t) does not vary at the endpoints makes the first term vanish. The path of the object follows that of the extremum for which the following is true: d dt ( ) L L ẋ x = 0. (6) This is the classical Lagrangian equation of motion. The form of the action, S = Ldt, hints at something deeper which we will come back to. Namely the fact that we have to consider neighbouring paths in order to find the actual path of the object, the path of least action. 3

5 3.2 The Quantum Mechanical Amplitude All paths contribute in the trajectory from a to b, and this must be accounted for, that is, we cannot just say that they do and not actually include them! In terms of quantum mechanics, this means adding up amplitudes to go from a to b. But how shall we characterise the differences of the contributions? The amplitudes themselves contribute equally in terms of magnitudes. The differences comes in when we look at the phases of the amplitudes. Next problem: how do we characterise the phases? It s easier than one would think. The action S must have units of in the quantum mechanical description, and S for any given path is the phase. The probability amplitude to go from x a (t a ) x b (t b ) is the absolute square of the amplitude from a to b. That is, P (b, a) = K(b, a) 2. Now we have to express these amplitudes in terms of the different paths of the particle. Naturally we turn to the sum of contributions φ[x(t)] from each possible path. K(b, a) = φ[x(t)]. (7) paths from a to b Earlier we stated that the only difference between the paths is the phase governed by S in units of, and so we must express the contributions in terms of these phases. φ[x(t)] = const e (i/ )S[x(t)], (8) where the constant is the usual constant we re used to from quantum mechanics. That is, we must choose it so that K is normalised correctly. There s nothing new or alternative about the action in the exponent, it is the same as before. The more mathematical explanation for what it is we mean when we sum over all possible paths will be revisited later. 3.3 The Classical Limit There might seem to be a problem with one of the statements made in the former section. The fact that each trajectory contributes equally in terms of the magnitude of the amplitudes may suggest that no one path stands out in the classical limit. After all, we want to find a path that distinguishes itself from the others by being an extremum of S. By classical limit we mean the case where the action is extremely large due to the large dimensions compared to, so in this approximation the angle S/ is also very large. The likelyhood of the real or imaginary part of φ to be plus or minus is the same, i.e., the probability for the minus sign is the same as for the plus sign. One the 4

6 classical scale, the dimensions are so large that the changes in S are small, but on the quantum mechanical scale the changes are enormous. Due to the large phases on the classical scale, our cosine or sine will oscillate rapidly between the plus and minus values which produces a total contribution of 0. I.e., for each positive contribution a negative contribution infinitesimally 2 close to it cancels out the positive one. This is the case for paths where a small change produces some change in S. In the case of the extremum, the action is unchanged. In the region of the extremum, the paths are nearly in phase S cl / and so they do not cancel out. Because the paths in the vicinity of the extremum path x(t) do not cancel, we need only consider these as important contributions. Then, in the classical limit, the only path of significance is that for which the action is an extremum, namely x(t). In this way, the classical motion arises from the quantum mechanical motion. Normally the end points are fixed, but what would happen to the phase if we changed the end point (x b, t b ) just a little bit? The simple answer is that the phase would change a lot and the amplitude K(b, a) would see rapid changes. This discussion is really only interesting on the classical scale where S >>. On the scale of quantum mechanics we still have to carefully add up all the contributions from eq. (7). 3.4 The Sum Over All Paths Analogy with the Riemann Integral Like the ordinary Riemann integral, we have to sum over all contributions in order to obtain the full description of the path from a to b 3, and so in a sense we have a clear picture of what it means to sum over all the paths. However, we cannot really use the definition in (10) to calculate anything. The expression is too inconcise to work with, and so we need a more quantitative description of the sum over all paths. The Riemannian integration scheme is something most physics and mathematics students know well, so it is natural to build up our understanding of the mathematical description soon to be given by considering this more familiar concept. The area under some curve is proportional to the ordinates with equal spacing h between them. That is, A i f(x i ) (9) 2 In the classical limit 3 In the case of the Riemann integral the sum goes over the ordinates, of course. If you haven t already googled what an ordinate is, it just means the height of the y-coordinate. 5

7 where x i is the i th point at each ordinate. Now, this is a more precise description than simply saying that the area A under some curve a sum over the ordinates of the curve, so we re getting closer! The integral is what we get when we take the limit of the sum. However, as the sum is expressed right now the limit would not exist. That is, as we continually sum up smaller and smaller pieces as h gets smaller which means that the limit doesn t exist. To solve this problem we multiply the sum by a normalising factor which in Riemann s case is simply h. I.e., [ ] A = lim h i f(x i ). (10) Constructing the Sum The idea behind visiting the realm of ordinary Riemannian integration is that we can follow a similar approach when defining the sum over all paths. There are however some notable differences. Instead of the element of length, h, we will make use of the independent time variable and divide it into steps of width ɛ. So in analogy to the steps x i of length h, we now have steps t i of width ɛ in time between the points a and b. Also, for each time t i there is a position x i and so the paths are constructed by drawing a straight line between each point from a to b. This might be better explained if we use the phrase time-slicing. The parameter separating the time slices is defined as ɛ = t i+1 t i, and so if we multiply the N steps by ɛ we get Nɛ = t b t a, i.e. the entire time interval between the two end points. Given this information, it is clear that t 0 = t a, t N = t b x 0 = x a, x N = x b. (11) As in the classical theory, the end points are fixed and so we only integrate between 1 and N 1. We can now go on to our first step in defining a path integral by constructing multiple integrals over all paths x i where i = 1,..., N 1, namely K(b, a)... φ[x(t)]dx 1 dx 2... dx N 1. (12) The observant reader will notice a similarity between this equation and equation (9), including the problem of making ɛ smaller. As a reminder, the limit does not exist if we do not include some normalising factor. It is not as easy to define such a factor in this case though. At least not a general one. An easy way out, for now, 6

8 is to mathemagically cook up a factor in the case where the Lagrangian takes the form as in (2) 4. That is, A N where A = (2πi ɛ/m) 1/2. This yields 1 K(b, a) = lim... e (i/ )S[b,a] dx 1 dx 2 ɛ 0 A A A... dx N 1 (13) A where S[b, a] is the usual expression for the action integrated from t a t b. So the action is a line integral over the linear paths between the points x i. One might understand this better if a graphical explanation is presented, which is what I have tried to do in figure(so and so) below The Path Integral Example: The Free Particle 3.5 Events Occurring in Succession The Rule for Two Events Extension to Several Events 4 Feynman and Hibbs proves this later in the book(sec. 4-1), but I won t bother with that in this text. The idea is the imporant thing! 7