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Path integrals Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 October 29, 2008 Sourendu Gupta (TIFR Graduate School) Path integrals QM I 1 / 20 Outline 1 The time-evolution operator 2 Feynman’s path integral 3 The free particle: doing the path integral 4 References Sourendu Gupta (TIFR Graduate School) Path integrals QM I 2 / 20 The time-evolution operator Outline 1 The time-evolution operator 2 Feynman’s path integral 3 The free particle: doing the path integral 4 References Sourendu Gupta (TIFR Graduate School) Path integrals QM I 3 / 20 The time-evolution operator The unitary operator When the Hamiltonian of a quantum system is H, then time evolution of any quantum state in the Hilbert space on which H acts is |ψ(t) = U(t, t0 )|ψt0 , U(t, t0 ) = exp[−iH(t − t0 )/ ]. We have assumed, as everywhere in this course, that H is time-independent. The evolution operator U is unitary, since H is Hermitean. Also, the evolution operators form a group, since U(t, t) = 1, U(t, t ′′ ) = U(t, t ′ )U(t ′ , t ′′ ), U(t, t ′ )−1 = U(t ′ , t), and multiplication of operators is associative. The generator of inﬁnitesimal time evolution is H. The group is the group U(1), since the action on each eigenstate of H is to multiply it by a phase. These eigenstates carry irreducible representations. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 4 / 20 The time-evolution operator Slicing the time interval Using the group property, we can write N−1 U(tﬁn , tin ) = U(ti + δt, ti ), where δt = (tﬁn − tin )/N. i=0 If the state at time ti is |ψi , then N−1 ψ(tﬁn )|U(tﬁn , tin )|ψ(tin ) = ψi+1 |U(ti + δt, ti )|ψi , {ψi } i=0 where |ψ(tﬁn ) = |ψN+1 and |ψ(tin ) = |ψ0 . The sum over all intermediate states is called a path integral. If the states |ψi are eigenstates of the Hamiltonian then ψi+1 |U(ti + δt, ti )|ψi = e−iEi δt/ δψi+1 ,ψi . However, if we choose a complete set of states which does not coincide with the eigenstates of the Hamiltonian, then the path integral may be non-trivial. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 5 / 20 The time-evolution operator A path integral δτ ψ1 ψ ψ0 i+1 ψ i ψ N+1 t0 t1 ti ti+1 tN+1 Sourendu Gupta (TIFR Graduate School) Path integrals QM I 6 / 20 The time-evolution operator A two-state path integral For a two-state system take the Hamiltonian to be H = i hi σi with all four hi real. Then the inﬁnitesimal evolution operator is iHδt 1 − iδt (h0 + h3 ) − iδt (h1 + ih2 ) U(δt) = 1 − + O(δt 2 ) = . − iδt (h1 − ih2 ) 1 − iδt (h0 − h3 ) Each of these elements is a phase factor associated with the path taken during the interval δt. A path contributing to the transition matrix element αN |U(tN , t0 )|α0 is the sequence of intermediate states |α(ti ) . The contribution of each path to the transition matrix element is UαN αN−1 UαN−1 αN−2 · · · Uα2 α1 Uα1 α0 . The sum over paths is a sum over intermediate states. This is exactly the same as doing the matrix multiplication needed to get the transition amplitude. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 7 / 20 The time-evolution operator A two-state path integral For a two-state system take the Hamiltonian to be H = i hi σi with all four hi real. Then the inﬁnitesimal evolution operator is iHδt 1 − iδt (h0 + h3 ) − iδt (h1 + ih2 ) U(δt) = 1 − + O(δt 2 ) = . − iδt (h1 − ih2 ) 1 − iδt (h0 − h3 ) Each of these elements is a phase factor associated with the path taken during the interval δt. A path contributing to the transition matrix element αN |U(tN , t0 )|α0 is the sequence of intermediate states |α(ti ) . The contribution of each path to the transition matrix element is UαN ,α0 = UαN αN−1 UαN−1 αN−2 · · · Uα2 α1 Uα1 α0 . αN−1 ,··· ,α1 The sum over paths is a sum over intermediate states. This is exactly the same as doing the matrix multiplication needed to get the transition amplitude. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 7 / 20 The time-evolution operator A diﬀerent take on a two-state path integral Another way of doing the two-state path integral is to introduce the unitary matrix V which diagonalizes the Hamiltonian, i.e., V † HV is diagonal. Then e−iE0 δt/ 0 U(δt) = V † V. 0 e−iE1 δt/ The sum over intermediate states is then diagonal, and the V s act only on the initial and ﬁnal states to give 0 0 1 1 αN |U(tN , t0 )|α0 = (αN )∗ α0 e−iE0 T / + (αN )∗ α0 e−iE1 T / , where T = tN − t0 ; therefore the path integral can be used to get the energy levels of the system. This is best done in Euclidean time, i.e.changing t → it. Then the above expression becomes αN |U(tN , t0 )|α0 → (αN )∗ α0 e−E0 T / , 0 0 when E1 > E0 and T (E1 − E0 ) ≫ . This gives the lowest eigenvalue of H. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 8 / 20 The time-evolution operator Using the path integral to compute energies 1 Choose δt and N (T = Nδt). Fix |α0 and |αN . 2 Choose a random path in Hilbert space, i.e., a random sequence of N quantum states |αi for 1 ≤ i < N. 3 Compute the product of Euclidean factors δαi ,αi+1 − δtHαi ,αi+1 / along the path. Call this A. 4 Repeat the above two steps many times and ﬁnd A . 5 Increase N and repeat the above three steps until the exponential behaviour manifests itself. 6 The exponential slope gives the lowest energy eigenvalue, E0 . This method is not the most eﬃcient technique in quantum mechanics on ﬁnite dimensional Hilbert spaces (matrix multiplication is cheaper). However, for inﬁnite dimensional Hilbert spaces, and in quantum ﬁeld theory, often this is the best possible method. This is one of the techniques of lattice quantum theory. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 9 / 20 Feynman’s path integral Outline 1 The time-evolution operator 2 Feynman’s path integral 3 The free particle: doing the path integral 4 References Sourendu Gupta (TIFR Graduate School) Path integrals QM I 10 / 20 Feynman’s path integral Using position eigenstates Choose to work with the position basis states |xi . Then the sum over intermediate states becomes integrals over the positions—- N−1 x(tﬁn )|U(tﬁn , tin )|x(tin ) = dxi xi+1 |e−iT δt/ |xi e−iV (x)δt/ . i=1 We have used the decomposition, H = T + V . One way to evaluate the matrix element involving the kinetic energy is to insert complete sets of eigenvectors of momentum. Then ′” 2 δx “ ′ − iT δt iδt − 2m p 2 −2mp x−x imδt x| e |x = dpe δt ∝ exp . 2 δt The Gaussian integral can be performed after completing squares and shifting. The only dependence on x and x ′ is in the factor shown. (Check) Sourendu Gupta (TIFR Graduate School) Path integrals QM I 11 / 20 Feynman’s path integral Expression in terms of the action Putting all this together, the transition matrix element turns out to be N−1 N−1 2 iδt m dxi dxi exp − V (xi ) . 2 dt i=1 i=0 The expression within braces is the Lagrangian of the system. The product of exponentials is just the Riemann integral when the time step is taken to zero, N−1 iδt i exp L(xi ) −→ exp dtL[x(t)] = eiS[x]/ , i=0 where the action is deﬁned to be S[x] = dtL[x(t)]. Feynman’s path integral is then Z = x(tﬁn )|U(tﬁn , tin )|x(tin ) = DxeiS[x]/ , where the paths join the given points x(tin ) and x(tﬁn ). Sourendu Gupta (TIFR Graduate School) Path integrals QM I 12 / 20 Feynman’s path integral Classical from quantum ... if we move the path ... by a small amount δx, small on the classical scale, the change in S is likewise small on the classical scale, but not when measured in the tiny unit . These small changes in path will, generally, make enormous changes in phase, and our cosine or sine will oscillate exceedingly rapidly between plus and minus values. The total contribution will then add to zero; for if one makes a positive contribution, another inﬁnitesimally close (on a classical scale) makes an equal negative contribution, so that no net contribution arises. ... But for the special path x, for which S is an extremum, a small change in path produces, in the ﬁrst order at least, no change in S. All the contributions from the paths in this region are nearly in phase, at phase Scl , and do not cancel out. Therefore, only for paths in the vicinity of x can be get important contributions, and in the classical limit we need only consider this particular trajectory as being of importance. (from “Quantum mechanics and Path Integrals”, (1965) by R. P. Feynman and A. R. Hibbs, p 30) Sourendu Gupta (TIFR Graduate School) Path integrals QM I 13 / 20 Feynman’s path integral Problems 1 Evaluate the Gaussian integral over momenta carefully and ﬁnd the normalization of Feynman’s path integral. 2 For a free particle show that the action Scl corresponding to the classical motion is Scl = m(xﬁn − xin )2 /2(tﬁn − tin ). 