Path integral formulation. Evolution operator in real space for harmonic oscillator Let us have another look at the dynamic solution for an arbitrary quantum state written as X −itEn χ(x; t) = h njχ(t = 0)i e n(x) ; n in terms of the energy eigenbasis. We can rewrite it identically as Z 0 0 0 0 X −itEn ∗ 0 χ(x; t) = dx U(x; t; x ; t = 0)χ(x ; t = 0) ;U(x; t; x ; t = 0) = e n(x) n(x ) ; n where all information about a particular quantum system is encoded in the U-function. If the first expression forces you to do the calculation for every state χ from scratch by doing all integrals h njχ(t = 0)i and then a sum over states, the second formulation allows you do the sum for a given system only once and then use U-function to get the final result by doing only one integral. [This method is known in math. as the Green's-function method.] There are many alternative ways to get the U-function; the most obvious one is to do the sum over the basis as in the second equation, e.g. for the harmonic oscillator one has to evaluate 1 0 X −it!(n+1=2) 2 −[(x0)2+x2]m!=2 p 0 p U(x; t; x ; t = 0) = e Nn e Hn( m!x )Hn( m!x) ; n=0 This is doable with some effort. One may also solve an appropriately formulated differential equation for U. Here I will discuss yet another method invented by Feynman|the path integral formulation of quantum mechanics. Path integrals offer a remarkable new way to look at things in quantum mechanical systems making the most intuitive link between the classical and quantum behavior. They do not resemble Schr¨odingerEquation at all, but ultimately lead to exactly the same final results. One starts from the most basic expression for time evolution ^ χ(t) = e−iHtχ(t = 0) ; and writes it in the real-space representation as (all integrals and variables are D-dimensional) Z ^ Z ^ χ(x; t) = hxjχ(t)i = dx0hxje−iHtjx0ihx0jχ(t = 0)i = dx0hxje−iHtjx0iχ(x0; t = 0) ; leading to ^ U(x; t; x0; t = 0) = hxje−iHtjx0i : Next, we slice the time interval (0; t) into large number N ! 1 of very short intervals of duration dτ = t=N ! 0 and write identically N ^ Y ^ ^ ^ ^ hxje−iHtjx0i = hxj e−iHdτ jx0i = hxje−iHdτ e−iHdτ : : : e−iHdτ jx0i = j=1 Z Z Z −iHdτ^ −iHdτ^ −iHdτ^ −iHdτ^ 0 dxN−1··· dx2··· dx1hxje jxN−1i hxN−1je jxN−2i ::: hx2je jx1i hx1je jx i ; 1 where in the last expression we used completeness relation to express the calculation of matrix elements over an arbitrary time t in terms of short time evolution only. The advantage of doing so is obvious: in the limit of dτ ! 0 we can write approximately ^ ^ ^ e−iHdτ ≈ e−iV dτ e−iKdτ ; where V^ = V (x) is the potential energy operator, and K^ = p2=2m is the kinetic energy operator. Corrections to the approximate relation above are proportional to dτ 2 and can be neglected in the 0 considered limit. It also makes sense to introduce notation x ≡ x0, and x = xN to label all space points with one ordered index. After we split each exponential into two, we face the following expression (real-space represen- tation is an eigenbasis of the potential energy operator) Z Z 1 Y −iV (xj )dτ −iKdτ^ dxN−1··· dx1 e hxjje jxj−1i : (1) j=N The only matrix elements left to calculate refer to the particle propagation in free space, i.e. no potential, which are obviously easy to deal with (using now completeness relation in momentum space) ^ Z dp 2 Z dp 2 hyje−iKdτ jxi = hyjpi e−i(p =2m)dτ hpjxi = e−i(p =2m)dτ+ip(y−x) : (2π)D (2π)D By evaluating the Gaussian integral we get for the propagation in free space ^ m D=2 2 hyje−iKdτ jxi = ei(y−x) m=2dτ : 2πidτ Substituting this expression back to Eq. (1) we get Z Z 1 m DN=2 Y 2 U(x ; t; x ; t = 0) = dx ··· dx e−iV (xj )dτ+i(xj −xj−1) m=2dτ : (2) N 0 2πidτ N−1 1 j=N To complete the calculation we take the limit of dτ ! 