D. L. Rubin September, 011 These notes are based on Sakurai,.4, Gottfried and Yan,.7, Shankar 8 & 1, and Richard MacKenzie s Vietnam School of Physics lecture notes arxiv:quanth/0004090v1 1 Path Integral Suppose we have the propagator Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e iht f t 0 / x 0 We can just as easily take two steps Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e iht f t 1 / e iht 1 t 0 / x 0 or we could divide the total time T into N steps, with δ T/N. Then Kx f, t f, x 0, t 0 x f, t f x 0, t 0 x f e ihδ/ e ihδ/... x 0 and then we could insert the identity everywhere along the path. Kx f, t f, x 0, t 0 x f e ihδ/ dx x x e ihδ/...... dx 1 x 1 x 1 e ihδ/ x 0 dx Kx f, t f, x, t dx 1 Kx 1, t 1, x 0, t 0 The amplitude is the sum of all N-legged paths. dx N x N x N... dx N Kx, t, x N, t N... A paths A paths paths dx 1 dx...dx, A path K xn,x K x,x N... 1
Let s consider the j th term. Kx j+1, x j x j+1 e ihδ/ x j Since δ is small we can expand the exponential and we have Kx j+1, x j x j+1 1 ihδ/ + Oδ x j Then we can insert the identity dp j p j p j and Kx j+1, x j dp j x j+1 1 ih δ p j p j x j dp j x j+1 p j p j x j i x j+1 H δ p j p j x j dp j x j+1 p j p j x j i δ p j m + V x j+1 x j+1 p j p j x j dpj e ix j+1 x j p j / i δ π p j m + V x j+1e ix j+1 x j p j / dpj π eix j+1 x j p j / exp i δ p j m + V x j+1 dpj π eix j+1 x j p j / exp i δ H Now we write x j+1 x j /δ ẋ j and we have dpj Kx j+1, x j π eiδẋ jp j / e i δ H There are N such factors in the amplitude so dp j A path π exp i δ ẋ j p j Hx j, p j That s the amplitude for one path. Now integrate over all paths Kx N, x 0 j1 dx j dp j π exp i δ ẋ j p j Hx j, p j
As N the sum becomes an integral over all time and we write Kx N, x 0 Dxt Dpt exp i δ T dtẋp Hx, p This is the phase space path integral. If the Hamiltonian has the standard form H p + V x then we can integrate each of the terms in the sum m Kx N, x 0 We use j1 dx j exp i δ e αx βx dx 0 V x j π α eβ /4α dp j π exp i δ ẋ j p j p j m where the above holds for pure imaginary α if it is regarded as a limit, namely if α a + ib, a > 0 it is the limit as a 0. This is what it looks like dp π exp i δ p m ẋp m e i δ πi δ mẋ / Putting it all together K j1 dx j exp i δ N/ πiδ j1 V x j dx j exp i δ ẋ j m πiδ exp i δ mẋ j V x j The sum is an approximation of the action of a path passing through the points x 0, x 1, x,... K Dxte is[xt] is the configuration space path integral. 1.1 Free particle path integral The configuration space path integral for a free particle is N/ [ iδ ẋ j K dx j exp πiδ j1 3 ]
K πiδ N/ K πiδ K j1 N/ πiδ N/ j1 j1 dx j exp dx j exp [ iδ m [ im δ ] xj+1 x j δ x j+1 x j ] [ im dx j exp xn x + x x N +... + x 1 x 0 ] δ where x 0 and x N are initial and final points. The integrals are Gaussian and can be evaluated exactly but since they are coupled it ain t pretty. Let s see if we can figure it out. First let s define y i δ 1 x i. Then K K πiδ N/ j1 1 δ dyj exp [ i y N y + y y N +... + y 1 y 0 ] m N/ / δ dy j exp [ i y N y + y y N +... + y 1 y 0 ] πiδ m j1 Let s do the y 1 integration first. dy 1 exp i y y 1 + y 1 y 0 dy 1 exp iy + y 0 + y 1 y 1 y + y 0 dy 1 exp i y 1 y + y 0 / exp iy + y0 exp i y + y 0 dz exp 1 i z exp i y y 0 iπ exp i y y 0 Next we do the y integration. iπ y 3 y + 1 y y 0 iπ dy exp 1 i dy exp 1 3y i y y 3 + y 0 exp 1 i y 3 + 1 y 0 4