3 For a harmonic oscillator with T = tﬁn − tin show that mω 2 2 Scl = (xin + xﬁn ) cos ωT − 2xin xﬁn . 2 sin ωT 4 Find Scl for a particle moving under a constant force F . 5 Formulate the problem of many particles in the path integral language. 6 How would you ﬁnd the lowest energy bound state in a potential using a Monte Carlo computation of the path integral in Euclidean time? Sourendu Gupta (TIFR Graduate School) Path integrals QM I 14 / 20 The free particle: doing the path integral Outline 1 The time-evolution operator 2 Feynman’s path integral 3 The free particle: doing the path integral 4 References Sourendu Gupta (TIFR Graduate School) Path integrals QM I 15 / 20 The free particle: doing the path integral Time slicing The path integral for the free particle is tﬁn 2 i m dx Z= Dx exp dt 2 tin dt The most straightforward method for doing this is to use the route we have taken before. Divide the time interval into pieces of length δt and use the coordinates xi at ti = tin + iδt to write the derivative as a ﬁnite diﬀerence. The resulting integral is quadratic in the xi , and hence the problem reduces to a set of Gaussian integrals. One can perform each integral separately using Gauss’ formula ∞ 2 /(2a2 ) dxe−x = (2π)a. −∞ Does this integral converge for all complex a? The multivariate version of Gauss’ formula can be written down most compactly in the form 1 dxi=1 exp − xT Ax = (2π)N/2 (Det A)−1/2 . N 2 Sourendu Gupta (TIFR Graduate School) Path integrals QM I 16 / 20 The free particle: doing the path integral Expansion in quantum ﬂuctuations The classical path is d 2 xc xﬁn − xin = 0, giving xc (t) = xin + (t − tin ). dt 2 tﬁn − tin Decompose an arbitrary path in the form x(t) = xc (t) + q(t), where q(tin ) = q(tﬁn ) = 0. The quantity q(t) corresponds to quantum ﬂuctuations of the path around its classical value. Since the classical path is an extremum of the action, one can show that tﬁn 2 m dq S = Scl + dt . 2 tin dt As a result, the partition function factors into the form Z = Zc Zq . Since Zq again contains only quadratic integrals, one can use the Gaussian integration formula all over again. Note that the quadratic form is diagonalized through Fourier transformation. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 17 / 20 The free particle: doing the path integral The result For a free particle, the partition function is m im (xﬁn − xin )2 Z= exp . 2πi (tﬁn − tin ) 2 tﬁn − tin o Note that this result can also be obtained from Schr¨dinger’s equation. One can check that the exponential comes from factor Zc and the square root in the prefactor is the result of performing the integral over the quantum ﬂuctuations. The Euclidean continuation of this result (t → it) shows that the distribution is exactly that which results from a random walk. When a random walker is released from xin at time tin and its position, xﬁn is measured at time tﬁn , the probability distribution of xﬁn is given by D D (xﬁn − xin )2 exp − . 2π(tﬁn − tin ) 2 tﬁn − tin √ The mean distance travelled by the random walker grows as tﬁn − tin . This is o closely related to the fact that Schr¨dinger’s equation goes to the diﬀusion equation with this analytic continuation of the time. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 18 / 20 References Outline 1 The time-evolution operator 2 Feynman’s path integral 3 The free particle: doing the path integral 4 References Sourendu Gupta (TIFR Graduate School) Path integrals QM I 19 / 20 References References 1 The Principles of Quantum Mechanics by P. A. M. Dirac. This contains the observation that the short-time evolution operator is the exponential of the action. 2 Quantum Mechanics and Path Integrals by R. P. Feynman and A. R. Hibbs. Chapters 2 and 3 contain material related to this lecture. 3 R. P. Feynman, Rev. Mod. Phys., 20 (1948) 367. This paper is the ﬁrst published version of Feynman’s theory of path integrals, which he developed for his thesis. 4 M. Kac, Trans. Am. Math. Soc., 65 (1949) 1. This paper is the ﬁrst application of path integrals to quantum statistical mechanics. 5 Techniques and Applications of Path Integration, L. S. Schulman. This is another standard textbook of path integral methods. Sourendu Gupta (TIFR Graduate School) Path integrals QM I 20 / 20

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