0. At this point we introduce the notion of a `trajectory' xN ; xN−1; : : : ; xj; : : : ; x0 ≡ x(t = Ndτ); x((N − 1)dτ); : : : ; x(jdτ); : : : ; x(0) ≡ [x(τ)] ; to describe the sequence of coordinates starting from x0 and followed by x1, etc. all the way to xN . By the same token, the combination [xj −xj−1]/dτ = [x(jdτ)−x((j−1)dτ)]/dτ = dx(τ)/dτ ≡ x_(τ) will be called particle velocity along the trajectory. Then 1 8 N 9 Y 2 < X = e−iV (xj )dτ+i(xj −xj−1) m=2dτ = exp i dτmx_ 2(τ)=2 − V (x(τ)) = j=N : j=1 ; Z t exp i dτmx_ 2(τ)=2 − V (x(τ)) (3) 0 2 With standard simplified notation for the set of integrals over all possible particle trajectories 0 between points x0 = x and xN = x (this is where the path-integral name comes from) Z x(t)=x m DN=2 Z Z Dx ··· = dxN−1··· dx1 ::: (4) x(0)=x0 2πidτ we finally arrive at Feynman's formulation of quantum mechanics Z x(t)=x U(x; t; x0; t = 0) = Dx eiS ; (5) x(0)=x0 where S is the classical action Z t Z t S[x(τ)] = dτ L[x(τ); x_(τ)] = dτmx_ 2(τ)=2 − V (x(τ)) ; (6) 0 0 related to the classical Lagrangian in the usual way! This expression is straightfor- wardly generalized to many-body t 0 systems (with only one extra point to care - for identical parti- cles the sets of coordinates at the initial and final times can be con- nected in a number of ways and all of them have to be accounted for in the path-integral). This is illustrated for the two parti- cle example in the figure; the sign rule distinguishes between bosons and fermions. t Remarkably, we have our classical picture of particles moving along classical trajectories with well defined coordinates and trajectories back! Not quite so. There is no principle of minimal action attached to this calculation and you see that individually ANY trajectory is as good as any other because they all contribute with exactly the same amplitude, only phases are different. It means that individual trajectories barely make any sense. However, packs of trajectories with large and dissimilar phases cancel individual contributions very efficiently while packs with nearly identical phases contribute in 'unison'. Thus if typical phases accumulated along trajectories are large, one has to look for trajectories which minimize action; in this case all nearby trajectories have nearly identical phases and their combined contribution to the answer is dominating. This is how classical equations of motion emerge in Feynman's picture. Before I complete this calculation for the harmonic oscillator let me also mention one connection to statistical mechanics. The main quantity of interest is the partition function X Z = e−βEn : n 3 We write it identically as X ^ X Z Z ^ Z = hnje−βH jni ≡ hnjxi dx dx0hxje−βH jx0ihx0jni ≡ n n Z Z ^ Z ^ dx dx0hx0jxihxje−βH jx0i = dxhxje−βH jxi ; which is nothing but the trace of the matrix (same in any representation). Up to change of notations β = it we have to deal with the same expression as in dynamic evolution, but now in 'imaginary time' t = −iβ. With this we arrive at I Z β Z = Dx exp − dτmx_ 2(τ)=2 + V (x(τ)) : (7) x(0)=x(β) 0 This expression is extremely well suited for Monte Carlo simulations of bosonic systems, and is also a powerful theoretical tool to deal with quantum statistics questions. Now back to harmonic oscillator. In this case the Lagrangian is quadratic in x(τ) and all integrals are of the Gaussian form. This property leads to the exact procedure (ignoring pre- exponential factors for a while): • Solve for the optimal classical trajectory subject to boundary conditions x(t = 0) = x0, x(t) = x which is the same calculation as in classical mechanics based on the minimal action principle x − x0 cos(!t) ! x¨ + !2x = 0 ! x(τ) = x0 cos(!τ) + sin(!τ) : sin(!t) R t 2 2 2 • Substitute this solution to the action S = 0 dτ mx_ (τ)=2 − m! x (τ)=2 to get ( ) m!2 Z t x − x0 cos(!t) 2 x − x0 cos(!t) 2 S = dτ − x0 sin(!τ) + cos(!τ) − x0 cos(!τ) + sin(!τ) = 2 0 sin(!t) sin(!t) ( ) m!2 Z t x − x0 cos(!t)2 x − x0 cos(!t) dτ − (x0)2 + cos(2!τ) − 2x0 sin(2!τ) = 2 0 sin(!t) sin(!t) integrate and re-arrange terms ( ) m! x − x0 cos(!t)2 x − x0 cos(!t) − (x0)2 + sin(!t) cos(!t) − 2x0 sin2(!t) = 2 sin(!t) sin(!t) m! S = cos(!t)x2 + (x0)2 − 2xx0 : 2 sin(!t) With this result we obtain U(x; t; x0; t = 0) = f(t) eiS ; where the pre-exponential factor can be determined as follows: Use identity ^ Z ^ ^ hxje−itH jx0i = dyhxje−itH=2jyihyje−itH=2jx0i ; 4 to obtain a relation (the integral over y is Gaussian) 1 s 0 Z 0 iπ sin(!t=2) 0 f(t)eiS(x;x ;t) = f 2(t=2) dy ei[S(x;y;t=2)+S(y;x ;t=2)] = f 2(t=2) eiS(x;x ;t) ; −∞ m! cos(!t=2) with the solution (check by substituting) r m! f(t) = : 2πi sin(!t) The final result reads r m! m! U(x; t; x0; t = 0) = exp i cos(!t)x2 + (x0)2 − 2xx0 : (8) 2πi sin(!t) 2 sin(!t) With this we may solve any dynamics (and thermodynamics for t = −iβ) problem in harmonic oscillator.