It looks like it goes to iπ dz exp 1 z y 3 + y 0 /3 exp 1 3 i i y 3 + 1 y 0 iπ iπ 3 exp 1 y3 + y 0 /3 exp 1 i i y 3 + 1 y 0 iπ exp 1 3 3i y 3 y 0 iπ Putting it all together we have N exp 1Ni y N y 0 N/ / δ iπ K lim N πiδ m N 1 exp 1Ni y n y 0 The answer is N/ / 1 πiδ K lim e imx x /Nδ N πiδ N m m 1/ K lim e imx x /Nδ N πinδ Since Nδ T we have 1/ K e imx x /T πit which is of course the same as we calculated directly. Now on further investigation we see that 1 K exp i m xn x 0 m 1 T exp i T L cl dt πit T πit 0 Cute huh. The coordinate space path integral for the free particle, the sum of the action through every possible point in space, reduces to simply the classical action. The propagator reduces to two factors, one being the phase exp i S cl 5
1. Harmonic oscillator path integral The coordinate space path integral for the harmonic oscillator is Now let s write K N/ πiδ j dx j exp iδ ẋ j 1 mω x j xt x cl t + yt, dx dy, ẋ ẋ cl + ẏ Then ẋ 1 mω x j ẋ cl 1 mω x cl + L ẋ ẋ cl ẏ + L x x cl y + ẏ 1 mω y Let s look at the middle term and convert the sum to an integral L dt x y + d L dt ẋ y d L dt ẋ y L ẋ y t N t0 + dt The first term is zero because y 0 yt N 0. So N/ i K lim exp N πiδ S cl j1 dy j exp iδ L x d dt ẏ L y 0 ẋ 1 mω y j The PI over y is independent of the endpoints. It is zero at each end. It will depend only on the total time T and if K exp i S mω cly T, Y T πi 1/ xt A cosωt + B sin ωt, x N A cos ωt + B, x 0 A Then B x N x 0 cos ωt / 1 S cl mẋ 1 mx dt 6
1 1 dt ωa sin ωt + ωb cos ωt mω A cos ωt + B sin ωt dt mω A sin ωt + B cos ωt AB sin ωt cos ωt A cos ωt + B sin ωt + AB sin ωt cos ωt 1 dt mω B A cos ωt AB sin ωt mω 4 B A sin ωt + AB cos ωt T 0 mω 4 B A + ABcos ωt 1 mω 4 x N x 0 cos ωt x 0 sin ωt sin ωt cos ωt 1 +x 0 x n x 0 cos ωt mω 4 x N + x 0 cos ωt x N x 0 cos ωt +x 0 x n x 0 cos ωt cos ωt 1 mω 4 x N + x 0 cos ωt x N x 0 cos ωt cos ωt 1 +x 0 x n x 0 cos ωt cos ωt sin ωt cos ωt mω 4 x N cos ωt + x 1 0 cos ωt 4x N x 0 mω x N + x 0 cos ωt x N x 0 1.3 Principle of Least Action Consider the configuration space path integral K Dxte is[xt]/. It says that a particle going from initial to final position and time takes all possible paths. The classical path is included but it gets no special mention. Every path has precisely unit magnitude. The contributions from the classical path and the totally wild path are the same. It turns out that the amplitudes interfere with each 7
other in a very special way. Consider two neighboring paths xt and x t and let x t xt + ηt, with ηt small. Then we can write the action S[x ] S[x + η] S[x] + dtηt δs[x] δxt + Oη The contribution of the two paths to the PI is A e is[x]/ 1 + exp i dtηt δs[x] δxt The phase difference between the two paths is 1 dtηt δs[x]. Smaller larger phase δxt difference. Even paths that are very close together will have large phase difference for small and on average they will interfere destructively. This is true except for one exceptional path, that which extremizes the action, namely the classical path x c t. For this path S[x c + η] S[x c ] + Oη. The classical path and a close neighbor will have actions which differ by much less than two randomly chosen but equally close paths. If the problem is classical action, paths near the classical path will on average interfere constructively small phase difference whereas for random paths the interference will be on average destructive. Classically, the particles motion is governed by the principle that the action is stationary. 8